Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 357869 11 pageshttpdxdoiorg1011552013357869

Research ArticleStochastic Differential Equations withMulti-Markovian Switching

Meng Liu1 and Ke Wang2

1 School of Mathematical Science Huaiyin Normal University Huairsquoan 223300 China2Department of Mathematics Harbin Institute of Technology Weihai 264209 China

Correspondence should be addressed to Meng Liu liumeng0557sinacom

Received 31 August 2012 Revised 2 March 2013 Accepted 5 March 2013

Academic Editor Jose L Gracia

Copyright copy 2013 M Liu and K WangThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with stochastic differential equations (SDEs) with multi-Markovian switching The existence anduniqueness of solution are investigated and the pth moment of the solution is estimated The classical theory of SDEs with singleMarkovian switching is extended

1 Introduction

Stochastic modeling has played an important role in manybranches of industry and science SDEs with single continu-ous-time Markovian chain have been used to model manypractical systems where they may experience abrupt changesin their parameters and structure caused by phenomenasuch as abrupt environment disturbances SDEs with singleMarkovian switching can be denoted by

119889119909 (119905) = 119891 (119909 (119905) 119905 120574 (119905)) 119889119905

+ 119892 (119909 (119905) 119905 120574 (119905)) 119889119861 (119905) 1199050le 119905 le 119879

(1)

with initial conditions 119909(1199050) = 119909

0isin 1198712

F1199050

and 120574(1199050) = 120574

0

where 120574(119905) is a right-continuous homogenous Markovianchain on the probability space taking values in a finite statespace S = 1 2 119873 and is F

119905-adapted but independent

of the Brownian motion 119861(119905) and

119891 R119899times R+times S 997888rarr R

119899

119892 R119899times R+times S 997888rarr R

119899+119898

(2)

Owing to their theoretical and practical significance (1)has received great attention and has been recently studiedextensively andwe heremention Skorokhod [1] andMao andYuan [2] among many others

However in the real world the condition that coefficients119891 and 119892 in (1) are perturbed by the same Markovian chainis too restrictive For example in the classical Black-Scholesmodel the asset price is given by a geometric Brownianmotion

119889119883 (119905) = 120583119883 (119905) 119889119905 + ]119883(119905) 119889119861 (119905) (3)

where 120583 is the rate of the return of the underlying assert ]is the volatility and 119861(119905) is a scalar Brownian motion Sincethere is strong evidence to indicate that 120583 is not a constant butis a Markovian jump process (see eg [3 4]) many authorsproposed the following model

119889119883 (119905) = 120583 (120574 (119905))119883 (119905) 119889119905 + ] (120574 (119905))119883 (119905) 119889119861 (119905) (4)

However many stochastic factors that affect 120583 are differentfrom those that affect ] Then the following model is moreappropriate than model (105) to describe this problem

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (5)

where 120574119894(119905) is a right-continuous homogenous Markovian

chain taking values in a finite state space 119894 = 1 2 Anotherexample is the stochastic Lotka-Volterra model with singleMarkovian switching which has received great attentionand has been studied extensively recently (see eg [5ndash12])

2 Journal of Applied Mathematics

For the sake of convenience we take the following two-dimensional competitive model as an example

1198891199091= 1199091[11990310

(120574 (119905)) minus 11988611

(120574 (119905)) 1199091 minus 11988612

(120574 (119905)) 1199092] 119889119905

+ 1205721(120574 (119905)) 119909

11198891198611(119905)

1198891199092= 1199092[11990320

(120574 (119905)) minus 11988621

(120574 (119905)) 1199091minus 11988622

(120574 (119905)) 1199092] 119889119905

+ 1205722(120574 (119905)) 11990921198891198612 (119905)

(6)

where 119909119894is the size of 119894th species at time 119905 119903

1198940(119895) represents

the growth rate of 119894th species in regime 119895 for 119894 = 1 2119895 isin S and 119861

1and 119861

2are independent standard Brownian

motions However there are many stochastic factors thataffect some coefficients intensely but have little impact onother coefficients in (6) For example suppose that thestochastic factor is rain falls and 119909

1is able to endure a damp

weather while 1199092is fond of a dry environment then the rain

falls will affect 1199092intensely but have little impact on 119909

1 Thus

a more appropriate model is governed by

1198891199091= 1199091[11990310

(12057410

(119905)) minus 11988611

(12057411

(119905)) 1199091minus 11988612

(12057412

(119905)) 1199092] 119889119905

+ 1205721(1205741 (119905)) 11990911198891198611 (119905)

1198891199092= 1199092[11990320

(12057420

(119905)) minus 11988621

(12057421

(119905)) 1199091minus 11988622

(12057422

(119905)) 1199092] 119889119905

+ 1205722(1205742 (119905)) 11990921198891198612 (119905)

(7)

where 120574119894119895(119905) and 120574

119896(119905) are right-continuous homogenous

Markovian chains taking values in finite state spaces S119894119895for

119894 = 1 2 119895 = 0 1 2 and S119896for 119896 = 1 2 respectively

Thus the above examples show that the study of thefollowing SDEs with multi-Markovian switchings is essentialand is of great importance fromboth theoretical and practicalpoints

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741(119905)) 119889119905

+ 119892 (119909 (119905) 119905 1205742 (119905)) 119889119861 (119905) 1199050le 119905 le 119879

(8)

with initial conditions 119909(1199050) = 119909

0isin 1198712

F0

and 120574119894(1199050) where

120574119894(119905) is a right-continuous homogenous Markovian chain

on the probality space taking values in a finite state spaceS119894= 1 2 119873

119894 and isF

119905-adapted but independent of the

Brownian motion 119861119894(119905) 119894 = 1 2 and

119891 R119899times R+times S1997888rarr R

119899

119892 R119899times R+times S2997888rarr R

119899+119898

(9)

Equation (8) can be regarded as the result of the 1198731times 1198732

equations

119889119909 (119905) = 119891 (119909 (119905) 119905 119894) 119889119905

+ 119892 (119909 (119905) 119905 119895) 119889119861 (119905) 119894 isin S1 119895 isin S

2

(10)

switching among each other according to the movement ofthe Markovian chains It is important for us to discover the

properties of the system (8) and to find out whether thepresence of two Markovian switchings affects some knownresults The first step and the foundation of those studies areto establish the theorems for the existence and uniqueness ofthe solution to system (8) So in this paper we will give sometheorems for the existence and uniqueness of the solution tosystem (8) and study some properties of this solution Thetheory developed in this paper is the foundation for furtherstudy and can be applied in many different and complicatedsituations and hence the importance of the results in thispaper is clear

It should be pointed out that the theory developed in thispaper can be generalized to cope with the more general SDEswith more Markovian chains

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741 (119905) 120574119899 (119905)) 119889119905

+ 119892 (119909 (119905) 119905 120574119899+1

(119905) 120574119899+119898

(119905)) 119889119861 (119905)

(11)

The reason we concentrate on (8) rather than (11) is to avoidthe notations becoming too complicated Once the theorydeveloped in this paper is established the reader should beable to cope with the more general (11) without any difficulty

The remaining part of this paper is as follows In Section 2the sufficient criteria for existence anduniqueness of solutionlocal solution andmaximal local solution will be establishedrespectively In Section 3 the 119871

119875-estimates of the solutionwill be given In Section 4 we will introduce an example toillustrate our main result Finally we will close the paper withconclusions in Section 5

2 SDEs with Markovian Chains

Throughout this paper let (ΩF F119905119905isin119877+

P) be a completeprobability space Let 119861(119905) = (119861

1(119905) 119861

119898(119905))119879 be an 119898-

dimensional Brownian motion defined on the probabilityspace

In this section we will consider (8) Let 120574(119905) = (1205741(119905)

1205742(119905)) We impose a hypothesis(H1) 120574

1(119905) is independent of 120574

2(119905)

Then 120574(119905) is a homogenous vector Markovian chain withtransition probabilities

119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (119904) = (119894

1 1198942)

= 119875 120574 (119905) = (1198951 1198952) | 120574 (0) = (119894

1 1198942)

= 1198751198941119894211989511198952

(119905)

(12)

where (1198941 1198942) (1198951 1198952) isin S = (1 1) (1 2) (1119873

2) (2 1)

(1198731 1198732)

Now we will prepare some lemmas which are importantfor further study

Lemma 1 1198751198941119894211989511198952

(119905) has the following properties

(i) 1198751198941119894211989511198952

(119905) ge 0 for (1198941 1198942) (1198951 1198952) isin S

(ii) sum(11989511198952)isinS 1198751198941119894211989511198952

(119905) = 1 for (1198941 1198942) isin S

(iii) 1198751198941119894211989511198952

(0) = 12057511989411198951

sdot 12057511989421198952

where 120575119894119896119895119896

= 1 if 119894119896

= 119895119896

otherwise 120575119894119896119895119896

= 0 119896 = 1 2

Journal of Applied Mathematics 3

(iv) (Chapman-Kolmogorov equation) For 119904 119905 ge 0 and(1198941 1198942) (1198951 1198952) isin S

1198751198941119894211989511198952

(119904 + 119905)

= sum

(1198961 1198962)isinS

1198751198941119894211989611198962

(119905) 1198751198961119896211989511198952

(119904) (13)

Proof The proofs of (i) (ii) and (iii) are obvious Now let usprove (iv)

1198751198941119894211989511198952

(119904 + 119905)

= 119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (0) = (119894

1 1198942)

= sum

(11989611198962)isinS

119875 120574 (119904 + 119905) = (1198951 1198952)

120574 (119905) = (1198961 1198962) | 120574 (0) = (119894

1 1198942)

= sum

(11989611198962)isinS

(119875 120574 (119904 + 119905) = (1198951 1198952)

120574 (119905) = (1198961 1198962) 120574 (0) = (119894

1 1198942)

times(119875 120574 (119905) = (1198961 1198962) 120574 (0) = (119894

1 1198942))minus1

)

times119875 120574 (119905) = (119896

1 1198962) 120574 (0) = (119894

1 1198942)

119875 120574 (0) = (1198941 1198942)

= sum

(11989611198962)isinS

119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (119905) = (119896

1 1198962)

120574 (0) = (1198941 1198942)

times 119875 120574 (119905) = (1198961 1198962) | 120574 (0) = (119894

1 1198942)

= sum

(1198961 1198962)isinS

119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (119905) = (119896

1 1198962)

times 119875 120574 (119905) = (1198961 1198962) | 120574 (0) = (119894

1 1198942)

= sum

(1198961 1198962)isinS

1198751198961119896211989511198952

(119904) 1198751198941119894211989611198962

(119905)

(14)

This completes the proof

Now we impose another hypothesis which is calledstandard condition

(H2) lim119905rarr0

1198751198941119894211989511198952

(119905) = 1205751198941119894211989511198952

Lemma 2 Under Assumption (H2) for all 119905 ge 0 (1198941 1198942) isin S

one has 1198751198941119894211989411198942

(119905) gt 0

Proof From 1198751198941119894211989411198942

(0) gt 0 and (H2) we know that forarbitrary fixed 119905 gt 0 we have

1198751198941119894211989411198942

(119905

119899) gt 0 (15)

for sufficient large 119899 Then making use of Chapman-Kolm-ogorov equation

1198751198941119894211989411198942

(119904 + 119905) = sum

(1198961 1198962)isinS

11987511989411198942119896111198962

(119905) 1198751198961119896211989411198942

(119904)

ge 1198751198941119894211989411198942

(119905) 1198751198941119894211989411198942

(119904)

(16)

gives

1198751198941119894211989411198942

(119905) ge (1198751198941119894211989411198942

(119905

119899))

119899

gt 0 (17)

which is the desired assertion

Lemma 3 Under Assumption (H2) for all (1198941 1198942) isin S

minus1199021198941119894211989411198942

= lim119905rarr0

1 minus 1198751198941119894211989411198942

(119905)

119905(18)

exists (but may be infin)

Proof Define 120601(119905) = minus ln1198751198941119894211989411198942

(119905) ge 0 Then making use of(16) gives

120601 (119904 + 119905) le 120601 (119904) + 120601 (119905) (19)

Set minus1199021198941119894211989411198942

= sup119905gt0

(120601(119905)119905) It is easy to see that

0 le minus1199021198941119894211989411198942

le infin lim sup119905rarr0

120601 (119905)

119905le minus1199021198941119894211989411198942

(20)

Now we will assert

lim inf119905rarr0

120601 (119905)

119905ge minus1199021198941119894211989411198942

(21)

In fact for 0 lt ℎ lt 119905 exist119899 0 le 120576 le ℎ such that 119905 = 119899ℎ + 120576Applying (19) yields

120601 (119905)

119905ge

119899ℎ

120576

120601 (ℎ)

ℎ+

120601 (120576)

119905 (22)

Note that 120576 rarr 0 119899ℎ120576 rarr 1 120601(120576) rarr 0 whenever ℎ rarr 0+

then for all 119905 gt 0 we have

120601 (119905)

119905le lim infℎrarr0

120601 (ℎ)

ℎ (23)

This implies minus1199021198941119894211989411198942

le lim inf119905rarr0

(120601(119905)119905) Thus

minus1199021198941119894211989411198942

= lim119905rarr0

120601 (119905)

119905 (24)

Using the definition of 120601(119905) gives

lim119905rarr0

1 minus 1198751198941119894211989411198942

(119905)

119905= lim119905rarr0

1 minus exp 120601 (119905)

120601 (119905)

120601 (119905)

119905= minus1199021198941119894211989411198942

(25)

which is the required assertion

4 Journal of Applied Mathematics

Lemma 4 Under Assumption (H2) for (1198941 1198942) (1198951 1198952) isin

S (1198941 1198942) = (1198951 1198952)

1199021198941119894211989511198952

= 1198751015840

1198941119894211989511198952

(0) = lim119905rarr0

1198751198941119894211989511198952

(119905)

119905(26)

exists and is finite

Proof By (H2) we note that for all 0 lt 120576 lt 13 exist0 lt 120575 lt 1such that

1198751198941119894211989411198942

(119905) gt 1 minus 120576 1198751198951119895211989511198952

(119905) gt 1 minus 120576

1198751198951119895211989411198942

(119905) lt 120576

(27)

provided 0 lt 119905 le 120575For forall0 le ℎ lt 119905 set 119899 = ⟨119905ℎ⟩ where ⟨119886⟩ = max

119899le119886119899 isin

Z Let

119875(11989511198952)

1198941119894211989611198962

(ℎ) = 1198751198941119894211989611198962

(ℎ)

119875(11989511198952)

1198941119894211989611198962

(119898ℎ)

= sum

(11990311199032) = (1198951 1198952)

119875(11989511198952)

1198941119894211990311199032

((119898 minus 1) ℎ) 1198751199031119903211989611198962

(ℎ)

(28)

where 119875(11989511198952)

1198941119894211989611198962

(119898ℎ)means that the probability of the 120574(119905)willnot reach to (119895

1 1198952) at times ℎ 2ℎ (119898 minus 1)ℎ but will reach

to (1198961 1198962) at time 119898ℎ Note that if ℎ le 119905 le 120575 then

120576 gt 1 minus 1198751198941119894211989411198942

(119905)

= sum

(1198961 1198962) = (1198941 1198942)

1198751198941119894211989611198962

(119905) ge 1198751198941119894211989511198952

(119905)

ge

119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ) 1198751198951119895211989511198952

(119905 minus 119898ℎ)

ge (1 minus 120576)

119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ)

(29)

which indicates119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ) le120576

1 minus 120576 (30)

Then making use of

1198751198941119894211989411198942

(119898ℎ) = 119875(11989511198952)

1198941119894211989411198942

(119898ℎ)

+

119898minus1

sum

119897=1

119875(11989511198952)

1198941119894211989511198952

(119897ℎ) 1198751198951119895211989411198942

((119898 minus 119897) ℎ)

(31)

we obtain

119875(11989511198952)

1198941119894211989411198942

(119898ℎ) ge 1198751198941119894211989411198942

(119898ℎ) minus

119898minus1

sum

119897=1

119875(11989511198952)

1198941119894211989511198952

(119897ℎ)

ge 1 minus 120576 minus120576

1 minus 120576

(32)

Consequently

1198751198941119894211989511198952

(119905)

gt

119899

sum

119898=1

119875(11989511198952)

1198941119894211989411198942

((119898 minus 1) ℎ) 1198751198941119894211989511198952

(ℎ) 1198751198951119895211989511198952

(119905 minus 119898ℎ)

ge 119899 (1 minus 120576 minus120576

1 minus 120576)1198751198941119894211989511198952

(ℎ) (1 minus 120576)

ge 119899 (1 minus 3120576) 1198751198941119894211989511198952

(ℎ)

(33)

Dividing both sides of the above inequality by ℎ and noting119899ℎ rarr 119905 whenever ℎ rarr 0 yield

lim supℎrarr0

1198751198941119894211989511198952

(ℎ)

ℎle

1

1 minus 3120576

1198751198941119894211989511198952

(119905)

119905lt infin (34)

Then letting 119905 rarr 0 gives

lim supℎrarr0

1198751198941119894211989511198952

(ℎ)

ℎle

1

1 minus 3120576lim inf119905rarr0

1198751198941119894211989511198952

(119905)

119905 (35)

and the required assertion follows immediately by letting120576 rarr 0 This completes the proof

Set 120574(119905) = (1205741(119905) 1205742(119905)) then it is easy to see that

almost every sample path of 120574(119905) is a right continuous stepfunction Now letting P(119905) = (119875

1198941119894211989511198952

(119905))11987311198732times11987311198732

Q =

(1199021198941119894211989511198952

)11987311198732times11987311198732

= P1015840(0) Then by Chapman-Kolmogorovequation

P (119905 + ℎ) = P (119905)P (ℎ) = P (ℎ)P (119905) (36)

we have

P (119905 + ℎ) minus P (119905)

ℎ= P (119905) [

P (ℎ) minus Iℎ

] = [P (ℎ) minus I

ℎ]P (119905)

(37)

Letting ℎ rarr 0 and taking limits give

P1015840(119905) = P (119905)Q

P1015840(119905) = QP (119905)

(38)

Note that

P (0) = I (39)

Then by solving the ordinary differential equations (38) and(39) we obtain the following lemma

Lemma 5 For P(119905) and Q one has

P (119905) = exp Q119905 (40)

We are now in the position to give the sufficient condi-tions for the existence and uniqueness of the solution of (8)For this end let us first give the definition of the solution

Journal of Applied Mathematics 5

Definition 6 An R119899-valued stochastic process 119909(119905)1199050le119905le119879

iscalled a solution of (8) if it has the following properties

(i) 119909(119905)1199050le119905le119879

is continuous andF119905-adapted

(ii) 119891(119909(119905) 119905 1205741(119905))1199050le119905le119879

isinL1([1199050 119879]R119899)while 119892(119909(119905)

119905 1205742(119905))1199050le119905le119879

isin L2([1199050 119879]R119899times119898)

(iii) for any 119905 isin [1199050 119879] equation

119909 (119905) = 119909 (1199050) + int

119905

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(41)

holds with probability 1

A solution 119909(119905)1199050le119905le119879

is said to be unique if any othersolution (119905)

1199050le119905le119879

is indistinguishable from 119909(119905)1199050le119905le119879

Now we can give our main results in this section

Theorem 7 Assume that there exist two positive constants 119870

and 119870 such that(Lipschitz condition) for all 119909 119910 isin R119899 119905 isin [119905

0 119879] and

(1198941 1198942) isin S

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le 119870

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(42)

(Linear growth condition) for all (119909 119905 (1198941 1198942)) isin R119899 times

[1199050 119879] times S

1003816100381610038161003816119891 (119909 119905 1198941)10038161003816100381610038162or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (43)

Then there exists a unique solution 119909(119905) to (8) and moreoverthe solution obeys

119864( sup1199050le119905le119879

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(44)

Proof Recall that almost every sample path of 120574(119905) =

(1205741(119905) 1205742(119905)) is a right continuous step function with a finite

number of jumps on [1199050 119879] Thus there exists a sequence of

stopping times 120591119896119896ge0

such that

(i) for almost every 120596 isin Ω there is a finite 119896120596for 1199050

=

1205910lt 1205911lt sdot sdot sdot lt 120591

119896= 119879 and 120591

119896= 119879 if 119896 gt 119896

120596

(ii) both 1205741(sdot) and 120574

2(sdot) in 120574(sdot) are constants on interval

[[120591119896 120591119896+1

[[ namely

1205741 (119905) = 120574

1(120591119896) on 120591

119896le 119905 le 120591

119896+1forall119896 ge 0

1205742(119905) = 120574

2(120591119896) on 120591

119896le 119905 le 120591

119896+1forall119896 ge 0

(45)

First of all let us consider (8) on 119905 isin [[1205910 1205911[[ then (8)

becomes119889119909 (119905) = 119891 (119909 (119905) 119905 120574

1(1199050)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(1199050)) 119889119861 (119905)

(46)

with initial conditions 119909(1199050) = 119909

0 120574(1199050) = (120574

1(1199050) 1205741(1199050))

Then by the theory of SDEs we obtain that (46) has a uniquesolution which obeys 119909(120591

1) isin 1198712

F1205911

(ΩR119899) We next consider(8) on 119905 isin [[120591

1 1205912[[ which becomes

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741 (1205911)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(1205911)) 119889119861 (119905)

(47)

Again by the theory of SDEs (47) has a unique solutionwhichobeys 119909(120591

1) isin 119871

2

F1205912

(ΩR119899) Repeating this procedure weconclude that (8) has a unique solution 119909(119905) on [119905

0 119879]

Now let us prove (44) For every 119896 ge 1 define thestopping time

120591119896= 119879 and inf 119905 isin [119905

0 119879] |119909 (119905)| ge 119896 (48)

It is obvious that 120591119896uarr 119879 as Set 119909

119896(119905) = 119909(119905and120591

119896) for 119905 isin [119905

0 119879]

Then 119909119896(119905) obeys the equation

119909119896(119905) = 119909

0+ int

119905

1199050

119891 (119909119896(119904) 119904 120574

1(119904)) 119868[[1199050120591119896]](119904) 119889119904

+ int

119905

1199050

119892 (119909119896(119904) 119904 120574

2(119904)) 119868[[1199050120591119896]](119904) 119889119861 (119904)

(49)

Making use of the elementary inequality |119886+119887+119888|2le 3(|119886|

2+

|119887|2+ |119888|2) the Holder inequality and (43) we can see that

1003816100381610038161003816119909119896 (119905)10038161003816100381610038162= 3

1003816100381610038161003816119909010038161003816100381610038162+ 3119870 (119879 minus 119905

0) int

119905

1199050

(1 +1003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

+ 3

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199050

119892 (119909119896(119904) 119904 120574

2(119904)) 119868[[1199050120591119896]](119904) 119889119861 (119904)

100381610038161003816100381610038161003816100381610038161003816

2

(50)

Thus applying the Doob martingale inequality and (43) wecan further show that

119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

= 311986410038161003816100381610038161199090

10038161003816100381610038162+ 3119870 (119879 minus 119905

0) int

119905

1199050

(1 + 1198641003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

+ 12119864int

119905

1199050

1003816100381610038161003816119892 (119909119896(119904) 119904 120574

2(119904))

10038161003816100381610038162119868[[1199050120591119896]](119904) 119889119904

le 311986410038161003816100381610038161199090

10038161003816100381610038162+ 3119870 (119879 minus 119905

0+ 4)int

119905

1199050

(1 + 1198641003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

(51)

That is to say

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162) le 1 + 3119864

1003816100381610038161003816119909010038161003816100381610038162+ 3119870 (119879 minus 119905

0+ 4)

times int

119905

1199050

[1 + 119864 sup1199050le119903le119904

1003816100381610038161003816119909119896 (119903)10038161003816100381610038162]119889119904

(52)

6 Journal of Applied Mathematics

Using the Gronwall inequality leads to

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(53)

Consequently

1 + 119864( sup1199050le119905le120591119896

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(54)

Then the required inequality (44) follows immediately byletting 119896 rarr infin

Condition (42) indicates that the coefficients 119891(119909 119905 1198941)

and 119892(119909 119905 1198942) do not change faster than a linear function of

119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894

1) and 119892(119909 119905 119894

2) in 119909 for all 119905 isin [119905

0 119879] Then functions

that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution

Definition 8 Let120590infinbe a stopping time such that 119905

0le 120590infin

le 119879

as An R119899-valuedF119905-adapted continuous stochastic process

119909(119905)1199050le119905lt120590infin

is called a local solution of (8) if 119909(1199050) = 1199090and

moreover there is a nondecreasing sequence 120590119896119896ge1

such that1199050le 120590119896 uarr 120590infin

as and

119909 (119905) = 119909 (1199050) + int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(55)

holds for any 119905 isin [1199050 119879) and 119896 ge 1 with probability one If

furthermore

lim sup119905rarr+120590

infin

|119909 (119905)| = infin whenever 120590infin

lt 119879 (56)

then it is called a maximal local solution and 120590infinis called the

explosion time Amaximal local solution 119909(119905) 1199050le 119905 lt 120590

infin

is said to be unique if any other maximal local solution 119909(119905)

1199050le 119905 lt 120590

infin is indistinguishable from it namely 120590

infin= 120590infin

and 119909(119905) = 119909(119905) for 1199050le 119905 lt 120590

infinwith probability one

Definition 9 (local Lipschitz condition) For every integer119896 ge 1 there exists a positive constant ℎ

119896such that for all

119905 isin [1199050 119879] 119894 = (119894

1 1198942) isin S and those 119909 119910 isin R119899 with

|119909| or |119910| le 119896

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le ℎ119896

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(57)

The following theorem shows the existence of uniquemaximal local solution under the local Lipschitz conditionwithout the linear growth condition

Theorem 10 Under condition (57) there exists a uniquemaximal local solution of (8)

Proof Define functions

119891119896(119909 119905 119894

1) =

119891 (119909 119905 1198941) if |119909| le 119896

119891(119896119909

|119909| 119905 1198941) if |119909| gt 119896

119892119896(119909 119905 119894

2) =

119892 (119909 119905 1198942) if |119909| le 119896

119892 (119896119909

|119909| 119905 1198942) if |119909| gt 119896

(58)

Then 119891119896and 119892

119896satisfy the Lipschitz condition and the linear

growth condition Thus by Theorem 7 there is a uniquesolution 119909

119896(119905) of the equation

119889119909119896(119905) = 119891

119896(119909119896(119905) 119905 120574

1(119905)) 119889119905

+ 119892119896(119909119896(119905) 119905 120574

2(119905)) 119889119861 (119905)

(59)

with the initial conditions 119909119896(1199050) = 119909

0and 120574(119905

0) = (120574

1(1199050)

1205742(1199050)) Define the stopping times

120590119896= 119879 and inf 119905 isin [119905

0 119879]

1003816100381610038161003816119909119896 (119905)1003816100381610038161003816 ge 119896 (60)

Clearly if 1199050le 119905 le 120590

119896

119909119896(119905) = 119909

119896+1(119905) (61)

which indicates that 120590119896is increasing so 120590

119896has its limit 120590

infin=

lim119896rarr+infin

120590119896 Define 119909(119905) 119905

0le 119905 lt 120590

infin by

119909 (119905) = 119909119896(119905) 119905 isin [[120590

119896minus1 120590119896[[ 119896 ge 1 (62)

where 1205900= 1199050 Applying (61) one can show that 119909(119905 and 120590

119896) =

119909119896(119905 and 120590119896) It then follows from (59) that

119909 (119905 and 120590119896) = 1199090+ int

119905and120590119896

1199050

119891119896(119909 (119904) 119904 120574

1(119904)) 119889119904

+ int

119905and120590119896

1199050

119892119896(119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

= 1199090+ int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

(63)

for any 119905 isin [1199050 119879) and 119896 ge 1 It is easy to see that if 120590

infinle 119879

then

lim sup119905rarr120590infin

|119909 (119905)|

ge lim sup119896rarr+infin

1003816100381610038161003816119909 (120590119896)1003816100381610038161003816 = lim sup119896rarr+infin

1003816100381610038161003816119909119896 (120590119896)1003816100381610038161003816 = infin

(64)

Therefore 119909(119905) 1199050le 119905 lt 120590

infin is a maximal local solution

Now we will prove the uniqueness Let 119909(119905) 1199050

le 119905 lt

120590infin

be another maximal local solution Define

120590119896= 120590infin

and inf 119905 isin [[1199050 120590infin

[[ 119909 (119905) ge 119896 (65)

Journal of Applied Mathematics 7

Then 120590119896

rarr 120590infin

as and

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590119896and 120590119896]] = 1 forall119896 ge 1 (66)

Letting 119896 rarr infin gives

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590infin

and 120590infin

[[ = 1 (67)

In order to complete the proof we need only to show that120590infin

= 120590infin

as In fact for almost every 120596 isin 120590infin

lt 120590infin

wehave

1003816100381610038161003816119909 (120590infin

120596)1003816100381610038161003816

= lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = infin(68)

which contradicts the fact that 119909(119905 120596) is continuous on 119905 isin

[1199050 120590infin

(120596)) This implies 120590infin

ge 120590infin

as In the same way onecan show 120590

infinle 120590infinasThus wemust have 120590

infin= 120590infinasThis

completes the proof

In many situations we often consider an SDE on [1199050infin)

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741(119905)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(119905)) 119889119861 (119905) 119905

0le 119905 lt infin

(69)

with initial data 119909(1199050) = 119909

0and 120574(119905

0) If the assumption of

the existence-and-uniqueness theorem holds on every finitesubinterval [119905

0 119879] of [119905

0infin) then (69) has a unique solution

119909(119905) on the entire interval [1199050infin) Such a solution is called

a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R

+times S to R which is

given by

119871119881 (119909 119905 (1198941 1198942))

= 119881119905(119909 119905 (119894

1 1198942)) + 119881

119909(119909 119905 (119894

1 1198942)) 119891 (119909 119905 119894

1)

+ 05 trace [119892119879 (119909 119905 1198942) 119881119909119909

(119909 119905 (1198941 1198942)) 119892 (119909 119905 119894

2)]

+ sum

(11989511198952)isinS

1199021198941119894211989511198952

119881 (119909 119905 (1198951 1198952))

(70)

where 119881(119909 119905 (1198941 1198942)) isin 119862

21(R119899 times R

+times SR

+) and

119881119905(119909 119905 (119894

1 1198942)) =

120597119881 (119909 119905 (1198941 1198942))

120597119905

119881119909(119909 119905 (119894

1 1198942))

= (120597119881 (119909 119905 (119894

1 1198942))

1205971199091

120597119881 (119909 119905 (119894

1 1198942))

120597119909119899

)

119881119909119909

(119909 119905 (1198941 1198942)) = (

1205972119881 (119909 119905 (119894

1 1198942))

120597119909119894120597119909119895

)

119899times119899

(71)

Theorem 11 Assume that the local Lipschitz condition (57)holds Assume also that there is a function 119881(119909 119905 (119894

1 1198942)) isin

11986221

(R119899 times R+times SR

+) and a constant 120579 gt 0 such that

lim|119909|rarrinfin

( inf(119905(1198941 1198942))isinR+timesS

119881 (119909 119905 (1198941 1198942))) = infin (72)

119871119881 (119909 119905 (1198941 1198942)) le 120579 (1 + 119881 (119909 119905 (119894

1 1198942)))

forall (119909 119905 (1198941 1198942)) isin R

119899times R+times S

(73)

Then there exists a unique global solution 119909(119905) to (69)

Proof We need only to prove the theorem for any initialcondition 119909

0isin R119899 and (120574

1(1199050) 1205742(1199050)) isin S FromTheorem 10

we know that the local Lipschitz condition guarantees theexistence of the unique maximal solution 119909(119905) on [119905

0 120590infin

)where 120590

infinis the explosion time We need only to show 120590

infin=

infin as If this is not true then we can find a pair of positiveconstants 120576 and 119879 such that

119875 120590infin

le 119879 gt 2120576 (74)

For each integer 119896 ge 1 define the stopping time

120590119896= inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (75)

Since 120590119896

rarr 120590infinalmost surely we can find a sufficiently large

integer 1198960for

119875 120590119896le 119879 gt 120576 forall119896 ge 119896

0 (76)

Fix any 119896 ge 1198960 then for any 119905

0le 119905 le 119879 by virtue of the

generalized Ito formula (see eg [1])

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

= 119881 (1199090 1199050 1205740) + 119864int

119905and120590119896

1199050

119871119881 (119909 (119904) 119904 120574 (119904)) 119889119904

le 119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(77)

Making use of the Gronwall inequality gives

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(78)

Therefore

119864 (119868120590119896le119879

119881 (119909 (120590119896) 120590119896 120574 (120590119896)))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(79)

At the same time set

119892119896= inf 119881 (119909 119905 (119894

1 1198942)) |119909| ge 119896 119905 isin [119905

0 119879] (119894

1 1198942) isin S

(80)

8 Journal of Applied Mathematics

Then (72) means 119892119896

rarr infin It follows from (76) and (79) that

[119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

ge 119892119896119875 120590119896le 119879 ge 120576119892

119896

(81)

Letting 119896 rarr infin yields a contradiction that is to say120590infin

= infinThe proof is complete

3 119871119875-Estimates

In the previous section we have investigated the existenceand uniqueness of the solution to (8) In this section as abovelet 119909(119905) 119905

0le 119905 le 119879 be the unique solution of (8) with initial

conditions 119909(1199050) = 1199090and 120574(119905

0) and we will estimate the 119901th

moment of the solution

Theorem 12 Assume that there is a function 119881(119909 119905 (1198941 1198942)) isin

11986221

(R119899 times R+times SR

+) and positive constants 119901 120578 120579 such that

for all (119909 119905 (1198941 1198942)) isin R119899 times R

+times S

120578|119909|119901le 119881 (119909 119905 (119894

1 1198942)) (82)

119871119881 (119909 119905 (1198941 1198942)) le 120579119881 (119909 119905 (119894

1 1198942)) (83)

Assume also the initial condition 119909(1199050) = 119909

0and 120574(119905

0) obeys

119864119881(1199090 1199050 120574(1199050)) lt infin then one has

119864|119909 (119905)|119901

le119864119881 (119909

0 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

120578 forall119905 isin [119905

0 119879]

(84)

Proof For each integer 119896 ge 1 define the stopping time

120590119896= 119879 and inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (85)

Thus 120590119896

rarr 119879 as Using the generalized Itorsquos formula and(83) we obtain that for 119905 isin [119905

0 119879]

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050))

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(86)

Then the Gronwall inequality indicates

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

(87)

for all 119905 isin [1199050 119879] By virtue of condition (83) we obtain the

required assertion (84)

Corollary 13 Assume 119901 ge 2 and 1199090isin 119871119901

F1199050

(ΩR119899) Assumealso that there exists a constant 120579 gt 0 such that for all(119909 119905 (119894

1 1198942)) isin R119899 times R

+times S

119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 120579 (1 + |119909|

2) (88)

Then one has

119864|119909 (119905)|119901

le 2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119905 minus 119905

0) forall119905 isin [119905

0 119879]

(89)

Proof Define 119881(119909 119905 (1198941 1198942)) = (1 + |119909|

2)1199012 Making use of

(88) yields

119871119881 (119909 119905 (1198941 1198942))

= 119901(1 + |119909|2)(119901minus2)2

119909119879119891 (119909 119905 119894

1)

+119901

2(1 + |119909|

2)(119901minus2)21003816100381610038161003816119892 (119909 119905 119894

2)10038161003816100381610038162

+119901 (119901 minus 2)

2(1 + |119909|

2)(119901minus4)210038161003816100381610038161003816

119909119879119892 (119909 119905 119894

2)10038161003816100381610038161003816

2

le 119901(1 + |119909|2)(119901minus2)2

[119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162]

le 119901120579(1 + |119909|2)1199012

(90)

Then byTheorem 12 we get

119864(1 + |119909 (119905)|2)1199012

le 119864(1 +10038161003816100381610038161199090

10038161003816100381610038162)1199012

exp 119901120579 (119905 minus 1199050)

(91)

and the required assertion (89) followsIt is useful to point out that if the linear growth condition

(43) is satisfied then (88) is fulfilled with 120579 = radic119870 + 119870(119901 minus

1)2 Now we will show the other important properties of thesolution

Theorem 14 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume alsothat the linear growth condition (43) holds Then one has

119864|119909 (119905) minus 119909 (119904)|119901le 119862(119905 minus 119904)

1199012 (92)

where

119862 = 2119901minus2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119879 minus 119905

0)

times ([2 (119879 minus 1199050)]1199012

+ [119901 (119901 minus 1)]1199012

)

(93)

and 120579 = radic119870+119870(119901minus1)2 Particularly the 119901th moment of thesolution is continuous on [119905

0 119879]

Journal of Applied Mathematics 9

Proof Applying the elementary inequality |119886 + 119887|119901

le

2119901minus1

(|119886|119901+ |119887|119901) the Holder inequality and the linear growth

condition we can derive that

119864|119909 (119905) minus 119909 (119904)|119901

le 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119891 (119909 (119906) 119906 1205741(119906)) 119889119906

10038161003816100381610038161003816100381610038161003816

119901

+ 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119892 (119909 (119906) 119906 1205742(119906)) 119889119861 (119906)

10038161003816100381610038161003816100381610038161003816

119901

le [2 (119905 minus 119904)]119901minus1

119864int

119905

119904

1003816100381610038161003816119891 (119909 (119906) 119906 1205741(119906))

1003816100381610038161003816119901119889119906

+ 05[2119901 (119901 minus 1)]1199012

(119905 minus 119904)(119901minus2)2

119864

times int

119905

119904

1003816100381610038161003816119892 (119909 (119906) 119906 1205742 (119906))1003816100381610038161003816119901119889119906

le 1198621(119905 minus 119904)

(119901minus2)2int

119905

119904

119864(1 + |119909 (119906)|2)1199012

119889119906

(94)

where1198621= 2(119901minus2)2

1198701199012

([2(119879minus 1199050)]1199012

+ [119901(119901minus1)]1199012

) Using(91) yields

119864|119909 (119905) minus 119909 (119904)|119901

le 1198621(119905 minus 119904)

(119901minus2)2

times int

119905

119904

2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119906 minus 119905

0) 119889119906

le 11986212(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901)

times exp 119901120579 (119879 minus 1199050) (119905 minus 119904)

(1199012)

(95)

which is the desired inequality

Theorem 15 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume thatthere is a119870 gt 0 such that for all (119909 119905 (119894

1 1198942)) isin R119899times[119905

0 119879]timesS

119909119879 1003816100381610038161003816119891 (119909 119905 119894

1)1003816100381610038161003816 or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (96)

Then

119864( sup1199050le119905le119879

|119909 (119905)|119901)

le (1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 2119901 (10119901 + 1)119870 (119879 minus 119905

0)

(97)

Proof Making use of the generalized Itorsquos formula andcondition (96) we have

|119909 (119905)|119901

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905)

+ int

119905

1199050

119901|119909 (119904)|119901minus2

[119909119879(119904) 119891 (119909 (119904) 119904 120574

1(119904))

+119901 minus 1

2

1003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162] 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 05119901 (119901 + 1)119870

times int

119905

1199050

|119909 (119904)|119901minus2

(1 + |119909 (119904)|2) 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 119901 (119901 + 1)119870int

119905

1199050

(1 + |119909 (119904)|119901) 119889119904

(98)

where

119872(119905) = int

119905

1199050

119901|119909 (119904)|119901minus2

119909119879(119904) 119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904) (99)

Therefore for any 1199051isin [1199050 119879]

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 119864

100381610038161003816100381611990901003816100381610038161003816119901+ 119864( sup

1199050le119905le1199051

|119872 (119905)|)

+ 119901 (119901 + 1)119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(100)

At the same time applying the well-known Burkholder-Davis-Gundy inequality (see eg [2]) gives

119864( sup1199050le119905le1199051

|119872 (119905)|)

le 3119864(int

1199051

1199050

1199012|119909 (119904)|

2119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162119889119904)

05

le 3119864( sup1199050le119905le1199051

|119909 (119905)|119901

timesint

1199051

1199050

1199012|119909 (119904)|

119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742(119904))

10038161003816100381610038162119889119904)

05

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 45119864

times int

1199051

1199050

1199012119870|119909 (119904)|

119901minus2(1 + |119909 (119904)|

2) 119889119904

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 9119901

2119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(101)

10 Journal of Applied Mathematics

Substituting the above inequality into (100) gives

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 2119864

100381610038161003816100381611990901003816100381610038161003816119901

+ 2119901 (10119901 + 1)119870int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(102)

Thus

1 + 119864( sup1199050le119905le1199051

|119909 (119905)|119901)

le 1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901+ 2119901 (10119901 + 1)119870

times int

1199051

1199050

(1 + sup1199050le119905le119904

|119909 (119905)|119901)119889119904

(103)

Then the required assertion follows from the Gronwallinequality

Up to now we have discussed the 119871119901-estimates for the

solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies

119864|119909 (119905)|119901le [119864|119909 (119905)|

2]05119901

(104)

4 Example

Consider the following Black-Scholes model

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (105)

where 1205741(119905) is a right-continuous homogenous Markovian

chain taking values in finite state spaces S1= 1 2 and 120574

2(119905)

is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S

2= 1 2 3 120583(119894) = 119894 119894 = 1 2

](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution

5 Conclusions and Further Research

This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis

(H1)1015840 both 1205741(119905) and 120574

2(119905) are right-continuous homoge-

nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a

homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper

can be established similarly It is easy to see that if 1205741(119905) equiv

1205742(119905) and 120574

1(119905) is a right-continuous homogenous Markovian

chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574

2(119905) (8) will reduce to the classical SDEs with

single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions

Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress

Acknowledgments

The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)

References

[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989

[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006

[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004

[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002

[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009

[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007

[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009

[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

Submit your manuscripts athttpwwwhindawicom

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

For the sake of convenience we take the following two-dimensional competitive model as an example

1198891199091= 1199091[11990310

(120574 (119905)) minus 11988611

(120574 (119905)) 1199091 minus 11988612

(120574 (119905)) 1199092] 119889119905

+ 1205721(120574 (119905)) 119909

11198891198611(119905)

1198891199092= 1199092[11990320

(120574 (119905)) minus 11988621

(120574 (119905)) 1199091minus 11988622

(120574 (119905)) 1199092] 119889119905

+ 1205722(120574 (119905)) 11990921198891198612 (119905)

(6)

where 119909119894is the size of 119894th species at time 119905 119903

1198940(119895) represents

the growth rate of 119894th species in regime 119895 for 119894 = 1 2119895 isin S and 119861

1and 119861

2are independent standard Brownian

motions However there are many stochastic factors thataffect some coefficients intensely but have little impact onother coefficients in (6) For example suppose that thestochastic factor is rain falls and 119909

1is able to endure a damp

weather while 1199092is fond of a dry environment then the rain

falls will affect 1199092intensely but have little impact on 119909

1 Thus

a more appropriate model is governed by

1198891199091= 1199091[11990310

(12057410

(119905)) minus 11988611

(12057411

(119905)) 1199091minus 11988612

(12057412

(119905)) 1199092] 119889119905

+ 1205721(1205741 (119905)) 11990911198891198611 (119905)

1198891199092= 1199092[11990320

(12057420

(119905)) minus 11988621

(12057421

(119905)) 1199091minus 11988622

(12057422

(119905)) 1199092] 119889119905

+ 1205722(1205742 (119905)) 11990921198891198612 (119905)

(7)

where 120574119894119895(119905) and 120574

119896(119905) are right-continuous homogenous

Markovian chains taking values in finite state spaces S119894119895for

119894 = 1 2 119895 = 0 1 2 and S119896for 119896 = 1 2 respectively

Thus the above examples show that the study of thefollowing SDEs with multi-Markovian switchings is essentialand is of great importance fromboth theoretical and practicalpoints

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741(119905)) 119889119905

+ 119892 (119909 (119905) 119905 1205742 (119905)) 119889119861 (119905) 1199050le 119905 le 119879

(8)

with initial conditions 119909(1199050) = 119909

0isin 1198712

F0

and 120574119894(1199050) where

120574119894(119905) is a right-continuous homogenous Markovian chain

on the probality space taking values in a finite state spaceS119894= 1 2 119873

119894 and isF

119905-adapted but independent of the

Brownian motion 119861119894(119905) 119894 = 1 2 and

119891 R119899times R+times S1997888rarr R

119899

119892 R119899times R+times S2997888rarr R

119899+119898

(9)

Equation (8) can be regarded as the result of the 1198731times 1198732

equations

119889119909 (119905) = 119891 (119909 (119905) 119905 119894) 119889119905

+ 119892 (119909 (119905) 119905 119895) 119889119861 (119905) 119894 isin S1 119895 isin S

2

(10)

switching among each other according to the movement ofthe Markovian chains It is important for us to discover the

properties of the system (8) and to find out whether thepresence of two Markovian switchings affects some knownresults The first step and the foundation of those studies areto establish the theorems for the existence and uniqueness ofthe solution to system (8) So in this paper we will give sometheorems for the existence and uniqueness of the solution tosystem (8) and study some properties of this solution Thetheory developed in this paper is the foundation for furtherstudy and can be applied in many different and complicatedsituations and hence the importance of the results in thispaper is clear

It should be pointed out that the theory developed in thispaper can be generalized to cope with the more general SDEswith more Markovian chains

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741 (119905) 120574119899 (119905)) 119889119905

+ 119892 (119909 (119905) 119905 120574119899+1

(119905) 120574119899+119898

(119905)) 119889119861 (119905)

(11)

The reason we concentrate on (8) rather than (11) is to avoidthe notations becoming too complicated Once the theorydeveloped in this paper is established the reader should beable to cope with the more general (11) without any difficulty

The remaining part of this paper is as follows In Section 2the sufficient criteria for existence anduniqueness of solutionlocal solution andmaximal local solution will be establishedrespectively In Section 3 the 119871

119875-estimates of the solutionwill be given In Section 4 we will introduce an example toillustrate our main result Finally we will close the paper withconclusions in Section 5

2 SDEs with Markovian Chains

Throughout this paper let (ΩF F119905119905isin119877+

P) be a completeprobability space Let 119861(119905) = (119861

1(119905) 119861

119898(119905))119879 be an 119898-

dimensional Brownian motion defined on the probabilityspace

In this section we will consider (8) Let 120574(119905) = (1205741(119905)

1205742(119905)) We impose a hypothesis(H1) 120574

1(119905) is independent of 120574

2(119905)

Then 120574(119905) is a homogenous vector Markovian chain withtransition probabilities

119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (119904) = (119894

1 1198942)

= 119875 120574 (119905) = (1198951 1198952) | 120574 (0) = (119894

1 1198942)

= 1198751198941119894211989511198952

(119905)

(12)

where (1198941 1198942) (1198951 1198952) isin S = (1 1) (1 2) (1119873

2) (2 1)

(1198731 1198732)

Now we will prepare some lemmas which are importantfor further study

Lemma 1 1198751198941119894211989511198952

(119905) has the following properties

(i) 1198751198941119894211989511198952

(119905) ge 0 for (1198941 1198942) (1198951 1198952) isin S

(ii) sum(11989511198952)isinS 1198751198941119894211989511198952

(119905) = 1 for (1198941 1198942) isin S

(iii) 1198751198941119894211989511198952

(0) = 12057511989411198951

sdot 12057511989421198952

where 120575119894119896119895119896

= 1 if 119894119896

= 119895119896

otherwise 120575119894119896119895119896

= 0 119896 = 1 2

Journal of Applied Mathematics 3

(iv) (Chapman-Kolmogorov equation) For 119904 119905 ge 0 and(1198941 1198942) (1198951 1198952) isin S

1198751198941119894211989511198952

(119904 + 119905)

= sum

(1198961 1198962)isinS

1198751198941119894211989611198962

(119905) 1198751198961119896211989511198952

(119904) (13)

Proof The proofs of (i) (ii) and (iii) are obvious Now let usprove (iv)

1198751198941119894211989511198952

(119904 + 119905)

= 119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (0) = (119894

1 1198942)

= sum

(11989611198962)isinS

119875 120574 (119904 + 119905) = (1198951 1198952)

120574 (119905) = (1198961 1198962) | 120574 (0) = (119894

1 1198942)

= sum

(11989611198962)isinS

(119875 120574 (119904 + 119905) = (1198951 1198952)

120574 (119905) = (1198961 1198962) 120574 (0) = (119894

1 1198942)

times(119875 120574 (119905) = (1198961 1198962) 120574 (0) = (119894

1 1198942))minus1

)

times119875 120574 (119905) = (119896

1 1198962) 120574 (0) = (119894

1 1198942)

119875 120574 (0) = (1198941 1198942)

= sum

(11989611198962)isinS

119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (119905) = (119896

1 1198962)

120574 (0) = (1198941 1198942)

times 119875 120574 (119905) = (1198961 1198962) | 120574 (0) = (119894

1 1198942)

= sum

(1198961 1198962)isinS

119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (119905) = (119896

1 1198962)

times 119875 120574 (119905) = (1198961 1198962) | 120574 (0) = (119894

1 1198942)

= sum

(1198961 1198962)isinS

1198751198961119896211989511198952

(119904) 1198751198941119894211989611198962

(119905)

(14)

This completes the proof

Now we impose another hypothesis which is calledstandard condition

(H2) lim119905rarr0

1198751198941119894211989511198952

(119905) = 1205751198941119894211989511198952

Lemma 2 Under Assumption (H2) for all 119905 ge 0 (1198941 1198942) isin S

one has 1198751198941119894211989411198942

(119905) gt 0

Proof From 1198751198941119894211989411198942

(0) gt 0 and (H2) we know that forarbitrary fixed 119905 gt 0 we have

1198751198941119894211989411198942

(119905

119899) gt 0 (15)

for sufficient large 119899 Then making use of Chapman-Kolm-ogorov equation

1198751198941119894211989411198942

(119904 + 119905) = sum

(1198961 1198962)isinS

11987511989411198942119896111198962

(119905) 1198751198961119896211989411198942

(119904)

ge 1198751198941119894211989411198942

(119905) 1198751198941119894211989411198942

(119904)

(16)

gives

1198751198941119894211989411198942

(119905) ge (1198751198941119894211989411198942

(119905

119899))

119899

gt 0 (17)

which is the desired assertion

Lemma 3 Under Assumption (H2) for all (1198941 1198942) isin S

minus1199021198941119894211989411198942

= lim119905rarr0

1 minus 1198751198941119894211989411198942

(119905)

119905(18)

exists (but may be infin)

Proof Define 120601(119905) = minus ln1198751198941119894211989411198942

(119905) ge 0 Then making use of(16) gives

120601 (119904 + 119905) le 120601 (119904) + 120601 (119905) (19)

Set minus1199021198941119894211989411198942

= sup119905gt0

(120601(119905)119905) It is easy to see that

0 le minus1199021198941119894211989411198942

le infin lim sup119905rarr0

120601 (119905)

119905le minus1199021198941119894211989411198942

(20)

Now we will assert

lim inf119905rarr0

120601 (119905)

119905ge minus1199021198941119894211989411198942

(21)

In fact for 0 lt ℎ lt 119905 exist119899 0 le 120576 le ℎ such that 119905 = 119899ℎ + 120576Applying (19) yields

120601 (119905)

119905ge

119899ℎ

120576

120601 (ℎ)

ℎ+

120601 (120576)

119905 (22)

Note that 120576 rarr 0 119899ℎ120576 rarr 1 120601(120576) rarr 0 whenever ℎ rarr 0+

then for all 119905 gt 0 we have

120601 (119905)

119905le lim infℎrarr0

120601 (ℎ)

ℎ (23)

This implies minus1199021198941119894211989411198942

le lim inf119905rarr0

(120601(119905)119905) Thus

minus1199021198941119894211989411198942

= lim119905rarr0

120601 (119905)

119905 (24)

Using the definition of 120601(119905) gives

lim119905rarr0

1 minus 1198751198941119894211989411198942

(119905)

119905= lim119905rarr0

1 minus exp 120601 (119905)

120601 (119905)

120601 (119905)

119905= minus1199021198941119894211989411198942

(25)

which is the required assertion

4 Journal of Applied Mathematics

Lemma 4 Under Assumption (H2) for (1198941 1198942) (1198951 1198952) isin

S (1198941 1198942) = (1198951 1198952)

1199021198941119894211989511198952

= 1198751015840

1198941119894211989511198952

(0) = lim119905rarr0

1198751198941119894211989511198952

(119905)

119905(26)

exists and is finite

Proof By (H2) we note that for all 0 lt 120576 lt 13 exist0 lt 120575 lt 1such that

1198751198941119894211989411198942

(119905) gt 1 minus 120576 1198751198951119895211989511198952

(119905) gt 1 minus 120576

1198751198951119895211989411198942

(119905) lt 120576

(27)

provided 0 lt 119905 le 120575For forall0 le ℎ lt 119905 set 119899 = ⟨119905ℎ⟩ where ⟨119886⟩ = max

119899le119886119899 isin

Z Let

119875(11989511198952)

1198941119894211989611198962

(ℎ) = 1198751198941119894211989611198962

(ℎ)

119875(11989511198952)

1198941119894211989611198962

(119898ℎ)

= sum

(11990311199032) = (1198951 1198952)

119875(11989511198952)

1198941119894211990311199032

((119898 minus 1) ℎ) 1198751199031119903211989611198962

(ℎ)

(28)

where 119875(11989511198952)

1198941119894211989611198962

(119898ℎ)means that the probability of the 120574(119905)willnot reach to (119895

1 1198952) at times ℎ 2ℎ (119898 minus 1)ℎ but will reach

to (1198961 1198962) at time 119898ℎ Note that if ℎ le 119905 le 120575 then

120576 gt 1 minus 1198751198941119894211989411198942

(119905)

= sum

(1198961 1198962) = (1198941 1198942)

1198751198941119894211989611198962

(119905) ge 1198751198941119894211989511198952

(119905)

ge

119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ) 1198751198951119895211989511198952

(119905 minus 119898ℎ)

ge (1 minus 120576)

119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ)

(29)

which indicates119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ) le120576

1 minus 120576 (30)

Then making use of

1198751198941119894211989411198942

(119898ℎ) = 119875(11989511198952)

1198941119894211989411198942

(119898ℎ)

+

119898minus1

sum

119897=1

119875(11989511198952)

1198941119894211989511198952

(119897ℎ) 1198751198951119895211989411198942

((119898 minus 119897) ℎ)

(31)

we obtain

119875(11989511198952)

1198941119894211989411198942

(119898ℎ) ge 1198751198941119894211989411198942

(119898ℎ) minus

119898minus1

sum

119897=1

119875(11989511198952)

1198941119894211989511198952

(119897ℎ)

ge 1 minus 120576 minus120576

1 minus 120576

(32)

Consequently

1198751198941119894211989511198952

(119905)

gt

119899

sum

119898=1

119875(11989511198952)

1198941119894211989411198942

((119898 minus 1) ℎ) 1198751198941119894211989511198952

(ℎ) 1198751198951119895211989511198952

(119905 minus 119898ℎ)

ge 119899 (1 minus 120576 minus120576

1 minus 120576)1198751198941119894211989511198952

(ℎ) (1 minus 120576)

ge 119899 (1 minus 3120576) 1198751198941119894211989511198952

(ℎ)

(33)

Dividing both sides of the above inequality by ℎ and noting119899ℎ rarr 119905 whenever ℎ rarr 0 yield

lim supℎrarr0

1198751198941119894211989511198952

(ℎ)

ℎle

1

1 minus 3120576

1198751198941119894211989511198952

(119905)

119905lt infin (34)

Then letting 119905 rarr 0 gives

lim supℎrarr0

1198751198941119894211989511198952

(ℎ)

ℎle

1

1 minus 3120576lim inf119905rarr0

1198751198941119894211989511198952

(119905)

119905 (35)

and the required assertion follows immediately by letting120576 rarr 0 This completes the proof

Set 120574(119905) = (1205741(119905) 1205742(119905)) then it is easy to see that

almost every sample path of 120574(119905) is a right continuous stepfunction Now letting P(119905) = (119875

1198941119894211989511198952

(119905))11987311198732times11987311198732

Q =

(1199021198941119894211989511198952

)11987311198732times11987311198732

= P1015840(0) Then by Chapman-Kolmogorovequation

P (119905 + ℎ) = P (119905)P (ℎ) = P (ℎ)P (119905) (36)

we have

P (119905 + ℎ) minus P (119905)

ℎ= P (119905) [

P (ℎ) minus Iℎ

] = [P (ℎ) minus I

ℎ]P (119905)

(37)

Letting ℎ rarr 0 and taking limits give

P1015840(119905) = P (119905)Q

P1015840(119905) = QP (119905)

(38)

Note that

P (0) = I (39)

Then by solving the ordinary differential equations (38) and(39) we obtain the following lemma

Lemma 5 For P(119905) and Q one has

P (119905) = exp Q119905 (40)

We are now in the position to give the sufficient condi-tions for the existence and uniqueness of the solution of (8)For this end let us first give the definition of the solution

Journal of Applied Mathematics 5

Definition 6 An R119899-valued stochastic process 119909(119905)1199050le119905le119879

iscalled a solution of (8) if it has the following properties

(i) 119909(119905)1199050le119905le119879

is continuous andF119905-adapted

(ii) 119891(119909(119905) 119905 1205741(119905))1199050le119905le119879

isinL1([1199050 119879]R119899)while 119892(119909(119905)

119905 1205742(119905))1199050le119905le119879

isin L2([1199050 119879]R119899times119898)

(iii) for any 119905 isin [1199050 119879] equation

119909 (119905) = 119909 (1199050) + int

119905

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(41)

holds with probability 1

A solution 119909(119905)1199050le119905le119879

is said to be unique if any othersolution (119905)

1199050le119905le119879

is indistinguishable from 119909(119905)1199050le119905le119879

Now we can give our main results in this section

Theorem 7 Assume that there exist two positive constants 119870

and 119870 such that(Lipschitz condition) for all 119909 119910 isin R119899 119905 isin [119905

0 119879] and

(1198941 1198942) isin S

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le 119870

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(42)

(Linear growth condition) for all (119909 119905 (1198941 1198942)) isin R119899 times

[1199050 119879] times S

1003816100381610038161003816119891 (119909 119905 1198941)10038161003816100381610038162or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (43)

Then there exists a unique solution 119909(119905) to (8) and moreoverthe solution obeys

119864( sup1199050le119905le119879

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(44)

Proof Recall that almost every sample path of 120574(119905) =

(1205741(119905) 1205742(119905)) is a right continuous step function with a finite

number of jumps on [1199050 119879] Thus there exists a sequence of

stopping times 120591119896119896ge0

such that

(i) for almost every 120596 isin Ω there is a finite 119896120596for 1199050

=

1205910lt 1205911lt sdot sdot sdot lt 120591

119896= 119879 and 120591

119896= 119879 if 119896 gt 119896

120596

(ii) both 1205741(sdot) and 120574

2(sdot) in 120574(sdot) are constants on interval

[[120591119896 120591119896+1

[[ namely

1205741 (119905) = 120574

1(120591119896) on 120591

119896le 119905 le 120591

119896+1forall119896 ge 0

1205742(119905) = 120574

2(120591119896) on 120591

119896le 119905 le 120591

119896+1forall119896 ge 0

(45)

First of all let us consider (8) on 119905 isin [[1205910 1205911[[ then (8)

becomes119889119909 (119905) = 119891 (119909 (119905) 119905 120574

1(1199050)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(1199050)) 119889119861 (119905)

(46)

with initial conditions 119909(1199050) = 119909

0 120574(1199050) = (120574

1(1199050) 1205741(1199050))

Then by the theory of SDEs we obtain that (46) has a uniquesolution which obeys 119909(120591

1) isin 1198712

F1205911

(ΩR119899) We next consider(8) on 119905 isin [[120591

1 1205912[[ which becomes

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741 (1205911)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(1205911)) 119889119861 (119905)

(47)

Again by the theory of SDEs (47) has a unique solutionwhichobeys 119909(120591

1) isin 119871

2

F1205912

(ΩR119899) Repeating this procedure weconclude that (8) has a unique solution 119909(119905) on [119905

0 119879]

Now let us prove (44) For every 119896 ge 1 define thestopping time

120591119896= 119879 and inf 119905 isin [119905

0 119879] |119909 (119905)| ge 119896 (48)

It is obvious that 120591119896uarr 119879 as Set 119909

119896(119905) = 119909(119905and120591

119896) for 119905 isin [119905

0 119879]

Then 119909119896(119905) obeys the equation

119909119896(119905) = 119909

0+ int

119905

1199050

119891 (119909119896(119904) 119904 120574

1(119904)) 119868[[1199050120591119896]](119904) 119889119904

+ int

119905

1199050

119892 (119909119896(119904) 119904 120574

2(119904)) 119868[[1199050120591119896]](119904) 119889119861 (119904)

(49)

Making use of the elementary inequality |119886+119887+119888|2le 3(|119886|

2+

|119887|2+ |119888|2) the Holder inequality and (43) we can see that

1003816100381610038161003816119909119896 (119905)10038161003816100381610038162= 3

1003816100381610038161003816119909010038161003816100381610038162+ 3119870 (119879 minus 119905

0) int

119905

1199050

(1 +1003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

+ 3

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199050

119892 (119909119896(119904) 119904 120574

2(119904)) 119868[[1199050120591119896]](119904) 119889119861 (119904)

100381610038161003816100381610038161003816100381610038161003816

2

(50)

Thus applying the Doob martingale inequality and (43) wecan further show that

119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

= 311986410038161003816100381610038161199090

10038161003816100381610038162+ 3119870 (119879 minus 119905

0) int

119905

1199050

(1 + 1198641003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

+ 12119864int

119905

1199050

1003816100381610038161003816119892 (119909119896(119904) 119904 120574

2(119904))

10038161003816100381610038162119868[[1199050120591119896]](119904) 119889119904

le 311986410038161003816100381610038161199090

10038161003816100381610038162+ 3119870 (119879 minus 119905

0+ 4)int

119905

1199050

(1 + 1198641003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

(51)

That is to say

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162) le 1 + 3119864

1003816100381610038161003816119909010038161003816100381610038162+ 3119870 (119879 minus 119905

0+ 4)

times int

119905

1199050

[1 + 119864 sup1199050le119903le119904

1003816100381610038161003816119909119896 (119903)10038161003816100381610038162]119889119904

(52)

6 Journal of Applied Mathematics

Using the Gronwall inequality leads to

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(53)

Consequently

1 + 119864( sup1199050le119905le120591119896

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(54)

Then the required inequality (44) follows immediately byletting 119896 rarr infin

Condition (42) indicates that the coefficients 119891(119909 119905 1198941)

and 119892(119909 119905 1198942) do not change faster than a linear function of

119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894

1) and 119892(119909 119905 119894

2) in 119909 for all 119905 isin [119905

0 119879] Then functions

that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution

Definition 8 Let120590infinbe a stopping time such that 119905

0le 120590infin

le 119879

as An R119899-valuedF119905-adapted continuous stochastic process

119909(119905)1199050le119905lt120590infin

is called a local solution of (8) if 119909(1199050) = 1199090and

moreover there is a nondecreasing sequence 120590119896119896ge1

such that1199050le 120590119896 uarr 120590infin

as and

119909 (119905) = 119909 (1199050) + int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(55)

holds for any 119905 isin [1199050 119879) and 119896 ge 1 with probability one If

furthermore

lim sup119905rarr+120590

infin

|119909 (119905)| = infin whenever 120590infin

lt 119879 (56)

then it is called a maximal local solution and 120590infinis called the

explosion time Amaximal local solution 119909(119905) 1199050le 119905 lt 120590

infin

is said to be unique if any other maximal local solution 119909(119905)

1199050le 119905 lt 120590

infin is indistinguishable from it namely 120590

infin= 120590infin

and 119909(119905) = 119909(119905) for 1199050le 119905 lt 120590

infinwith probability one

Definition 9 (local Lipschitz condition) For every integer119896 ge 1 there exists a positive constant ℎ

119896such that for all

119905 isin [1199050 119879] 119894 = (119894

1 1198942) isin S and those 119909 119910 isin R119899 with

|119909| or |119910| le 119896

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le ℎ119896

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(57)

The following theorem shows the existence of uniquemaximal local solution under the local Lipschitz conditionwithout the linear growth condition

Theorem 10 Under condition (57) there exists a uniquemaximal local solution of (8)

Proof Define functions

119891119896(119909 119905 119894

1) =

119891 (119909 119905 1198941) if |119909| le 119896

119891(119896119909

|119909| 119905 1198941) if |119909| gt 119896

119892119896(119909 119905 119894

2) =

119892 (119909 119905 1198942) if |119909| le 119896

119892 (119896119909

|119909| 119905 1198942) if |119909| gt 119896

(58)

Then 119891119896and 119892

119896satisfy the Lipschitz condition and the linear

growth condition Thus by Theorem 7 there is a uniquesolution 119909

119896(119905) of the equation

119889119909119896(119905) = 119891

119896(119909119896(119905) 119905 120574

1(119905)) 119889119905

+ 119892119896(119909119896(119905) 119905 120574

2(119905)) 119889119861 (119905)

(59)

with the initial conditions 119909119896(1199050) = 119909

0and 120574(119905

0) = (120574

1(1199050)

1205742(1199050)) Define the stopping times

120590119896= 119879 and inf 119905 isin [119905

0 119879]

1003816100381610038161003816119909119896 (119905)1003816100381610038161003816 ge 119896 (60)

Clearly if 1199050le 119905 le 120590

119896

119909119896(119905) = 119909

119896+1(119905) (61)

which indicates that 120590119896is increasing so 120590

119896has its limit 120590

infin=

lim119896rarr+infin

120590119896 Define 119909(119905) 119905

0le 119905 lt 120590

infin by

119909 (119905) = 119909119896(119905) 119905 isin [[120590

119896minus1 120590119896[[ 119896 ge 1 (62)

where 1205900= 1199050 Applying (61) one can show that 119909(119905 and 120590

119896) =

119909119896(119905 and 120590119896) It then follows from (59) that

119909 (119905 and 120590119896) = 1199090+ int

119905and120590119896

1199050

119891119896(119909 (119904) 119904 120574

1(119904)) 119889119904

+ int

119905and120590119896

1199050

119892119896(119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

= 1199090+ int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

(63)

for any 119905 isin [1199050 119879) and 119896 ge 1 It is easy to see that if 120590

infinle 119879

then

lim sup119905rarr120590infin

|119909 (119905)|

ge lim sup119896rarr+infin

1003816100381610038161003816119909 (120590119896)1003816100381610038161003816 = lim sup119896rarr+infin

1003816100381610038161003816119909119896 (120590119896)1003816100381610038161003816 = infin

(64)

Therefore 119909(119905) 1199050le 119905 lt 120590

infin is a maximal local solution

Now we will prove the uniqueness Let 119909(119905) 1199050

le 119905 lt

120590infin

be another maximal local solution Define

120590119896= 120590infin

and inf 119905 isin [[1199050 120590infin

[[ 119909 (119905) ge 119896 (65)

Journal of Applied Mathematics 7

Then 120590119896

rarr 120590infin

as and

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590119896and 120590119896]] = 1 forall119896 ge 1 (66)

Letting 119896 rarr infin gives

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590infin

and 120590infin

[[ = 1 (67)

In order to complete the proof we need only to show that120590infin

= 120590infin

as In fact for almost every 120596 isin 120590infin

lt 120590infin

wehave

1003816100381610038161003816119909 (120590infin

120596)1003816100381610038161003816

= lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = infin(68)

which contradicts the fact that 119909(119905 120596) is continuous on 119905 isin

[1199050 120590infin

(120596)) This implies 120590infin

ge 120590infin

as In the same way onecan show 120590

infinle 120590infinasThus wemust have 120590

infin= 120590infinasThis

completes the proof

In many situations we often consider an SDE on [1199050infin)

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741(119905)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(119905)) 119889119861 (119905) 119905

0le 119905 lt infin

(69)

with initial data 119909(1199050) = 119909

0and 120574(119905

0) If the assumption of

the existence-and-uniqueness theorem holds on every finitesubinterval [119905

0 119879] of [119905

0infin) then (69) has a unique solution

119909(119905) on the entire interval [1199050infin) Such a solution is called

a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R

+times S to R which is

given by

119871119881 (119909 119905 (1198941 1198942))

= 119881119905(119909 119905 (119894

1 1198942)) + 119881

119909(119909 119905 (119894

1 1198942)) 119891 (119909 119905 119894

1)

+ 05 trace [119892119879 (119909 119905 1198942) 119881119909119909

(119909 119905 (1198941 1198942)) 119892 (119909 119905 119894

2)]

+ sum

(11989511198952)isinS

1199021198941119894211989511198952

119881 (119909 119905 (1198951 1198952))

(70)

where 119881(119909 119905 (1198941 1198942)) isin 119862

21(R119899 times R

+times SR

+) and

119881119905(119909 119905 (119894

1 1198942)) =

120597119881 (119909 119905 (1198941 1198942))

120597119905

119881119909(119909 119905 (119894

1 1198942))

= (120597119881 (119909 119905 (119894

1 1198942))

1205971199091

120597119881 (119909 119905 (119894

1 1198942))

120597119909119899

)

119881119909119909

(119909 119905 (1198941 1198942)) = (

1205972119881 (119909 119905 (119894

1 1198942))

120597119909119894120597119909119895

)

119899times119899

(71)

Theorem 11 Assume that the local Lipschitz condition (57)holds Assume also that there is a function 119881(119909 119905 (119894

1 1198942)) isin

11986221

(R119899 times R+times SR

+) and a constant 120579 gt 0 such that

lim|119909|rarrinfin

( inf(119905(1198941 1198942))isinR+timesS

119881 (119909 119905 (1198941 1198942))) = infin (72)

119871119881 (119909 119905 (1198941 1198942)) le 120579 (1 + 119881 (119909 119905 (119894

1 1198942)))

forall (119909 119905 (1198941 1198942)) isin R

119899times R+times S

(73)

Then there exists a unique global solution 119909(119905) to (69)

Proof We need only to prove the theorem for any initialcondition 119909

0isin R119899 and (120574

1(1199050) 1205742(1199050)) isin S FromTheorem 10

we know that the local Lipschitz condition guarantees theexistence of the unique maximal solution 119909(119905) on [119905

0 120590infin

)where 120590

infinis the explosion time We need only to show 120590

infin=

infin as If this is not true then we can find a pair of positiveconstants 120576 and 119879 such that

119875 120590infin

le 119879 gt 2120576 (74)

For each integer 119896 ge 1 define the stopping time

120590119896= inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (75)

Since 120590119896

rarr 120590infinalmost surely we can find a sufficiently large

integer 1198960for

119875 120590119896le 119879 gt 120576 forall119896 ge 119896

0 (76)

Fix any 119896 ge 1198960 then for any 119905

0le 119905 le 119879 by virtue of the

generalized Ito formula (see eg [1])

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

= 119881 (1199090 1199050 1205740) + 119864int

119905and120590119896

1199050

119871119881 (119909 (119904) 119904 120574 (119904)) 119889119904

le 119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(77)

Making use of the Gronwall inequality gives

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(78)

Therefore

119864 (119868120590119896le119879

119881 (119909 (120590119896) 120590119896 120574 (120590119896)))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(79)

At the same time set

119892119896= inf 119881 (119909 119905 (119894

1 1198942)) |119909| ge 119896 119905 isin [119905

0 119879] (119894

1 1198942) isin S

(80)

8 Journal of Applied Mathematics

Then (72) means 119892119896

rarr infin It follows from (76) and (79) that

[119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

ge 119892119896119875 120590119896le 119879 ge 120576119892

119896

(81)

Letting 119896 rarr infin yields a contradiction that is to say120590infin

= infinThe proof is complete

3 119871119875-Estimates

In the previous section we have investigated the existenceand uniqueness of the solution to (8) In this section as abovelet 119909(119905) 119905

0le 119905 le 119879 be the unique solution of (8) with initial

conditions 119909(1199050) = 1199090and 120574(119905

0) and we will estimate the 119901th

moment of the solution

Theorem 12 Assume that there is a function 119881(119909 119905 (1198941 1198942)) isin

11986221

(R119899 times R+times SR

+) and positive constants 119901 120578 120579 such that

for all (119909 119905 (1198941 1198942)) isin R119899 times R

+times S

120578|119909|119901le 119881 (119909 119905 (119894

1 1198942)) (82)

119871119881 (119909 119905 (1198941 1198942)) le 120579119881 (119909 119905 (119894

1 1198942)) (83)

Assume also the initial condition 119909(1199050) = 119909

0and 120574(119905

0) obeys

119864119881(1199090 1199050 120574(1199050)) lt infin then one has

119864|119909 (119905)|119901

le119864119881 (119909

0 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

120578 forall119905 isin [119905

0 119879]

(84)

Proof For each integer 119896 ge 1 define the stopping time

120590119896= 119879 and inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (85)

Thus 120590119896

rarr 119879 as Using the generalized Itorsquos formula and(83) we obtain that for 119905 isin [119905

0 119879]

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050))

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(86)

Then the Gronwall inequality indicates

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

(87)

for all 119905 isin [1199050 119879] By virtue of condition (83) we obtain the

required assertion (84)

Corollary 13 Assume 119901 ge 2 and 1199090isin 119871119901

F1199050

(ΩR119899) Assumealso that there exists a constant 120579 gt 0 such that for all(119909 119905 (119894

1 1198942)) isin R119899 times R

+times S

119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 120579 (1 + |119909|

2) (88)

Then one has

119864|119909 (119905)|119901

le 2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119905 minus 119905

0) forall119905 isin [119905

0 119879]

(89)

Proof Define 119881(119909 119905 (1198941 1198942)) = (1 + |119909|

2)1199012 Making use of

(88) yields

119871119881 (119909 119905 (1198941 1198942))

= 119901(1 + |119909|2)(119901minus2)2

119909119879119891 (119909 119905 119894

1)

+119901

2(1 + |119909|

2)(119901minus2)21003816100381610038161003816119892 (119909 119905 119894

2)10038161003816100381610038162

+119901 (119901 minus 2)

2(1 + |119909|

2)(119901minus4)210038161003816100381610038161003816

119909119879119892 (119909 119905 119894

2)10038161003816100381610038161003816

2

le 119901(1 + |119909|2)(119901minus2)2

[119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162]

le 119901120579(1 + |119909|2)1199012

(90)

Then byTheorem 12 we get

119864(1 + |119909 (119905)|2)1199012

le 119864(1 +10038161003816100381610038161199090

10038161003816100381610038162)1199012

exp 119901120579 (119905 minus 1199050)

(91)

and the required assertion (89) followsIt is useful to point out that if the linear growth condition

(43) is satisfied then (88) is fulfilled with 120579 = radic119870 + 119870(119901 minus

1)2 Now we will show the other important properties of thesolution

Theorem 14 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume alsothat the linear growth condition (43) holds Then one has

119864|119909 (119905) minus 119909 (119904)|119901le 119862(119905 minus 119904)

1199012 (92)

where

119862 = 2119901minus2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119879 minus 119905

0)

times ([2 (119879 minus 1199050)]1199012

+ [119901 (119901 minus 1)]1199012

)

(93)

and 120579 = radic119870+119870(119901minus1)2 Particularly the 119901th moment of thesolution is continuous on [119905

0 119879]

Journal of Applied Mathematics 9

Proof Applying the elementary inequality |119886 + 119887|119901

le

2119901minus1

(|119886|119901+ |119887|119901) the Holder inequality and the linear growth

condition we can derive that

119864|119909 (119905) minus 119909 (119904)|119901

le 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119891 (119909 (119906) 119906 1205741(119906)) 119889119906

10038161003816100381610038161003816100381610038161003816

119901

+ 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119892 (119909 (119906) 119906 1205742(119906)) 119889119861 (119906)

10038161003816100381610038161003816100381610038161003816

119901

le [2 (119905 minus 119904)]119901minus1

119864int

119905

119904

1003816100381610038161003816119891 (119909 (119906) 119906 1205741(119906))

1003816100381610038161003816119901119889119906

+ 05[2119901 (119901 minus 1)]1199012

(119905 minus 119904)(119901minus2)2

119864

times int

119905

119904

1003816100381610038161003816119892 (119909 (119906) 119906 1205742 (119906))1003816100381610038161003816119901119889119906

le 1198621(119905 minus 119904)

(119901minus2)2int

119905

119904

119864(1 + |119909 (119906)|2)1199012

119889119906

(94)

where1198621= 2(119901minus2)2

1198701199012

([2(119879minus 1199050)]1199012

+ [119901(119901minus1)]1199012

) Using(91) yields

119864|119909 (119905) minus 119909 (119904)|119901

le 1198621(119905 minus 119904)

(119901minus2)2

times int

119905

119904

2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119906 minus 119905

0) 119889119906

le 11986212(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901)

times exp 119901120579 (119879 minus 1199050) (119905 minus 119904)

(1199012)

(95)

which is the desired inequality

Theorem 15 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume thatthere is a119870 gt 0 such that for all (119909 119905 (119894

1 1198942)) isin R119899times[119905

0 119879]timesS

119909119879 1003816100381610038161003816119891 (119909 119905 119894

1)1003816100381610038161003816 or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (96)

Then

119864( sup1199050le119905le119879

|119909 (119905)|119901)

le (1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 2119901 (10119901 + 1)119870 (119879 minus 119905

0)

(97)

Proof Making use of the generalized Itorsquos formula andcondition (96) we have

|119909 (119905)|119901

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905)

+ int

119905

1199050

119901|119909 (119904)|119901minus2

[119909119879(119904) 119891 (119909 (119904) 119904 120574

1(119904))

+119901 minus 1

2

1003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162] 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 05119901 (119901 + 1)119870

times int

119905

1199050

|119909 (119904)|119901minus2

(1 + |119909 (119904)|2) 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 119901 (119901 + 1)119870int

119905

1199050

(1 + |119909 (119904)|119901) 119889119904

(98)

where

119872(119905) = int

119905

1199050

119901|119909 (119904)|119901minus2

119909119879(119904) 119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904) (99)

Therefore for any 1199051isin [1199050 119879]

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 119864

100381610038161003816100381611990901003816100381610038161003816119901+ 119864( sup

1199050le119905le1199051

|119872 (119905)|)

+ 119901 (119901 + 1)119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(100)

At the same time applying the well-known Burkholder-Davis-Gundy inequality (see eg [2]) gives

119864( sup1199050le119905le1199051

|119872 (119905)|)

le 3119864(int

1199051

1199050

1199012|119909 (119904)|

2119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162119889119904)

05

le 3119864( sup1199050le119905le1199051

|119909 (119905)|119901

timesint

1199051

1199050

1199012|119909 (119904)|

119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742(119904))

10038161003816100381610038162119889119904)

05

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 45119864

times int

1199051

1199050

1199012119870|119909 (119904)|

119901minus2(1 + |119909 (119904)|

2) 119889119904

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 9119901

2119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(101)

10 Journal of Applied Mathematics

Substituting the above inequality into (100) gives

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 2119864

100381610038161003816100381611990901003816100381610038161003816119901

+ 2119901 (10119901 + 1)119870int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(102)

Thus

1 + 119864( sup1199050le119905le1199051

|119909 (119905)|119901)

le 1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901+ 2119901 (10119901 + 1)119870

times int

1199051

1199050

(1 + sup1199050le119905le119904

|119909 (119905)|119901)119889119904

(103)

Then the required assertion follows from the Gronwallinequality

Up to now we have discussed the 119871119901-estimates for the

solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies

119864|119909 (119905)|119901le [119864|119909 (119905)|

2]05119901

(104)

4 Example

Consider the following Black-Scholes model

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (105)

where 1205741(119905) is a right-continuous homogenous Markovian

chain taking values in finite state spaces S1= 1 2 and 120574

2(119905)

is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S

2= 1 2 3 120583(119894) = 119894 119894 = 1 2

](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution

5 Conclusions and Further Research

This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis

(H1)1015840 both 1205741(119905) and 120574

2(119905) are right-continuous homoge-

nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a

homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper

can be established similarly It is easy to see that if 1205741(119905) equiv

1205742(119905) and 120574

1(119905) is a right-continuous homogenous Markovian

chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574

2(119905) (8) will reduce to the classical SDEs with

single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions

Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress

Acknowledgments

The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)

References

[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989

[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006

[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004

[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002

[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009

[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007

[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009

[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

(iv) (Chapman-Kolmogorov equation) For 119904 119905 ge 0 and(1198941 1198942) (1198951 1198952) isin S

1198751198941119894211989511198952

(119904 + 119905)

= sum

(1198961 1198962)isinS

1198751198941119894211989611198962

(119905) 1198751198961119896211989511198952

(119904) (13)

Proof The proofs of (i) (ii) and (iii) are obvious Now let usprove (iv)

1198751198941119894211989511198952

(119904 + 119905)

= 119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (0) = (119894

1 1198942)

= sum

(11989611198962)isinS

119875 120574 (119904 + 119905) = (1198951 1198952)

120574 (119905) = (1198961 1198962) | 120574 (0) = (119894

1 1198942)

= sum

(11989611198962)isinS

(119875 120574 (119904 + 119905) = (1198951 1198952)

120574 (119905) = (1198961 1198962) 120574 (0) = (119894

1 1198942)

times(119875 120574 (119905) = (1198961 1198962) 120574 (0) = (119894

1 1198942))minus1

)

times119875 120574 (119905) = (119896

1 1198962) 120574 (0) = (119894

1 1198942)

119875 120574 (0) = (1198941 1198942)

= sum

(11989611198962)isinS

119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (119905) = (119896

1 1198962)

120574 (0) = (1198941 1198942)

times 119875 120574 (119905) = (1198961 1198962) | 120574 (0) = (119894

1 1198942)

= sum

(1198961 1198962)isinS

119875 120574 (119904 + 119905) = (1198951 1198952) | 120574 (119905) = (119896

1 1198962)

times 119875 120574 (119905) = (1198961 1198962) | 120574 (0) = (119894

1 1198942)

= sum

(1198961 1198962)isinS

1198751198961119896211989511198952

(119904) 1198751198941119894211989611198962

(119905)

(14)

This completes the proof

Now we impose another hypothesis which is calledstandard condition

(H2) lim119905rarr0

1198751198941119894211989511198952

(119905) = 1205751198941119894211989511198952

Lemma 2 Under Assumption (H2) for all 119905 ge 0 (1198941 1198942) isin S

one has 1198751198941119894211989411198942

(119905) gt 0

Proof From 1198751198941119894211989411198942

(0) gt 0 and (H2) we know that forarbitrary fixed 119905 gt 0 we have

1198751198941119894211989411198942

(119905

119899) gt 0 (15)

for sufficient large 119899 Then making use of Chapman-Kolm-ogorov equation

1198751198941119894211989411198942

(119904 + 119905) = sum

(1198961 1198962)isinS

11987511989411198942119896111198962

(119905) 1198751198961119896211989411198942

(119904)

ge 1198751198941119894211989411198942

(119905) 1198751198941119894211989411198942

(119904)

(16)

gives

1198751198941119894211989411198942

(119905) ge (1198751198941119894211989411198942

(119905

119899))

119899

gt 0 (17)

which is the desired assertion

Lemma 3 Under Assumption (H2) for all (1198941 1198942) isin S

minus1199021198941119894211989411198942

= lim119905rarr0

1 minus 1198751198941119894211989411198942

(119905)

119905(18)

exists (but may be infin)

Proof Define 120601(119905) = minus ln1198751198941119894211989411198942

(119905) ge 0 Then making use of(16) gives

120601 (119904 + 119905) le 120601 (119904) + 120601 (119905) (19)

Set minus1199021198941119894211989411198942

= sup119905gt0

(120601(119905)119905) It is easy to see that

0 le minus1199021198941119894211989411198942

le infin lim sup119905rarr0

120601 (119905)

119905le minus1199021198941119894211989411198942

(20)

Now we will assert

lim inf119905rarr0

120601 (119905)

119905ge minus1199021198941119894211989411198942

(21)

In fact for 0 lt ℎ lt 119905 exist119899 0 le 120576 le ℎ such that 119905 = 119899ℎ + 120576Applying (19) yields

120601 (119905)

119905ge

119899ℎ

120576

120601 (ℎ)

ℎ+

120601 (120576)

119905 (22)

Note that 120576 rarr 0 119899ℎ120576 rarr 1 120601(120576) rarr 0 whenever ℎ rarr 0+

then for all 119905 gt 0 we have

120601 (119905)

119905le lim infℎrarr0

120601 (ℎ)

ℎ (23)

This implies minus1199021198941119894211989411198942

le lim inf119905rarr0

(120601(119905)119905) Thus

minus1199021198941119894211989411198942

= lim119905rarr0

120601 (119905)

119905 (24)

Using the definition of 120601(119905) gives

lim119905rarr0

1 minus 1198751198941119894211989411198942

(119905)

119905= lim119905rarr0

1 minus exp 120601 (119905)

120601 (119905)

120601 (119905)

119905= minus1199021198941119894211989411198942

(25)

which is the required assertion

4 Journal of Applied Mathematics

Lemma 4 Under Assumption (H2) for (1198941 1198942) (1198951 1198952) isin

S (1198941 1198942) = (1198951 1198952)

1199021198941119894211989511198952

= 1198751015840

1198941119894211989511198952

(0) = lim119905rarr0

1198751198941119894211989511198952

(119905)

119905(26)

exists and is finite

Proof By (H2) we note that for all 0 lt 120576 lt 13 exist0 lt 120575 lt 1such that

1198751198941119894211989411198942

(119905) gt 1 minus 120576 1198751198951119895211989511198952

(119905) gt 1 minus 120576

1198751198951119895211989411198942

(119905) lt 120576

(27)

provided 0 lt 119905 le 120575For forall0 le ℎ lt 119905 set 119899 = ⟨119905ℎ⟩ where ⟨119886⟩ = max

119899le119886119899 isin

Z Let

119875(11989511198952)

1198941119894211989611198962

(ℎ) = 1198751198941119894211989611198962

(ℎ)

119875(11989511198952)

1198941119894211989611198962

(119898ℎ)

= sum

(11990311199032) = (1198951 1198952)

119875(11989511198952)

1198941119894211990311199032

((119898 minus 1) ℎ) 1198751199031119903211989611198962

(ℎ)

(28)

where 119875(11989511198952)

1198941119894211989611198962

(119898ℎ)means that the probability of the 120574(119905)willnot reach to (119895

1 1198952) at times ℎ 2ℎ (119898 minus 1)ℎ but will reach

to (1198961 1198962) at time 119898ℎ Note that if ℎ le 119905 le 120575 then

120576 gt 1 minus 1198751198941119894211989411198942

(119905)

= sum

(1198961 1198962) = (1198941 1198942)

1198751198941119894211989611198962

(119905) ge 1198751198941119894211989511198952

(119905)

ge

119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ) 1198751198951119895211989511198952

(119905 minus 119898ℎ)

ge (1 minus 120576)

119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ)

(29)

which indicates119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ) le120576

1 minus 120576 (30)

Then making use of

1198751198941119894211989411198942

(119898ℎ) = 119875(11989511198952)

1198941119894211989411198942

(119898ℎ)

+

119898minus1

sum

119897=1

119875(11989511198952)

1198941119894211989511198952

(119897ℎ) 1198751198951119895211989411198942

((119898 minus 119897) ℎ)

(31)

we obtain

119875(11989511198952)

1198941119894211989411198942

(119898ℎ) ge 1198751198941119894211989411198942

(119898ℎ) minus

119898minus1

sum

119897=1

119875(11989511198952)

1198941119894211989511198952

(119897ℎ)

ge 1 minus 120576 minus120576

1 minus 120576

(32)

Consequently

1198751198941119894211989511198952

(119905)

gt

119899

sum

119898=1

119875(11989511198952)

1198941119894211989411198942

((119898 minus 1) ℎ) 1198751198941119894211989511198952

(ℎ) 1198751198951119895211989511198952

(119905 minus 119898ℎ)

ge 119899 (1 minus 120576 minus120576

1 minus 120576)1198751198941119894211989511198952

(ℎ) (1 minus 120576)

ge 119899 (1 minus 3120576) 1198751198941119894211989511198952

(ℎ)

(33)

Dividing both sides of the above inequality by ℎ and noting119899ℎ rarr 119905 whenever ℎ rarr 0 yield

lim supℎrarr0

1198751198941119894211989511198952

(ℎ)

ℎle

1

1 minus 3120576

1198751198941119894211989511198952

(119905)

119905lt infin (34)

Then letting 119905 rarr 0 gives

lim supℎrarr0

1198751198941119894211989511198952

(ℎ)

ℎle

1

1 minus 3120576lim inf119905rarr0

1198751198941119894211989511198952

(119905)

119905 (35)

and the required assertion follows immediately by letting120576 rarr 0 This completes the proof

Set 120574(119905) = (1205741(119905) 1205742(119905)) then it is easy to see that

almost every sample path of 120574(119905) is a right continuous stepfunction Now letting P(119905) = (119875

1198941119894211989511198952

(119905))11987311198732times11987311198732

Q =

(1199021198941119894211989511198952

)11987311198732times11987311198732

= P1015840(0) Then by Chapman-Kolmogorovequation

P (119905 + ℎ) = P (119905)P (ℎ) = P (ℎ)P (119905) (36)

we have

P (119905 + ℎ) minus P (119905)

ℎ= P (119905) [

P (ℎ) minus Iℎ

] = [P (ℎ) minus I

ℎ]P (119905)

(37)

Letting ℎ rarr 0 and taking limits give

P1015840(119905) = P (119905)Q

P1015840(119905) = QP (119905)

(38)

Note that

P (0) = I (39)

Then by solving the ordinary differential equations (38) and(39) we obtain the following lemma

Lemma 5 For P(119905) and Q one has

P (119905) = exp Q119905 (40)

We are now in the position to give the sufficient condi-tions for the existence and uniqueness of the solution of (8)For this end let us first give the definition of the solution

Journal of Applied Mathematics 5

Definition 6 An R119899-valued stochastic process 119909(119905)1199050le119905le119879

iscalled a solution of (8) if it has the following properties

(i) 119909(119905)1199050le119905le119879

is continuous andF119905-adapted

(ii) 119891(119909(119905) 119905 1205741(119905))1199050le119905le119879

isinL1([1199050 119879]R119899)while 119892(119909(119905)

119905 1205742(119905))1199050le119905le119879

isin L2([1199050 119879]R119899times119898)

(iii) for any 119905 isin [1199050 119879] equation

119909 (119905) = 119909 (1199050) + int

119905

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(41)

holds with probability 1

A solution 119909(119905)1199050le119905le119879

is said to be unique if any othersolution (119905)

1199050le119905le119879

is indistinguishable from 119909(119905)1199050le119905le119879

Now we can give our main results in this section

Theorem 7 Assume that there exist two positive constants 119870

and 119870 such that(Lipschitz condition) for all 119909 119910 isin R119899 119905 isin [119905

0 119879] and

(1198941 1198942) isin S

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le 119870

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(42)

(Linear growth condition) for all (119909 119905 (1198941 1198942)) isin R119899 times

[1199050 119879] times S

1003816100381610038161003816119891 (119909 119905 1198941)10038161003816100381610038162or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (43)

Then there exists a unique solution 119909(119905) to (8) and moreoverthe solution obeys

119864( sup1199050le119905le119879

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(44)

Proof Recall that almost every sample path of 120574(119905) =

(1205741(119905) 1205742(119905)) is a right continuous step function with a finite

number of jumps on [1199050 119879] Thus there exists a sequence of

stopping times 120591119896119896ge0

such that

(i) for almost every 120596 isin Ω there is a finite 119896120596for 1199050

=

1205910lt 1205911lt sdot sdot sdot lt 120591

119896= 119879 and 120591

119896= 119879 if 119896 gt 119896

120596

(ii) both 1205741(sdot) and 120574

2(sdot) in 120574(sdot) are constants on interval

[[120591119896 120591119896+1

[[ namely

1205741 (119905) = 120574

1(120591119896) on 120591

119896le 119905 le 120591

119896+1forall119896 ge 0

1205742(119905) = 120574

2(120591119896) on 120591

119896le 119905 le 120591

119896+1forall119896 ge 0

(45)

First of all let us consider (8) on 119905 isin [[1205910 1205911[[ then (8)

becomes119889119909 (119905) = 119891 (119909 (119905) 119905 120574

1(1199050)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(1199050)) 119889119861 (119905)

(46)

with initial conditions 119909(1199050) = 119909

0 120574(1199050) = (120574

1(1199050) 1205741(1199050))

Then by the theory of SDEs we obtain that (46) has a uniquesolution which obeys 119909(120591

1) isin 1198712

F1205911

(ΩR119899) We next consider(8) on 119905 isin [[120591

1 1205912[[ which becomes

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741 (1205911)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(1205911)) 119889119861 (119905)

(47)

Again by the theory of SDEs (47) has a unique solutionwhichobeys 119909(120591

1) isin 119871

2

F1205912

(ΩR119899) Repeating this procedure weconclude that (8) has a unique solution 119909(119905) on [119905

0 119879]

Now let us prove (44) For every 119896 ge 1 define thestopping time

120591119896= 119879 and inf 119905 isin [119905

0 119879] |119909 (119905)| ge 119896 (48)

It is obvious that 120591119896uarr 119879 as Set 119909

119896(119905) = 119909(119905and120591

119896) for 119905 isin [119905

0 119879]

Then 119909119896(119905) obeys the equation

119909119896(119905) = 119909

0+ int

119905

1199050

119891 (119909119896(119904) 119904 120574

1(119904)) 119868[[1199050120591119896]](119904) 119889119904

+ int

119905

1199050

119892 (119909119896(119904) 119904 120574

2(119904)) 119868[[1199050120591119896]](119904) 119889119861 (119904)

(49)

Making use of the elementary inequality |119886+119887+119888|2le 3(|119886|

2+

|119887|2+ |119888|2) the Holder inequality and (43) we can see that

1003816100381610038161003816119909119896 (119905)10038161003816100381610038162= 3

1003816100381610038161003816119909010038161003816100381610038162+ 3119870 (119879 minus 119905

0) int

119905

1199050

(1 +1003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

+ 3

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199050

119892 (119909119896(119904) 119904 120574

2(119904)) 119868[[1199050120591119896]](119904) 119889119861 (119904)

100381610038161003816100381610038161003816100381610038161003816

2

(50)

Thus applying the Doob martingale inequality and (43) wecan further show that

119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

= 311986410038161003816100381610038161199090

10038161003816100381610038162+ 3119870 (119879 minus 119905

0) int

119905

1199050

(1 + 1198641003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

+ 12119864int

119905

1199050

1003816100381610038161003816119892 (119909119896(119904) 119904 120574

2(119904))

10038161003816100381610038162119868[[1199050120591119896]](119904) 119889119904

le 311986410038161003816100381610038161199090

10038161003816100381610038162+ 3119870 (119879 minus 119905

0+ 4)int

119905

1199050

(1 + 1198641003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

(51)

That is to say

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162) le 1 + 3119864

1003816100381610038161003816119909010038161003816100381610038162+ 3119870 (119879 minus 119905

0+ 4)

times int

119905

1199050

[1 + 119864 sup1199050le119903le119904

1003816100381610038161003816119909119896 (119903)10038161003816100381610038162]119889119904

(52)

6 Journal of Applied Mathematics

Using the Gronwall inequality leads to

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(53)

Consequently

1 + 119864( sup1199050le119905le120591119896

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(54)

Then the required inequality (44) follows immediately byletting 119896 rarr infin

Condition (42) indicates that the coefficients 119891(119909 119905 1198941)

and 119892(119909 119905 1198942) do not change faster than a linear function of

119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894

1) and 119892(119909 119905 119894

2) in 119909 for all 119905 isin [119905

0 119879] Then functions

that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution

Definition 8 Let120590infinbe a stopping time such that 119905

0le 120590infin

le 119879

as An R119899-valuedF119905-adapted continuous stochastic process

119909(119905)1199050le119905lt120590infin

is called a local solution of (8) if 119909(1199050) = 1199090and

moreover there is a nondecreasing sequence 120590119896119896ge1

such that1199050le 120590119896 uarr 120590infin

as and

119909 (119905) = 119909 (1199050) + int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(55)

holds for any 119905 isin [1199050 119879) and 119896 ge 1 with probability one If

furthermore

lim sup119905rarr+120590

infin

|119909 (119905)| = infin whenever 120590infin

lt 119879 (56)

then it is called a maximal local solution and 120590infinis called the

explosion time Amaximal local solution 119909(119905) 1199050le 119905 lt 120590

infin

is said to be unique if any other maximal local solution 119909(119905)

1199050le 119905 lt 120590

infin is indistinguishable from it namely 120590

infin= 120590infin

and 119909(119905) = 119909(119905) for 1199050le 119905 lt 120590

infinwith probability one

Definition 9 (local Lipschitz condition) For every integer119896 ge 1 there exists a positive constant ℎ

119896such that for all

119905 isin [1199050 119879] 119894 = (119894

1 1198942) isin S and those 119909 119910 isin R119899 with

|119909| or |119910| le 119896

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le ℎ119896

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(57)

The following theorem shows the existence of uniquemaximal local solution under the local Lipschitz conditionwithout the linear growth condition

Theorem 10 Under condition (57) there exists a uniquemaximal local solution of (8)

Proof Define functions

119891119896(119909 119905 119894

1) =

119891 (119909 119905 1198941) if |119909| le 119896

119891(119896119909

|119909| 119905 1198941) if |119909| gt 119896

119892119896(119909 119905 119894

2) =

119892 (119909 119905 1198942) if |119909| le 119896

119892 (119896119909

|119909| 119905 1198942) if |119909| gt 119896

(58)

Then 119891119896and 119892

119896satisfy the Lipschitz condition and the linear

growth condition Thus by Theorem 7 there is a uniquesolution 119909

119896(119905) of the equation

119889119909119896(119905) = 119891

119896(119909119896(119905) 119905 120574

1(119905)) 119889119905

+ 119892119896(119909119896(119905) 119905 120574

2(119905)) 119889119861 (119905)

(59)

with the initial conditions 119909119896(1199050) = 119909

0and 120574(119905

0) = (120574

1(1199050)

1205742(1199050)) Define the stopping times

120590119896= 119879 and inf 119905 isin [119905

0 119879]

1003816100381610038161003816119909119896 (119905)1003816100381610038161003816 ge 119896 (60)

Clearly if 1199050le 119905 le 120590

119896

119909119896(119905) = 119909

119896+1(119905) (61)

which indicates that 120590119896is increasing so 120590

119896has its limit 120590

infin=

lim119896rarr+infin

120590119896 Define 119909(119905) 119905

0le 119905 lt 120590

infin by

119909 (119905) = 119909119896(119905) 119905 isin [[120590

119896minus1 120590119896[[ 119896 ge 1 (62)

where 1205900= 1199050 Applying (61) one can show that 119909(119905 and 120590

119896) =

119909119896(119905 and 120590119896) It then follows from (59) that

119909 (119905 and 120590119896) = 1199090+ int

119905and120590119896

1199050

119891119896(119909 (119904) 119904 120574

1(119904)) 119889119904

+ int

119905and120590119896

1199050

119892119896(119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

= 1199090+ int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

(63)

for any 119905 isin [1199050 119879) and 119896 ge 1 It is easy to see that if 120590

infinle 119879

then

lim sup119905rarr120590infin

|119909 (119905)|

ge lim sup119896rarr+infin

1003816100381610038161003816119909 (120590119896)1003816100381610038161003816 = lim sup119896rarr+infin

1003816100381610038161003816119909119896 (120590119896)1003816100381610038161003816 = infin

(64)

Therefore 119909(119905) 1199050le 119905 lt 120590

infin is a maximal local solution

Now we will prove the uniqueness Let 119909(119905) 1199050

le 119905 lt

120590infin

be another maximal local solution Define

120590119896= 120590infin

and inf 119905 isin [[1199050 120590infin

[[ 119909 (119905) ge 119896 (65)

Journal of Applied Mathematics 7

Then 120590119896

rarr 120590infin

as and

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590119896and 120590119896]] = 1 forall119896 ge 1 (66)

Letting 119896 rarr infin gives

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590infin

and 120590infin

[[ = 1 (67)

In order to complete the proof we need only to show that120590infin

= 120590infin

as In fact for almost every 120596 isin 120590infin

lt 120590infin

wehave

1003816100381610038161003816119909 (120590infin

120596)1003816100381610038161003816

= lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = infin(68)

which contradicts the fact that 119909(119905 120596) is continuous on 119905 isin

[1199050 120590infin

(120596)) This implies 120590infin

ge 120590infin

as In the same way onecan show 120590

infinle 120590infinasThus wemust have 120590

infin= 120590infinasThis

completes the proof

In many situations we often consider an SDE on [1199050infin)

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741(119905)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(119905)) 119889119861 (119905) 119905

0le 119905 lt infin

(69)

with initial data 119909(1199050) = 119909

0and 120574(119905

0) If the assumption of

the existence-and-uniqueness theorem holds on every finitesubinterval [119905

0 119879] of [119905

0infin) then (69) has a unique solution

119909(119905) on the entire interval [1199050infin) Such a solution is called

a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R

+times S to R which is

given by

119871119881 (119909 119905 (1198941 1198942))

= 119881119905(119909 119905 (119894

1 1198942)) + 119881

119909(119909 119905 (119894

1 1198942)) 119891 (119909 119905 119894

1)

+ 05 trace [119892119879 (119909 119905 1198942) 119881119909119909

(119909 119905 (1198941 1198942)) 119892 (119909 119905 119894

2)]

+ sum

(11989511198952)isinS

1199021198941119894211989511198952

119881 (119909 119905 (1198951 1198952))

(70)

where 119881(119909 119905 (1198941 1198942)) isin 119862

21(R119899 times R

+times SR

+) and

119881119905(119909 119905 (119894

1 1198942)) =

120597119881 (119909 119905 (1198941 1198942))

120597119905

119881119909(119909 119905 (119894

1 1198942))

= (120597119881 (119909 119905 (119894

1 1198942))

1205971199091

120597119881 (119909 119905 (119894

1 1198942))

120597119909119899

)

119881119909119909

(119909 119905 (1198941 1198942)) = (

1205972119881 (119909 119905 (119894

1 1198942))

120597119909119894120597119909119895

)

119899times119899

(71)

Theorem 11 Assume that the local Lipschitz condition (57)holds Assume also that there is a function 119881(119909 119905 (119894

1 1198942)) isin

11986221

(R119899 times R+times SR

+) and a constant 120579 gt 0 such that

lim|119909|rarrinfin

( inf(119905(1198941 1198942))isinR+timesS

119881 (119909 119905 (1198941 1198942))) = infin (72)

119871119881 (119909 119905 (1198941 1198942)) le 120579 (1 + 119881 (119909 119905 (119894

1 1198942)))

forall (119909 119905 (1198941 1198942)) isin R

119899times R+times S

(73)

Then there exists a unique global solution 119909(119905) to (69)

Proof We need only to prove the theorem for any initialcondition 119909

0isin R119899 and (120574

1(1199050) 1205742(1199050)) isin S FromTheorem 10

we know that the local Lipschitz condition guarantees theexistence of the unique maximal solution 119909(119905) on [119905

0 120590infin

)where 120590

infinis the explosion time We need only to show 120590

infin=

infin as If this is not true then we can find a pair of positiveconstants 120576 and 119879 such that

119875 120590infin

le 119879 gt 2120576 (74)

For each integer 119896 ge 1 define the stopping time

120590119896= inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (75)

Since 120590119896

rarr 120590infinalmost surely we can find a sufficiently large

integer 1198960for

119875 120590119896le 119879 gt 120576 forall119896 ge 119896

0 (76)

Fix any 119896 ge 1198960 then for any 119905

0le 119905 le 119879 by virtue of the

generalized Ito formula (see eg [1])

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

= 119881 (1199090 1199050 1205740) + 119864int

119905and120590119896

1199050

119871119881 (119909 (119904) 119904 120574 (119904)) 119889119904

le 119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(77)

Making use of the Gronwall inequality gives

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(78)

Therefore

119864 (119868120590119896le119879

119881 (119909 (120590119896) 120590119896 120574 (120590119896)))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(79)

At the same time set

119892119896= inf 119881 (119909 119905 (119894

1 1198942)) |119909| ge 119896 119905 isin [119905

0 119879] (119894

1 1198942) isin S

(80)

8 Journal of Applied Mathematics

Then (72) means 119892119896

rarr infin It follows from (76) and (79) that

[119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

ge 119892119896119875 120590119896le 119879 ge 120576119892

119896

(81)

Letting 119896 rarr infin yields a contradiction that is to say120590infin

= infinThe proof is complete

3 119871119875-Estimates

In the previous section we have investigated the existenceand uniqueness of the solution to (8) In this section as abovelet 119909(119905) 119905

0le 119905 le 119879 be the unique solution of (8) with initial

conditions 119909(1199050) = 1199090and 120574(119905

0) and we will estimate the 119901th

moment of the solution

Theorem 12 Assume that there is a function 119881(119909 119905 (1198941 1198942)) isin

11986221

(R119899 times R+times SR

+) and positive constants 119901 120578 120579 such that

for all (119909 119905 (1198941 1198942)) isin R119899 times R

+times S

120578|119909|119901le 119881 (119909 119905 (119894

1 1198942)) (82)

119871119881 (119909 119905 (1198941 1198942)) le 120579119881 (119909 119905 (119894

1 1198942)) (83)

Assume also the initial condition 119909(1199050) = 119909

0and 120574(119905

0) obeys

119864119881(1199090 1199050 120574(1199050)) lt infin then one has

119864|119909 (119905)|119901

le119864119881 (119909

0 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

120578 forall119905 isin [119905

0 119879]

(84)

Proof For each integer 119896 ge 1 define the stopping time

120590119896= 119879 and inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (85)

Thus 120590119896

rarr 119879 as Using the generalized Itorsquos formula and(83) we obtain that for 119905 isin [119905

0 119879]

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050))

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(86)

Then the Gronwall inequality indicates

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

(87)

for all 119905 isin [1199050 119879] By virtue of condition (83) we obtain the

required assertion (84)

Corollary 13 Assume 119901 ge 2 and 1199090isin 119871119901

F1199050

(ΩR119899) Assumealso that there exists a constant 120579 gt 0 such that for all(119909 119905 (119894

1 1198942)) isin R119899 times R

+times S

119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 120579 (1 + |119909|

2) (88)

Then one has

119864|119909 (119905)|119901

le 2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119905 minus 119905

0) forall119905 isin [119905

0 119879]

(89)

Proof Define 119881(119909 119905 (1198941 1198942)) = (1 + |119909|

2)1199012 Making use of

(88) yields

119871119881 (119909 119905 (1198941 1198942))

= 119901(1 + |119909|2)(119901minus2)2

119909119879119891 (119909 119905 119894

1)

+119901

2(1 + |119909|

2)(119901minus2)21003816100381610038161003816119892 (119909 119905 119894

2)10038161003816100381610038162

+119901 (119901 minus 2)

2(1 + |119909|

2)(119901minus4)210038161003816100381610038161003816

119909119879119892 (119909 119905 119894

2)10038161003816100381610038161003816

2

le 119901(1 + |119909|2)(119901minus2)2

[119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162]

le 119901120579(1 + |119909|2)1199012

(90)

Then byTheorem 12 we get

119864(1 + |119909 (119905)|2)1199012

le 119864(1 +10038161003816100381610038161199090

10038161003816100381610038162)1199012

exp 119901120579 (119905 minus 1199050)

(91)

and the required assertion (89) followsIt is useful to point out that if the linear growth condition

(43) is satisfied then (88) is fulfilled with 120579 = radic119870 + 119870(119901 minus

1)2 Now we will show the other important properties of thesolution

Theorem 14 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume alsothat the linear growth condition (43) holds Then one has

119864|119909 (119905) minus 119909 (119904)|119901le 119862(119905 minus 119904)

1199012 (92)

where

119862 = 2119901minus2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119879 minus 119905

0)

times ([2 (119879 minus 1199050)]1199012

+ [119901 (119901 minus 1)]1199012

)

(93)

and 120579 = radic119870+119870(119901minus1)2 Particularly the 119901th moment of thesolution is continuous on [119905

0 119879]

Journal of Applied Mathematics 9

Proof Applying the elementary inequality |119886 + 119887|119901

le

2119901minus1

(|119886|119901+ |119887|119901) the Holder inequality and the linear growth

condition we can derive that

119864|119909 (119905) minus 119909 (119904)|119901

le 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119891 (119909 (119906) 119906 1205741(119906)) 119889119906

10038161003816100381610038161003816100381610038161003816

119901

+ 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119892 (119909 (119906) 119906 1205742(119906)) 119889119861 (119906)

10038161003816100381610038161003816100381610038161003816

119901

le [2 (119905 minus 119904)]119901minus1

119864int

119905

119904

1003816100381610038161003816119891 (119909 (119906) 119906 1205741(119906))

1003816100381610038161003816119901119889119906

+ 05[2119901 (119901 minus 1)]1199012

(119905 minus 119904)(119901minus2)2

119864

times int

119905

119904

1003816100381610038161003816119892 (119909 (119906) 119906 1205742 (119906))1003816100381610038161003816119901119889119906

le 1198621(119905 minus 119904)

(119901minus2)2int

119905

119904

119864(1 + |119909 (119906)|2)1199012

119889119906

(94)

where1198621= 2(119901minus2)2

1198701199012

([2(119879minus 1199050)]1199012

+ [119901(119901minus1)]1199012

) Using(91) yields

119864|119909 (119905) minus 119909 (119904)|119901

le 1198621(119905 minus 119904)

(119901minus2)2

times int

119905

119904

2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119906 minus 119905

0) 119889119906

le 11986212(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901)

times exp 119901120579 (119879 minus 1199050) (119905 minus 119904)

(1199012)

(95)

which is the desired inequality

Theorem 15 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume thatthere is a119870 gt 0 such that for all (119909 119905 (119894

1 1198942)) isin R119899times[119905

0 119879]timesS

119909119879 1003816100381610038161003816119891 (119909 119905 119894

1)1003816100381610038161003816 or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (96)

Then

119864( sup1199050le119905le119879

|119909 (119905)|119901)

le (1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 2119901 (10119901 + 1)119870 (119879 minus 119905

0)

(97)

Proof Making use of the generalized Itorsquos formula andcondition (96) we have

|119909 (119905)|119901

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905)

+ int

119905

1199050

119901|119909 (119904)|119901minus2

[119909119879(119904) 119891 (119909 (119904) 119904 120574

1(119904))

+119901 minus 1

2

1003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162] 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 05119901 (119901 + 1)119870

times int

119905

1199050

|119909 (119904)|119901minus2

(1 + |119909 (119904)|2) 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 119901 (119901 + 1)119870int

119905

1199050

(1 + |119909 (119904)|119901) 119889119904

(98)

where

119872(119905) = int

119905

1199050

119901|119909 (119904)|119901minus2

119909119879(119904) 119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904) (99)

Therefore for any 1199051isin [1199050 119879]

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 119864

100381610038161003816100381611990901003816100381610038161003816119901+ 119864( sup

1199050le119905le1199051

|119872 (119905)|)

+ 119901 (119901 + 1)119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(100)

At the same time applying the well-known Burkholder-Davis-Gundy inequality (see eg [2]) gives

119864( sup1199050le119905le1199051

|119872 (119905)|)

le 3119864(int

1199051

1199050

1199012|119909 (119904)|

2119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162119889119904)

05

le 3119864( sup1199050le119905le1199051

|119909 (119905)|119901

timesint

1199051

1199050

1199012|119909 (119904)|

119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742(119904))

10038161003816100381610038162119889119904)

05

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 45119864

times int

1199051

1199050

1199012119870|119909 (119904)|

119901minus2(1 + |119909 (119904)|

2) 119889119904

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 9119901

2119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(101)

10 Journal of Applied Mathematics

Substituting the above inequality into (100) gives

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 2119864

100381610038161003816100381611990901003816100381610038161003816119901

+ 2119901 (10119901 + 1)119870int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(102)

Thus

1 + 119864( sup1199050le119905le1199051

|119909 (119905)|119901)

le 1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901+ 2119901 (10119901 + 1)119870

times int

1199051

1199050

(1 + sup1199050le119905le119904

|119909 (119905)|119901)119889119904

(103)

Then the required assertion follows from the Gronwallinequality

Up to now we have discussed the 119871119901-estimates for the

solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies

119864|119909 (119905)|119901le [119864|119909 (119905)|

2]05119901

(104)

4 Example

Consider the following Black-Scholes model

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (105)

where 1205741(119905) is a right-continuous homogenous Markovian

chain taking values in finite state spaces S1= 1 2 and 120574

2(119905)

is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S

2= 1 2 3 120583(119894) = 119894 119894 = 1 2

](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution

5 Conclusions and Further Research

This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis

(H1)1015840 both 1205741(119905) and 120574

2(119905) are right-continuous homoge-

nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a

homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper

can be established similarly It is easy to see that if 1205741(119905) equiv

1205742(119905) and 120574

1(119905) is a right-continuous homogenous Markovian

chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574

2(119905) (8) will reduce to the classical SDEs with

single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions

Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress

Acknowledgments

The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)

References

[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989

[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006

[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004

[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002

[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009

[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007

[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009

[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

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Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Lemma 4 Under Assumption (H2) for (1198941 1198942) (1198951 1198952) isin

S (1198941 1198942) = (1198951 1198952)

1199021198941119894211989511198952

= 1198751015840

1198941119894211989511198952

(0) = lim119905rarr0

1198751198941119894211989511198952

(119905)

119905(26)

exists and is finite

Proof By (H2) we note that for all 0 lt 120576 lt 13 exist0 lt 120575 lt 1such that

1198751198941119894211989411198942

(119905) gt 1 minus 120576 1198751198951119895211989511198952

(119905) gt 1 minus 120576

1198751198951119895211989411198942

(119905) lt 120576

(27)

provided 0 lt 119905 le 120575For forall0 le ℎ lt 119905 set 119899 = ⟨119905ℎ⟩ where ⟨119886⟩ = max

119899le119886119899 isin

Z Let

119875(11989511198952)

1198941119894211989611198962

(ℎ) = 1198751198941119894211989611198962

(ℎ)

119875(11989511198952)

1198941119894211989611198962

(119898ℎ)

= sum

(11990311199032) = (1198951 1198952)

119875(11989511198952)

1198941119894211990311199032

((119898 minus 1) ℎ) 1198751199031119903211989611198962

(ℎ)

(28)

where 119875(11989511198952)

1198941119894211989611198962

(119898ℎ)means that the probability of the 120574(119905)willnot reach to (119895

1 1198952) at times ℎ 2ℎ (119898 minus 1)ℎ but will reach

to (1198961 1198962) at time 119898ℎ Note that if ℎ le 119905 le 120575 then

120576 gt 1 minus 1198751198941119894211989411198942

(119905)

= sum

(1198961 1198962) = (1198941 1198942)

1198751198941119894211989611198962

(119905) ge 1198751198941119894211989511198952

(119905)

ge

119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ) 1198751198951119895211989511198952

(119905 minus 119898ℎ)

ge (1 minus 120576)

119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ)

(29)

which indicates119899

sum

119898=1

119875(11989511198952)

1198941119894211989511198952

(119898ℎ) le120576

1 minus 120576 (30)

Then making use of

1198751198941119894211989411198942

(119898ℎ) = 119875(11989511198952)

1198941119894211989411198942

(119898ℎ)

+

119898minus1

sum

119897=1

119875(11989511198952)

1198941119894211989511198952

(119897ℎ) 1198751198951119895211989411198942

((119898 minus 119897) ℎ)

(31)

we obtain

119875(11989511198952)

1198941119894211989411198942

(119898ℎ) ge 1198751198941119894211989411198942

(119898ℎ) minus

119898minus1

sum

119897=1

119875(11989511198952)

1198941119894211989511198952

(119897ℎ)

ge 1 minus 120576 minus120576

1 minus 120576

(32)

Consequently

1198751198941119894211989511198952

(119905)

gt

119899

sum

119898=1

119875(11989511198952)

1198941119894211989411198942

((119898 minus 1) ℎ) 1198751198941119894211989511198952

(ℎ) 1198751198951119895211989511198952

(119905 minus 119898ℎ)

ge 119899 (1 minus 120576 minus120576

1 minus 120576)1198751198941119894211989511198952

(ℎ) (1 minus 120576)

ge 119899 (1 minus 3120576) 1198751198941119894211989511198952

(ℎ)

(33)

Dividing both sides of the above inequality by ℎ and noting119899ℎ rarr 119905 whenever ℎ rarr 0 yield

lim supℎrarr0

1198751198941119894211989511198952

(ℎ)

ℎle

1

1 minus 3120576

1198751198941119894211989511198952

(119905)

119905lt infin (34)

Then letting 119905 rarr 0 gives

lim supℎrarr0

1198751198941119894211989511198952

(ℎ)

ℎle

1

1 minus 3120576lim inf119905rarr0

1198751198941119894211989511198952

(119905)

119905 (35)

and the required assertion follows immediately by letting120576 rarr 0 This completes the proof

Set 120574(119905) = (1205741(119905) 1205742(119905)) then it is easy to see that

almost every sample path of 120574(119905) is a right continuous stepfunction Now letting P(119905) = (119875

1198941119894211989511198952

(119905))11987311198732times11987311198732

Q =

(1199021198941119894211989511198952

)11987311198732times11987311198732

= P1015840(0) Then by Chapman-Kolmogorovequation

P (119905 + ℎ) = P (119905)P (ℎ) = P (ℎ)P (119905) (36)

we have

P (119905 + ℎ) minus P (119905)

ℎ= P (119905) [

P (ℎ) minus Iℎ

] = [P (ℎ) minus I

ℎ]P (119905)

(37)

Letting ℎ rarr 0 and taking limits give

P1015840(119905) = P (119905)Q

P1015840(119905) = QP (119905)

(38)

Note that

P (0) = I (39)

Then by solving the ordinary differential equations (38) and(39) we obtain the following lemma

Lemma 5 For P(119905) and Q one has

P (119905) = exp Q119905 (40)

We are now in the position to give the sufficient condi-tions for the existence and uniqueness of the solution of (8)For this end let us first give the definition of the solution

Journal of Applied Mathematics 5

Definition 6 An R119899-valued stochastic process 119909(119905)1199050le119905le119879

iscalled a solution of (8) if it has the following properties

(i) 119909(119905)1199050le119905le119879

is continuous andF119905-adapted

(ii) 119891(119909(119905) 119905 1205741(119905))1199050le119905le119879

isinL1([1199050 119879]R119899)while 119892(119909(119905)

119905 1205742(119905))1199050le119905le119879

isin L2([1199050 119879]R119899times119898)

(iii) for any 119905 isin [1199050 119879] equation

119909 (119905) = 119909 (1199050) + int

119905

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(41)

holds with probability 1

A solution 119909(119905)1199050le119905le119879

is said to be unique if any othersolution (119905)

1199050le119905le119879

is indistinguishable from 119909(119905)1199050le119905le119879

Now we can give our main results in this section

Theorem 7 Assume that there exist two positive constants 119870

and 119870 such that(Lipschitz condition) for all 119909 119910 isin R119899 119905 isin [119905

0 119879] and

(1198941 1198942) isin S

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le 119870

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(42)

(Linear growth condition) for all (119909 119905 (1198941 1198942)) isin R119899 times

[1199050 119879] times S

1003816100381610038161003816119891 (119909 119905 1198941)10038161003816100381610038162or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (43)

Then there exists a unique solution 119909(119905) to (8) and moreoverthe solution obeys

119864( sup1199050le119905le119879

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(44)

Proof Recall that almost every sample path of 120574(119905) =

(1205741(119905) 1205742(119905)) is a right continuous step function with a finite

number of jumps on [1199050 119879] Thus there exists a sequence of

stopping times 120591119896119896ge0

such that

(i) for almost every 120596 isin Ω there is a finite 119896120596for 1199050

=

1205910lt 1205911lt sdot sdot sdot lt 120591

119896= 119879 and 120591

119896= 119879 if 119896 gt 119896

120596

(ii) both 1205741(sdot) and 120574

2(sdot) in 120574(sdot) are constants on interval

[[120591119896 120591119896+1

[[ namely

1205741 (119905) = 120574

1(120591119896) on 120591

119896le 119905 le 120591

119896+1forall119896 ge 0

1205742(119905) = 120574

2(120591119896) on 120591

119896le 119905 le 120591

119896+1forall119896 ge 0

(45)

First of all let us consider (8) on 119905 isin [[1205910 1205911[[ then (8)

becomes119889119909 (119905) = 119891 (119909 (119905) 119905 120574

1(1199050)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(1199050)) 119889119861 (119905)

(46)

with initial conditions 119909(1199050) = 119909

0 120574(1199050) = (120574

1(1199050) 1205741(1199050))

Then by the theory of SDEs we obtain that (46) has a uniquesolution which obeys 119909(120591

1) isin 1198712

F1205911

(ΩR119899) We next consider(8) on 119905 isin [[120591

1 1205912[[ which becomes

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741 (1205911)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(1205911)) 119889119861 (119905)

(47)

Again by the theory of SDEs (47) has a unique solutionwhichobeys 119909(120591

1) isin 119871

2

F1205912

(ΩR119899) Repeating this procedure weconclude that (8) has a unique solution 119909(119905) on [119905

0 119879]

Now let us prove (44) For every 119896 ge 1 define thestopping time

120591119896= 119879 and inf 119905 isin [119905

0 119879] |119909 (119905)| ge 119896 (48)

It is obvious that 120591119896uarr 119879 as Set 119909

119896(119905) = 119909(119905and120591

119896) for 119905 isin [119905

0 119879]

Then 119909119896(119905) obeys the equation

119909119896(119905) = 119909

0+ int

119905

1199050

119891 (119909119896(119904) 119904 120574

1(119904)) 119868[[1199050120591119896]](119904) 119889119904

+ int

119905

1199050

119892 (119909119896(119904) 119904 120574

2(119904)) 119868[[1199050120591119896]](119904) 119889119861 (119904)

(49)

Making use of the elementary inequality |119886+119887+119888|2le 3(|119886|

2+

|119887|2+ |119888|2) the Holder inequality and (43) we can see that

1003816100381610038161003816119909119896 (119905)10038161003816100381610038162= 3

1003816100381610038161003816119909010038161003816100381610038162+ 3119870 (119879 minus 119905

0) int

119905

1199050

(1 +1003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

+ 3

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199050

119892 (119909119896(119904) 119904 120574

2(119904)) 119868[[1199050120591119896]](119904) 119889119861 (119904)

100381610038161003816100381610038161003816100381610038161003816

2

(50)

Thus applying the Doob martingale inequality and (43) wecan further show that

119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

= 311986410038161003816100381610038161199090

10038161003816100381610038162+ 3119870 (119879 minus 119905

0) int

119905

1199050

(1 + 1198641003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

+ 12119864int

119905

1199050

1003816100381610038161003816119892 (119909119896(119904) 119904 120574

2(119904))

10038161003816100381610038162119868[[1199050120591119896]](119904) 119889119904

le 311986410038161003816100381610038161199090

10038161003816100381610038162+ 3119870 (119879 minus 119905

0+ 4)int

119905

1199050

(1 + 1198641003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

(51)

That is to say

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162) le 1 + 3119864

1003816100381610038161003816119909010038161003816100381610038162+ 3119870 (119879 minus 119905

0+ 4)

times int

119905

1199050

[1 + 119864 sup1199050le119903le119904

1003816100381610038161003816119909119896 (119903)10038161003816100381610038162]119889119904

(52)

6 Journal of Applied Mathematics

Using the Gronwall inequality leads to

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(53)

Consequently

1 + 119864( sup1199050le119905le120591119896

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(54)

Then the required inequality (44) follows immediately byletting 119896 rarr infin

Condition (42) indicates that the coefficients 119891(119909 119905 1198941)

and 119892(119909 119905 1198942) do not change faster than a linear function of

119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894

1) and 119892(119909 119905 119894

2) in 119909 for all 119905 isin [119905

0 119879] Then functions

that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution

Definition 8 Let120590infinbe a stopping time such that 119905

0le 120590infin

le 119879

as An R119899-valuedF119905-adapted continuous stochastic process

119909(119905)1199050le119905lt120590infin

is called a local solution of (8) if 119909(1199050) = 1199090and

moreover there is a nondecreasing sequence 120590119896119896ge1

such that1199050le 120590119896 uarr 120590infin

as and

119909 (119905) = 119909 (1199050) + int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(55)

holds for any 119905 isin [1199050 119879) and 119896 ge 1 with probability one If

furthermore

lim sup119905rarr+120590

infin

|119909 (119905)| = infin whenever 120590infin

lt 119879 (56)

then it is called a maximal local solution and 120590infinis called the

explosion time Amaximal local solution 119909(119905) 1199050le 119905 lt 120590

infin

is said to be unique if any other maximal local solution 119909(119905)

1199050le 119905 lt 120590

infin is indistinguishable from it namely 120590

infin= 120590infin

and 119909(119905) = 119909(119905) for 1199050le 119905 lt 120590

infinwith probability one

Definition 9 (local Lipschitz condition) For every integer119896 ge 1 there exists a positive constant ℎ

119896such that for all

119905 isin [1199050 119879] 119894 = (119894

1 1198942) isin S and those 119909 119910 isin R119899 with

|119909| or |119910| le 119896

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le ℎ119896

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(57)

The following theorem shows the existence of uniquemaximal local solution under the local Lipschitz conditionwithout the linear growth condition

Theorem 10 Under condition (57) there exists a uniquemaximal local solution of (8)

Proof Define functions

119891119896(119909 119905 119894

1) =

119891 (119909 119905 1198941) if |119909| le 119896

119891(119896119909

|119909| 119905 1198941) if |119909| gt 119896

119892119896(119909 119905 119894

2) =

119892 (119909 119905 1198942) if |119909| le 119896

119892 (119896119909

|119909| 119905 1198942) if |119909| gt 119896

(58)

Then 119891119896and 119892

119896satisfy the Lipschitz condition and the linear

growth condition Thus by Theorem 7 there is a uniquesolution 119909

119896(119905) of the equation

119889119909119896(119905) = 119891

119896(119909119896(119905) 119905 120574

1(119905)) 119889119905

+ 119892119896(119909119896(119905) 119905 120574

2(119905)) 119889119861 (119905)

(59)

with the initial conditions 119909119896(1199050) = 119909

0and 120574(119905

0) = (120574

1(1199050)

1205742(1199050)) Define the stopping times

120590119896= 119879 and inf 119905 isin [119905

0 119879]

1003816100381610038161003816119909119896 (119905)1003816100381610038161003816 ge 119896 (60)

Clearly if 1199050le 119905 le 120590

119896

119909119896(119905) = 119909

119896+1(119905) (61)

which indicates that 120590119896is increasing so 120590

119896has its limit 120590

infin=

lim119896rarr+infin

120590119896 Define 119909(119905) 119905

0le 119905 lt 120590

infin by

119909 (119905) = 119909119896(119905) 119905 isin [[120590

119896minus1 120590119896[[ 119896 ge 1 (62)

where 1205900= 1199050 Applying (61) one can show that 119909(119905 and 120590

119896) =

119909119896(119905 and 120590119896) It then follows from (59) that

119909 (119905 and 120590119896) = 1199090+ int

119905and120590119896

1199050

119891119896(119909 (119904) 119904 120574

1(119904)) 119889119904

+ int

119905and120590119896

1199050

119892119896(119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

= 1199090+ int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

(63)

for any 119905 isin [1199050 119879) and 119896 ge 1 It is easy to see that if 120590

infinle 119879

then

lim sup119905rarr120590infin

|119909 (119905)|

ge lim sup119896rarr+infin

1003816100381610038161003816119909 (120590119896)1003816100381610038161003816 = lim sup119896rarr+infin

1003816100381610038161003816119909119896 (120590119896)1003816100381610038161003816 = infin

(64)

Therefore 119909(119905) 1199050le 119905 lt 120590

infin is a maximal local solution

Now we will prove the uniqueness Let 119909(119905) 1199050

le 119905 lt

120590infin

be another maximal local solution Define

120590119896= 120590infin

and inf 119905 isin [[1199050 120590infin

[[ 119909 (119905) ge 119896 (65)

Journal of Applied Mathematics 7

Then 120590119896

rarr 120590infin

as and

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590119896and 120590119896]] = 1 forall119896 ge 1 (66)

Letting 119896 rarr infin gives

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590infin

and 120590infin

[[ = 1 (67)

In order to complete the proof we need only to show that120590infin

= 120590infin

as In fact for almost every 120596 isin 120590infin

lt 120590infin

wehave

1003816100381610038161003816119909 (120590infin

120596)1003816100381610038161003816

= lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = infin(68)

which contradicts the fact that 119909(119905 120596) is continuous on 119905 isin

[1199050 120590infin

(120596)) This implies 120590infin

ge 120590infin

as In the same way onecan show 120590

infinle 120590infinasThus wemust have 120590

infin= 120590infinasThis

completes the proof

In many situations we often consider an SDE on [1199050infin)

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741(119905)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(119905)) 119889119861 (119905) 119905

0le 119905 lt infin

(69)

with initial data 119909(1199050) = 119909

0and 120574(119905

0) If the assumption of

the existence-and-uniqueness theorem holds on every finitesubinterval [119905

0 119879] of [119905

0infin) then (69) has a unique solution

119909(119905) on the entire interval [1199050infin) Such a solution is called

a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R

+times S to R which is

given by

119871119881 (119909 119905 (1198941 1198942))

= 119881119905(119909 119905 (119894

1 1198942)) + 119881

119909(119909 119905 (119894

1 1198942)) 119891 (119909 119905 119894

1)

+ 05 trace [119892119879 (119909 119905 1198942) 119881119909119909

(119909 119905 (1198941 1198942)) 119892 (119909 119905 119894

2)]

+ sum

(11989511198952)isinS

1199021198941119894211989511198952

119881 (119909 119905 (1198951 1198952))

(70)

where 119881(119909 119905 (1198941 1198942)) isin 119862

21(R119899 times R

+times SR

+) and

119881119905(119909 119905 (119894

1 1198942)) =

120597119881 (119909 119905 (1198941 1198942))

120597119905

119881119909(119909 119905 (119894

1 1198942))

= (120597119881 (119909 119905 (119894

1 1198942))

1205971199091

120597119881 (119909 119905 (119894

1 1198942))

120597119909119899

)

119881119909119909

(119909 119905 (1198941 1198942)) = (

1205972119881 (119909 119905 (119894

1 1198942))

120597119909119894120597119909119895

)

119899times119899

(71)

Theorem 11 Assume that the local Lipschitz condition (57)holds Assume also that there is a function 119881(119909 119905 (119894

1 1198942)) isin

11986221

(R119899 times R+times SR

+) and a constant 120579 gt 0 such that

lim|119909|rarrinfin

( inf(119905(1198941 1198942))isinR+timesS

119881 (119909 119905 (1198941 1198942))) = infin (72)

119871119881 (119909 119905 (1198941 1198942)) le 120579 (1 + 119881 (119909 119905 (119894

1 1198942)))

forall (119909 119905 (1198941 1198942)) isin R

119899times R+times S

(73)

Then there exists a unique global solution 119909(119905) to (69)

Proof We need only to prove the theorem for any initialcondition 119909

0isin R119899 and (120574

1(1199050) 1205742(1199050)) isin S FromTheorem 10

we know that the local Lipschitz condition guarantees theexistence of the unique maximal solution 119909(119905) on [119905

0 120590infin

)where 120590

infinis the explosion time We need only to show 120590

infin=

infin as If this is not true then we can find a pair of positiveconstants 120576 and 119879 such that

119875 120590infin

le 119879 gt 2120576 (74)

For each integer 119896 ge 1 define the stopping time

120590119896= inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (75)

Since 120590119896

rarr 120590infinalmost surely we can find a sufficiently large

integer 1198960for

119875 120590119896le 119879 gt 120576 forall119896 ge 119896

0 (76)

Fix any 119896 ge 1198960 then for any 119905

0le 119905 le 119879 by virtue of the

generalized Ito formula (see eg [1])

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

= 119881 (1199090 1199050 1205740) + 119864int

119905and120590119896

1199050

119871119881 (119909 (119904) 119904 120574 (119904)) 119889119904

le 119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(77)

Making use of the Gronwall inequality gives

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(78)

Therefore

119864 (119868120590119896le119879

119881 (119909 (120590119896) 120590119896 120574 (120590119896)))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(79)

At the same time set

119892119896= inf 119881 (119909 119905 (119894

1 1198942)) |119909| ge 119896 119905 isin [119905

0 119879] (119894

1 1198942) isin S

(80)

8 Journal of Applied Mathematics

Then (72) means 119892119896

rarr infin It follows from (76) and (79) that

[119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

ge 119892119896119875 120590119896le 119879 ge 120576119892

119896

(81)

Letting 119896 rarr infin yields a contradiction that is to say120590infin

= infinThe proof is complete

3 119871119875-Estimates

In the previous section we have investigated the existenceand uniqueness of the solution to (8) In this section as abovelet 119909(119905) 119905

0le 119905 le 119879 be the unique solution of (8) with initial

conditions 119909(1199050) = 1199090and 120574(119905

0) and we will estimate the 119901th

moment of the solution

Theorem 12 Assume that there is a function 119881(119909 119905 (1198941 1198942)) isin

11986221

(R119899 times R+times SR

+) and positive constants 119901 120578 120579 such that

for all (119909 119905 (1198941 1198942)) isin R119899 times R

+times S

120578|119909|119901le 119881 (119909 119905 (119894

1 1198942)) (82)

119871119881 (119909 119905 (1198941 1198942)) le 120579119881 (119909 119905 (119894

1 1198942)) (83)

Assume also the initial condition 119909(1199050) = 119909

0and 120574(119905

0) obeys

119864119881(1199090 1199050 120574(1199050)) lt infin then one has

119864|119909 (119905)|119901

le119864119881 (119909

0 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

120578 forall119905 isin [119905

0 119879]

(84)

Proof For each integer 119896 ge 1 define the stopping time

120590119896= 119879 and inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (85)

Thus 120590119896

rarr 119879 as Using the generalized Itorsquos formula and(83) we obtain that for 119905 isin [119905

0 119879]

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050))

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(86)

Then the Gronwall inequality indicates

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

(87)

for all 119905 isin [1199050 119879] By virtue of condition (83) we obtain the

required assertion (84)

Corollary 13 Assume 119901 ge 2 and 1199090isin 119871119901

F1199050

(ΩR119899) Assumealso that there exists a constant 120579 gt 0 such that for all(119909 119905 (119894

1 1198942)) isin R119899 times R

+times S

119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 120579 (1 + |119909|

2) (88)

Then one has

119864|119909 (119905)|119901

le 2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119905 minus 119905

0) forall119905 isin [119905

0 119879]

(89)

Proof Define 119881(119909 119905 (1198941 1198942)) = (1 + |119909|

2)1199012 Making use of

(88) yields

119871119881 (119909 119905 (1198941 1198942))

= 119901(1 + |119909|2)(119901minus2)2

119909119879119891 (119909 119905 119894

1)

+119901

2(1 + |119909|

2)(119901minus2)21003816100381610038161003816119892 (119909 119905 119894

2)10038161003816100381610038162

+119901 (119901 minus 2)

2(1 + |119909|

2)(119901minus4)210038161003816100381610038161003816

119909119879119892 (119909 119905 119894

2)10038161003816100381610038161003816

2

le 119901(1 + |119909|2)(119901minus2)2

[119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162]

le 119901120579(1 + |119909|2)1199012

(90)

Then byTheorem 12 we get

119864(1 + |119909 (119905)|2)1199012

le 119864(1 +10038161003816100381610038161199090

10038161003816100381610038162)1199012

exp 119901120579 (119905 minus 1199050)

(91)

and the required assertion (89) followsIt is useful to point out that if the linear growth condition

(43) is satisfied then (88) is fulfilled with 120579 = radic119870 + 119870(119901 minus

1)2 Now we will show the other important properties of thesolution

Theorem 14 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume alsothat the linear growth condition (43) holds Then one has

119864|119909 (119905) minus 119909 (119904)|119901le 119862(119905 minus 119904)

1199012 (92)

where

119862 = 2119901minus2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119879 minus 119905

0)

times ([2 (119879 minus 1199050)]1199012

+ [119901 (119901 minus 1)]1199012

)

(93)

and 120579 = radic119870+119870(119901minus1)2 Particularly the 119901th moment of thesolution is continuous on [119905

0 119879]

Journal of Applied Mathematics 9

Proof Applying the elementary inequality |119886 + 119887|119901

le

2119901minus1

(|119886|119901+ |119887|119901) the Holder inequality and the linear growth

condition we can derive that

119864|119909 (119905) minus 119909 (119904)|119901

le 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119891 (119909 (119906) 119906 1205741(119906)) 119889119906

10038161003816100381610038161003816100381610038161003816

119901

+ 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119892 (119909 (119906) 119906 1205742(119906)) 119889119861 (119906)

10038161003816100381610038161003816100381610038161003816

119901

le [2 (119905 minus 119904)]119901minus1

119864int

119905

119904

1003816100381610038161003816119891 (119909 (119906) 119906 1205741(119906))

1003816100381610038161003816119901119889119906

+ 05[2119901 (119901 minus 1)]1199012

(119905 minus 119904)(119901minus2)2

119864

times int

119905

119904

1003816100381610038161003816119892 (119909 (119906) 119906 1205742 (119906))1003816100381610038161003816119901119889119906

le 1198621(119905 minus 119904)

(119901minus2)2int

119905

119904

119864(1 + |119909 (119906)|2)1199012

119889119906

(94)

where1198621= 2(119901minus2)2

1198701199012

([2(119879minus 1199050)]1199012

+ [119901(119901minus1)]1199012

) Using(91) yields

119864|119909 (119905) minus 119909 (119904)|119901

le 1198621(119905 minus 119904)

(119901minus2)2

times int

119905

119904

2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119906 minus 119905

0) 119889119906

le 11986212(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901)

times exp 119901120579 (119879 minus 1199050) (119905 minus 119904)

(1199012)

(95)

which is the desired inequality

Theorem 15 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume thatthere is a119870 gt 0 such that for all (119909 119905 (119894

1 1198942)) isin R119899times[119905

0 119879]timesS

119909119879 1003816100381610038161003816119891 (119909 119905 119894

1)1003816100381610038161003816 or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (96)

Then

119864( sup1199050le119905le119879

|119909 (119905)|119901)

le (1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 2119901 (10119901 + 1)119870 (119879 minus 119905

0)

(97)

Proof Making use of the generalized Itorsquos formula andcondition (96) we have

|119909 (119905)|119901

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905)

+ int

119905

1199050

119901|119909 (119904)|119901minus2

[119909119879(119904) 119891 (119909 (119904) 119904 120574

1(119904))

+119901 minus 1

2

1003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162] 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 05119901 (119901 + 1)119870

times int

119905

1199050

|119909 (119904)|119901minus2

(1 + |119909 (119904)|2) 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 119901 (119901 + 1)119870int

119905

1199050

(1 + |119909 (119904)|119901) 119889119904

(98)

where

119872(119905) = int

119905

1199050

119901|119909 (119904)|119901minus2

119909119879(119904) 119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904) (99)

Therefore for any 1199051isin [1199050 119879]

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 119864

100381610038161003816100381611990901003816100381610038161003816119901+ 119864( sup

1199050le119905le1199051

|119872 (119905)|)

+ 119901 (119901 + 1)119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(100)

At the same time applying the well-known Burkholder-Davis-Gundy inequality (see eg [2]) gives

119864( sup1199050le119905le1199051

|119872 (119905)|)

le 3119864(int

1199051

1199050

1199012|119909 (119904)|

2119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162119889119904)

05

le 3119864( sup1199050le119905le1199051

|119909 (119905)|119901

timesint

1199051

1199050

1199012|119909 (119904)|

119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742(119904))

10038161003816100381610038162119889119904)

05

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 45119864

times int

1199051

1199050

1199012119870|119909 (119904)|

119901minus2(1 + |119909 (119904)|

2) 119889119904

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 9119901

2119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(101)

10 Journal of Applied Mathematics

Substituting the above inequality into (100) gives

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 2119864

100381610038161003816100381611990901003816100381610038161003816119901

+ 2119901 (10119901 + 1)119870int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(102)

Thus

1 + 119864( sup1199050le119905le1199051

|119909 (119905)|119901)

le 1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901+ 2119901 (10119901 + 1)119870

times int

1199051

1199050

(1 + sup1199050le119905le119904

|119909 (119905)|119901)119889119904

(103)

Then the required assertion follows from the Gronwallinequality

Up to now we have discussed the 119871119901-estimates for the

solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies

119864|119909 (119905)|119901le [119864|119909 (119905)|

2]05119901

(104)

4 Example

Consider the following Black-Scholes model

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (105)

where 1205741(119905) is a right-continuous homogenous Markovian

chain taking values in finite state spaces S1= 1 2 and 120574

2(119905)

is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S

2= 1 2 3 120583(119894) = 119894 119894 = 1 2

](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution

5 Conclusions and Further Research

This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis

(H1)1015840 both 1205741(119905) and 120574

2(119905) are right-continuous homoge-

nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a

homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper

can be established similarly It is easy to see that if 1205741(119905) equiv

1205742(119905) and 120574

1(119905) is a right-continuous homogenous Markovian

chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574

2(119905) (8) will reduce to the classical SDEs with

single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions

Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress

Acknowledgments

The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)

References

[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989

[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006

[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004

[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002

[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009

[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007

[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009

[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Definition 6 An R119899-valued stochastic process 119909(119905)1199050le119905le119879

iscalled a solution of (8) if it has the following properties

(i) 119909(119905)1199050le119905le119879

is continuous andF119905-adapted

(ii) 119891(119909(119905) 119905 1205741(119905))1199050le119905le119879

isinL1([1199050 119879]R119899)while 119892(119909(119905)

119905 1205742(119905))1199050le119905le119879

isin L2([1199050 119879]R119899times119898)

(iii) for any 119905 isin [1199050 119879] equation

119909 (119905) = 119909 (1199050) + int

119905

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(41)

holds with probability 1

A solution 119909(119905)1199050le119905le119879

is said to be unique if any othersolution (119905)

1199050le119905le119879

is indistinguishable from 119909(119905)1199050le119905le119879

Now we can give our main results in this section

Theorem 7 Assume that there exist two positive constants 119870

and 119870 such that(Lipschitz condition) for all 119909 119910 isin R119899 119905 isin [119905

0 119879] and

(1198941 1198942) isin S

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le 119870

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(42)

(Linear growth condition) for all (119909 119905 (1198941 1198942)) isin R119899 times

[1199050 119879] times S

1003816100381610038161003816119891 (119909 119905 1198941)10038161003816100381610038162or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (43)

Then there exists a unique solution 119909(119905) to (8) and moreoverthe solution obeys

119864( sup1199050le119905le119879

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(44)

Proof Recall that almost every sample path of 120574(119905) =

(1205741(119905) 1205742(119905)) is a right continuous step function with a finite

number of jumps on [1199050 119879] Thus there exists a sequence of

stopping times 120591119896119896ge0

such that

(i) for almost every 120596 isin Ω there is a finite 119896120596for 1199050

=

1205910lt 1205911lt sdot sdot sdot lt 120591

119896= 119879 and 120591

119896= 119879 if 119896 gt 119896

120596

(ii) both 1205741(sdot) and 120574

2(sdot) in 120574(sdot) are constants on interval

[[120591119896 120591119896+1

[[ namely

1205741 (119905) = 120574

1(120591119896) on 120591

119896le 119905 le 120591

119896+1forall119896 ge 0

1205742(119905) = 120574

2(120591119896) on 120591

119896le 119905 le 120591

119896+1forall119896 ge 0

(45)

First of all let us consider (8) on 119905 isin [[1205910 1205911[[ then (8)

becomes119889119909 (119905) = 119891 (119909 (119905) 119905 120574

1(1199050)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(1199050)) 119889119861 (119905)

(46)

with initial conditions 119909(1199050) = 119909

0 120574(1199050) = (120574

1(1199050) 1205741(1199050))

Then by the theory of SDEs we obtain that (46) has a uniquesolution which obeys 119909(120591

1) isin 1198712

F1205911

(ΩR119899) We next consider(8) on 119905 isin [[120591

1 1205912[[ which becomes

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741 (1205911)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(1205911)) 119889119861 (119905)

(47)

Again by the theory of SDEs (47) has a unique solutionwhichobeys 119909(120591

1) isin 119871

2

F1205912

(ΩR119899) Repeating this procedure weconclude that (8) has a unique solution 119909(119905) on [119905

0 119879]

Now let us prove (44) For every 119896 ge 1 define thestopping time

120591119896= 119879 and inf 119905 isin [119905

0 119879] |119909 (119905)| ge 119896 (48)

It is obvious that 120591119896uarr 119879 as Set 119909

119896(119905) = 119909(119905and120591

119896) for 119905 isin [119905

0 119879]

Then 119909119896(119905) obeys the equation

119909119896(119905) = 119909

0+ int

119905

1199050

119891 (119909119896(119904) 119904 120574

1(119904)) 119868[[1199050120591119896]](119904) 119889119904

+ int

119905

1199050

119892 (119909119896(119904) 119904 120574

2(119904)) 119868[[1199050120591119896]](119904) 119889119861 (119904)

(49)

Making use of the elementary inequality |119886+119887+119888|2le 3(|119886|

2+

|119887|2+ |119888|2) the Holder inequality and (43) we can see that

1003816100381610038161003816119909119896 (119905)10038161003816100381610038162= 3

1003816100381610038161003816119909010038161003816100381610038162+ 3119870 (119879 minus 119905

0) int

119905

1199050

(1 +1003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

+ 3

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199050

119892 (119909119896(119904) 119904 120574

2(119904)) 119868[[1199050120591119896]](119904) 119889119861 (119904)

100381610038161003816100381610038161003816100381610038161003816

2

(50)

Thus applying the Doob martingale inequality and (43) wecan further show that

119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

= 311986410038161003816100381610038161199090

10038161003816100381610038162+ 3119870 (119879 minus 119905

0) int

119905

1199050

(1 + 1198641003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

+ 12119864int

119905

1199050

1003816100381610038161003816119892 (119909119896(119904) 119904 120574

2(119904))

10038161003816100381610038162119868[[1199050120591119896]](119904) 119889119904

le 311986410038161003816100381610038161199090

10038161003816100381610038162+ 3119870 (119879 minus 119905

0+ 4)int

119905

1199050

(1 + 1198641003816100381610038161003816119909119896 (119904)

10038161003816100381610038162) 119889119904

(51)

That is to say

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162) le 1 + 3119864

1003816100381610038161003816119909010038161003816100381610038162+ 3119870 (119879 minus 119905

0+ 4)

times int

119905

1199050

[1 + 119864 sup1199050le119903le119904

1003816100381610038161003816119909119896 (119903)10038161003816100381610038162]119889119904

(52)

6 Journal of Applied Mathematics

Using the Gronwall inequality leads to

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(53)

Consequently

1 + 119864( sup1199050le119905le120591119896

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(54)

Then the required inequality (44) follows immediately byletting 119896 rarr infin

Condition (42) indicates that the coefficients 119891(119909 119905 1198941)

and 119892(119909 119905 1198942) do not change faster than a linear function of

119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894

1) and 119892(119909 119905 119894

2) in 119909 for all 119905 isin [119905

0 119879] Then functions

that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution

Definition 8 Let120590infinbe a stopping time such that 119905

0le 120590infin

le 119879

as An R119899-valuedF119905-adapted continuous stochastic process

119909(119905)1199050le119905lt120590infin

is called a local solution of (8) if 119909(1199050) = 1199090and

moreover there is a nondecreasing sequence 120590119896119896ge1

such that1199050le 120590119896 uarr 120590infin

as and

119909 (119905) = 119909 (1199050) + int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(55)

holds for any 119905 isin [1199050 119879) and 119896 ge 1 with probability one If

furthermore

lim sup119905rarr+120590

infin

|119909 (119905)| = infin whenever 120590infin

lt 119879 (56)

then it is called a maximal local solution and 120590infinis called the

explosion time Amaximal local solution 119909(119905) 1199050le 119905 lt 120590

infin

is said to be unique if any other maximal local solution 119909(119905)

1199050le 119905 lt 120590

infin is indistinguishable from it namely 120590

infin= 120590infin

and 119909(119905) = 119909(119905) for 1199050le 119905 lt 120590

infinwith probability one

Definition 9 (local Lipschitz condition) For every integer119896 ge 1 there exists a positive constant ℎ

119896such that for all

119905 isin [1199050 119879] 119894 = (119894

1 1198942) isin S and those 119909 119910 isin R119899 with

|119909| or |119910| le 119896

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le ℎ119896

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(57)

The following theorem shows the existence of uniquemaximal local solution under the local Lipschitz conditionwithout the linear growth condition

Theorem 10 Under condition (57) there exists a uniquemaximal local solution of (8)

Proof Define functions

119891119896(119909 119905 119894

1) =

119891 (119909 119905 1198941) if |119909| le 119896

119891(119896119909

|119909| 119905 1198941) if |119909| gt 119896

119892119896(119909 119905 119894

2) =

119892 (119909 119905 1198942) if |119909| le 119896

119892 (119896119909

|119909| 119905 1198942) if |119909| gt 119896

(58)

Then 119891119896and 119892

119896satisfy the Lipschitz condition and the linear

growth condition Thus by Theorem 7 there is a uniquesolution 119909

119896(119905) of the equation

119889119909119896(119905) = 119891

119896(119909119896(119905) 119905 120574

1(119905)) 119889119905

+ 119892119896(119909119896(119905) 119905 120574

2(119905)) 119889119861 (119905)

(59)

with the initial conditions 119909119896(1199050) = 119909

0and 120574(119905

0) = (120574

1(1199050)

1205742(1199050)) Define the stopping times

120590119896= 119879 and inf 119905 isin [119905

0 119879]

1003816100381610038161003816119909119896 (119905)1003816100381610038161003816 ge 119896 (60)

Clearly if 1199050le 119905 le 120590

119896

119909119896(119905) = 119909

119896+1(119905) (61)

which indicates that 120590119896is increasing so 120590

119896has its limit 120590

infin=

lim119896rarr+infin

120590119896 Define 119909(119905) 119905

0le 119905 lt 120590

infin by

119909 (119905) = 119909119896(119905) 119905 isin [[120590

119896minus1 120590119896[[ 119896 ge 1 (62)

where 1205900= 1199050 Applying (61) one can show that 119909(119905 and 120590

119896) =

119909119896(119905 and 120590119896) It then follows from (59) that

119909 (119905 and 120590119896) = 1199090+ int

119905and120590119896

1199050

119891119896(119909 (119904) 119904 120574

1(119904)) 119889119904

+ int

119905and120590119896

1199050

119892119896(119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

= 1199090+ int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

(63)

for any 119905 isin [1199050 119879) and 119896 ge 1 It is easy to see that if 120590

infinle 119879

then

lim sup119905rarr120590infin

|119909 (119905)|

ge lim sup119896rarr+infin

1003816100381610038161003816119909 (120590119896)1003816100381610038161003816 = lim sup119896rarr+infin

1003816100381610038161003816119909119896 (120590119896)1003816100381610038161003816 = infin

(64)

Therefore 119909(119905) 1199050le 119905 lt 120590

infin is a maximal local solution

Now we will prove the uniqueness Let 119909(119905) 1199050

le 119905 lt

120590infin

be another maximal local solution Define

120590119896= 120590infin

and inf 119905 isin [[1199050 120590infin

[[ 119909 (119905) ge 119896 (65)

Journal of Applied Mathematics 7

Then 120590119896

rarr 120590infin

as and

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590119896and 120590119896]] = 1 forall119896 ge 1 (66)

Letting 119896 rarr infin gives

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590infin

and 120590infin

[[ = 1 (67)

In order to complete the proof we need only to show that120590infin

= 120590infin

as In fact for almost every 120596 isin 120590infin

lt 120590infin

wehave

1003816100381610038161003816119909 (120590infin

120596)1003816100381610038161003816

= lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = infin(68)

which contradicts the fact that 119909(119905 120596) is continuous on 119905 isin

[1199050 120590infin

(120596)) This implies 120590infin

ge 120590infin

as In the same way onecan show 120590

infinle 120590infinasThus wemust have 120590

infin= 120590infinasThis

completes the proof

In many situations we often consider an SDE on [1199050infin)

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741(119905)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(119905)) 119889119861 (119905) 119905

0le 119905 lt infin

(69)

with initial data 119909(1199050) = 119909

0and 120574(119905

0) If the assumption of

the existence-and-uniqueness theorem holds on every finitesubinterval [119905

0 119879] of [119905

0infin) then (69) has a unique solution

119909(119905) on the entire interval [1199050infin) Such a solution is called

a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R

+times S to R which is

given by

119871119881 (119909 119905 (1198941 1198942))

= 119881119905(119909 119905 (119894

1 1198942)) + 119881

119909(119909 119905 (119894

1 1198942)) 119891 (119909 119905 119894

1)

+ 05 trace [119892119879 (119909 119905 1198942) 119881119909119909

(119909 119905 (1198941 1198942)) 119892 (119909 119905 119894

2)]

+ sum

(11989511198952)isinS

1199021198941119894211989511198952

119881 (119909 119905 (1198951 1198952))

(70)

where 119881(119909 119905 (1198941 1198942)) isin 119862

21(R119899 times R

+times SR

+) and

119881119905(119909 119905 (119894

1 1198942)) =

120597119881 (119909 119905 (1198941 1198942))

120597119905

119881119909(119909 119905 (119894

1 1198942))

= (120597119881 (119909 119905 (119894

1 1198942))

1205971199091

120597119881 (119909 119905 (119894

1 1198942))

120597119909119899

)

119881119909119909

(119909 119905 (1198941 1198942)) = (

1205972119881 (119909 119905 (119894

1 1198942))

120597119909119894120597119909119895

)

119899times119899

(71)

Theorem 11 Assume that the local Lipschitz condition (57)holds Assume also that there is a function 119881(119909 119905 (119894

1 1198942)) isin

11986221

(R119899 times R+times SR

+) and a constant 120579 gt 0 such that

lim|119909|rarrinfin

( inf(119905(1198941 1198942))isinR+timesS

119881 (119909 119905 (1198941 1198942))) = infin (72)

119871119881 (119909 119905 (1198941 1198942)) le 120579 (1 + 119881 (119909 119905 (119894

1 1198942)))

forall (119909 119905 (1198941 1198942)) isin R

119899times R+times S

(73)

Then there exists a unique global solution 119909(119905) to (69)

Proof We need only to prove the theorem for any initialcondition 119909

0isin R119899 and (120574

1(1199050) 1205742(1199050)) isin S FromTheorem 10

we know that the local Lipschitz condition guarantees theexistence of the unique maximal solution 119909(119905) on [119905

0 120590infin

)where 120590

infinis the explosion time We need only to show 120590

infin=

infin as If this is not true then we can find a pair of positiveconstants 120576 and 119879 such that

119875 120590infin

le 119879 gt 2120576 (74)

For each integer 119896 ge 1 define the stopping time

120590119896= inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (75)

Since 120590119896

rarr 120590infinalmost surely we can find a sufficiently large

integer 1198960for

119875 120590119896le 119879 gt 120576 forall119896 ge 119896

0 (76)

Fix any 119896 ge 1198960 then for any 119905

0le 119905 le 119879 by virtue of the

generalized Ito formula (see eg [1])

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

= 119881 (1199090 1199050 1205740) + 119864int

119905and120590119896

1199050

119871119881 (119909 (119904) 119904 120574 (119904)) 119889119904

le 119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(77)

Making use of the Gronwall inequality gives

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(78)

Therefore

119864 (119868120590119896le119879

119881 (119909 (120590119896) 120590119896 120574 (120590119896)))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(79)

At the same time set

119892119896= inf 119881 (119909 119905 (119894

1 1198942)) |119909| ge 119896 119905 isin [119905

0 119879] (119894

1 1198942) isin S

(80)

8 Journal of Applied Mathematics

Then (72) means 119892119896

rarr infin It follows from (76) and (79) that

[119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

ge 119892119896119875 120590119896le 119879 ge 120576119892

119896

(81)

Letting 119896 rarr infin yields a contradiction that is to say120590infin

= infinThe proof is complete

3 119871119875-Estimates

In the previous section we have investigated the existenceand uniqueness of the solution to (8) In this section as abovelet 119909(119905) 119905

0le 119905 le 119879 be the unique solution of (8) with initial

conditions 119909(1199050) = 1199090and 120574(119905

0) and we will estimate the 119901th

moment of the solution

Theorem 12 Assume that there is a function 119881(119909 119905 (1198941 1198942)) isin

11986221

(R119899 times R+times SR

+) and positive constants 119901 120578 120579 such that

for all (119909 119905 (1198941 1198942)) isin R119899 times R

+times S

120578|119909|119901le 119881 (119909 119905 (119894

1 1198942)) (82)

119871119881 (119909 119905 (1198941 1198942)) le 120579119881 (119909 119905 (119894

1 1198942)) (83)

Assume also the initial condition 119909(1199050) = 119909

0and 120574(119905

0) obeys

119864119881(1199090 1199050 120574(1199050)) lt infin then one has

119864|119909 (119905)|119901

le119864119881 (119909

0 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

120578 forall119905 isin [119905

0 119879]

(84)

Proof For each integer 119896 ge 1 define the stopping time

120590119896= 119879 and inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (85)

Thus 120590119896

rarr 119879 as Using the generalized Itorsquos formula and(83) we obtain that for 119905 isin [119905

0 119879]

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050))

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(86)

Then the Gronwall inequality indicates

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

(87)

for all 119905 isin [1199050 119879] By virtue of condition (83) we obtain the

required assertion (84)

Corollary 13 Assume 119901 ge 2 and 1199090isin 119871119901

F1199050

(ΩR119899) Assumealso that there exists a constant 120579 gt 0 such that for all(119909 119905 (119894

1 1198942)) isin R119899 times R

+times S

119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 120579 (1 + |119909|

2) (88)

Then one has

119864|119909 (119905)|119901

le 2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119905 minus 119905

0) forall119905 isin [119905

0 119879]

(89)

Proof Define 119881(119909 119905 (1198941 1198942)) = (1 + |119909|

2)1199012 Making use of

(88) yields

119871119881 (119909 119905 (1198941 1198942))

= 119901(1 + |119909|2)(119901minus2)2

119909119879119891 (119909 119905 119894

1)

+119901

2(1 + |119909|

2)(119901minus2)21003816100381610038161003816119892 (119909 119905 119894

2)10038161003816100381610038162

+119901 (119901 minus 2)

2(1 + |119909|

2)(119901minus4)210038161003816100381610038161003816

119909119879119892 (119909 119905 119894

2)10038161003816100381610038161003816

2

le 119901(1 + |119909|2)(119901minus2)2

[119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162]

le 119901120579(1 + |119909|2)1199012

(90)

Then byTheorem 12 we get

119864(1 + |119909 (119905)|2)1199012

le 119864(1 +10038161003816100381610038161199090

10038161003816100381610038162)1199012

exp 119901120579 (119905 minus 1199050)

(91)

and the required assertion (89) followsIt is useful to point out that if the linear growth condition

(43) is satisfied then (88) is fulfilled with 120579 = radic119870 + 119870(119901 minus

1)2 Now we will show the other important properties of thesolution

Theorem 14 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume alsothat the linear growth condition (43) holds Then one has

119864|119909 (119905) minus 119909 (119904)|119901le 119862(119905 minus 119904)

1199012 (92)

where

119862 = 2119901minus2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119879 minus 119905

0)

times ([2 (119879 minus 1199050)]1199012

+ [119901 (119901 minus 1)]1199012

)

(93)

and 120579 = radic119870+119870(119901minus1)2 Particularly the 119901th moment of thesolution is continuous on [119905

0 119879]

Journal of Applied Mathematics 9

Proof Applying the elementary inequality |119886 + 119887|119901

le

2119901minus1

(|119886|119901+ |119887|119901) the Holder inequality and the linear growth

condition we can derive that

119864|119909 (119905) minus 119909 (119904)|119901

le 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119891 (119909 (119906) 119906 1205741(119906)) 119889119906

10038161003816100381610038161003816100381610038161003816

119901

+ 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119892 (119909 (119906) 119906 1205742(119906)) 119889119861 (119906)

10038161003816100381610038161003816100381610038161003816

119901

le [2 (119905 minus 119904)]119901minus1

119864int

119905

119904

1003816100381610038161003816119891 (119909 (119906) 119906 1205741(119906))

1003816100381610038161003816119901119889119906

+ 05[2119901 (119901 minus 1)]1199012

(119905 minus 119904)(119901minus2)2

119864

times int

119905

119904

1003816100381610038161003816119892 (119909 (119906) 119906 1205742 (119906))1003816100381610038161003816119901119889119906

le 1198621(119905 minus 119904)

(119901minus2)2int

119905

119904

119864(1 + |119909 (119906)|2)1199012

119889119906

(94)

where1198621= 2(119901minus2)2

1198701199012

([2(119879minus 1199050)]1199012

+ [119901(119901minus1)]1199012

) Using(91) yields

119864|119909 (119905) minus 119909 (119904)|119901

le 1198621(119905 minus 119904)

(119901minus2)2

times int

119905

119904

2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119906 minus 119905

0) 119889119906

le 11986212(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901)

times exp 119901120579 (119879 minus 1199050) (119905 minus 119904)

(1199012)

(95)

which is the desired inequality

Theorem 15 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume thatthere is a119870 gt 0 such that for all (119909 119905 (119894

1 1198942)) isin R119899times[119905

0 119879]timesS

119909119879 1003816100381610038161003816119891 (119909 119905 119894

1)1003816100381610038161003816 or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (96)

Then

119864( sup1199050le119905le119879

|119909 (119905)|119901)

le (1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 2119901 (10119901 + 1)119870 (119879 minus 119905

0)

(97)

Proof Making use of the generalized Itorsquos formula andcondition (96) we have

|119909 (119905)|119901

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905)

+ int

119905

1199050

119901|119909 (119904)|119901minus2

[119909119879(119904) 119891 (119909 (119904) 119904 120574

1(119904))

+119901 minus 1

2

1003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162] 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 05119901 (119901 + 1)119870

times int

119905

1199050

|119909 (119904)|119901minus2

(1 + |119909 (119904)|2) 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 119901 (119901 + 1)119870int

119905

1199050

(1 + |119909 (119904)|119901) 119889119904

(98)

where

119872(119905) = int

119905

1199050

119901|119909 (119904)|119901minus2

119909119879(119904) 119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904) (99)

Therefore for any 1199051isin [1199050 119879]

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 119864

100381610038161003816100381611990901003816100381610038161003816119901+ 119864( sup

1199050le119905le1199051

|119872 (119905)|)

+ 119901 (119901 + 1)119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(100)

At the same time applying the well-known Burkholder-Davis-Gundy inequality (see eg [2]) gives

119864( sup1199050le119905le1199051

|119872 (119905)|)

le 3119864(int

1199051

1199050

1199012|119909 (119904)|

2119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162119889119904)

05

le 3119864( sup1199050le119905le1199051

|119909 (119905)|119901

timesint

1199051

1199050

1199012|119909 (119904)|

119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742(119904))

10038161003816100381610038162119889119904)

05

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 45119864

times int

1199051

1199050

1199012119870|119909 (119904)|

119901minus2(1 + |119909 (119904)|

2) 119889119904

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 9119901

2119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(101)

10 Journal of Applied Mathematics

Substituting the above inequality into (100) gives

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 2119864

100381610038161003816100381611990901003816100381610038161003816119901

+ 2119901 (10119901 + 1)119870int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(102)

Thus

1 + 119864( sup1199050le119905le1199051

|119909 (119905)|119901)

le 1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901+ 2119901 (10119901 + 1)119870

times int

1199051

1199050

(1 + sup1199050le119905le119904

|119909 (119905)|119901)119889119904

(103)

Then the required assertion follows from the Gronwallinequality

Up to now we have discussed the 119871119901-estimates for the

solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies

119864|119909 (119905)|119901le [119864|119909 (119905)|

2]05119901

(104)

4 Example

Consider the following Black-Scholes model

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (105)

where 1205741(119905) is a right-continuous homogenous Markovian

chain taking values in finite state spaces S1= 1 2 and 120574

2(119905)

is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S

2= 1 2 3 120583(119894) = 119894 119894 = 1 2

](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution

5 Conclusions and Further Research

This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis

(H1)1015840 both 1205741(119905) and 120574

2(119905) are right-continuous homoge-

nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a

homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper

can be established similarly It is easy to see that if 1205741(119905) equiv

1205742(119905) and 120574

1(119905) is a right-continuous homogenous Markovian

chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574

2(119905) (8) will reduce to the classical SDEs with

single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions

Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress

Acknowledgments

The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)

References

[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989

[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006

[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004

[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002

[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009

[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007

[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009

[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

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Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Using the Gronwall inequality leads to

1 + 119864( sup1199050le119904le119905

1003816100381610038161003816119909119896 (119904)10038161003816100381610038162)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(53)

Consequently

1 + 119864( sup1199050le119905le120591119896

|119909 (119905)|2)

le (1 + 311986410038161003816100381610038161199090

10038161003816100381610038162) exp 3119870 (119879 minus 119905

0) (119879 minus 119905

0+ 4)

(54)

Then the required inequality (44) follows immediately byletting 119896 rarr infin

Condition (42) indicates that the coefficients 119891(119909 119905 1198941)

and 119892(119909 119905 1198942) do not change faster than a linear function of

119909 as change in 119909 This means in particular the continuity of119891(119909 119905 119894

1) and 119892(119909 119905 119894

2) in 119909 for all 119905 isin [119905

0 119879] Then functions

that are discontinuous with respect to 119909 are excluded as thecoefficients Besides there are many functions that do notsatisfy the Lipschitz conditionThese imply that the Lipschitzcondition is too restrictive To improve this Lipschitz condi-tion let us introduce the concept of local solution

Definition 8 Let120590infinbe a stopping time such that 119905

0le 120590infin

le 119879

as An R119899-valuedF119905-adapted continuous stochastic process

119909(119905)1199050le119905lt120590infin

is called a local solution of (8) if 119909(1199050) = 1199090and

moreover there is a nondecreasing sequence 120590119896119896ge1

such that1199050le 120590119896 uarr 120590infin

as and

119909 (119905) = 119909 (1199050) + int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742(119904)) 119889119861 (119904)

(55)

holds for any 119905 isin [1199050 119879) and 119896 ge 1 with probability one If

furthermore

lim sup119905rarr+120590

infin

|119909 (119905)| = infin whenever 120590infin

lt 119879 (56)

then it is called a maximal local solution and 120590infinis called the

explosion time Amaximal local solution 119909(119905) 1199050le 119905 lt 120590

infin

is said to be unique if any other maximal local solution 119909(119905)

1199050le 119905 lt 120590

infin is indistinguishable from it namely 120590

infin= 120590infin

and 119909(119905) = 119909(119905) for 1199050le 119905 lt 120590

infinwith probability one

Definition 9 (local Lipschitz condition) For every integer119896 ge 1 there exists a positive constant ℎ

119896such that for all

119905 isin [1199050 119879] 119894 = (119894

1 1198942) isin S and those 119909 119910 isin R119899 with

|119909| or |119910| le 119896

1003816100381610038161003816119891 (119909 119905 1198941) minus 119891 (119910 119905 119894

1)10038161003816100381610038162

or1003816100381610038161003816119892 (119909 119905 119894

2) minus 119892 (119910 119905 119894

2)10038161003816100381610038162le ℎ119896

1003816100381610038161003816119909 minus 11991010038161003816100381610038162

(57)

The following theorem shows the existence of uniquemaximal local solution under the local Lipschitz conditionwithout the linear growth condition

Theorem 10 Under condition (57) there exists a uniquemaximal local solution of (8)

Proof Define functions

119891119896(119909 119905 119894

1) =

119891 (119909 119905 1198941) if |119909| le 119896

119891(119896119909

|119909| 119905 1198941) if |119909| gt 119896

119892119896(119909 119905 119894

2) =

119892 (119909 119905 1198942) if |119909| le 119896

119892 (119896119909

|119909| 119905 1198942) if |119909| gt 119896

(58)

Then 119891119896and 119892

119896satisfy the Lipschitz condition and the linear

growth condition Thus by Theorem 7 there is a uniquesolution 119909

119896(119905) of the equation

119889119909119896(119905) = 119891

119896(119909119896(119905) 119905 120574

1(119905)) 119889119905

+ 119892119896(119909119896(119905) 119905 120574

2(119905)) 119889119861 (119905)

(59)

with the initial conditions 119909119896(1199050) = 119909

0and 120574(119905

0) = (120574

1(1199050)

1205742(1199050)) Define the stopping times

120590119896= 119879 and inf 119905 isin [119905

0 119879]

1003816100381610038161003816119909119896 (119905)1003816100381610038161003816 ge 119896 (60)

Clearly if 1199050le 119905 le 120590

119896

119909119896(119905) = 119909

119896+1(119905) (61)

which indicates that 120590119896is increasing so 120590

119896has its limit 120590

infin=

lim119896rarr+infin

120590119896 Define 119909(119905) 119905

0le 119905 lt 120590

infin by

119909 (119905) = 119909119896(119905) 119905 isin [[120590

119896minus1 120590119896[[ 119896 ge 1 (62)

where 1205900= 1199050 Applying (61) one can show that 119909(119905 and 120590

119896) =

119909119896(119905 and 120590119896) It then follows from (59) that

119909 (119905 and 120590119896) = 1199090+ int

119905and120590119896

1199050

119891119896(119909 (119904) 119904 120574

1(119904)) 119889119904

+ int

119905and120590119896

1199050

119892119896(119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

= 1199090+ int

119905and120590119896

1199050

119891 (119909 (119904) 119904 1205741(119904)) 119889119904

+ int

119905and120590119896

1199050

119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904)

(63)

for any 119905 isin [1199050 119879) and 119896 ge 1 It is easy to see that if 120590

infinle 119879

then

lim sup119905rarr120590infin

|119909 (119905)|

ge lim sup119896rarr+infin

1003816100381610038161003816119909 (120590119896)1003816100381610038161003816 = lim sup119896rarr+infin

1003816100381610038161003816119909119896 (120590119896)1003816100381610038161003816 = infin

(64)

Therefore 119909(119905) 1199050le 119905 lt 120590

infin is a maximal local solution

Now we will prove the uniqueness Let 119909(119905) 1199050

le 119905 lt

120590infin

be another maximal local solution Define

120590119896= 120590infin

and inf 119905 isin [[1199050 120590infin

[[ 119909 (119905) ge 119896 (65)

Journal of Applied Mathematics 7

Then 120590119896

rarr 120590infin

as and

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590119896and 120590119896]] = 1 forall119896 ge 1 (66)

Letting 119896 rarr infin gives

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590infin

and 120590infin

[[ = 1 (67)

In order to complete the proof we need only to show that120590infin

= 120590infin

as In fact for almost every 120596 isin 120590infin

lt 120590infin

wehave

1003816100381610038161003816119909 (120590infin

120596)1003816100381610038161003816

= lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = infin(68)

which contradicts the fact that 119909(119905 120596) is continuous on 119905 isin

[1199050 120590infin

(120596)) This implies 120590infin

ge 120590infin

as In the same way onecan show 120590

infinle 120590infinasThus wemust have 120590

infin= 120590infinasThis

completes the proof

In many situations we often consider an SDE on [1199050infin)

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741(119905)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(119905)) 119889119861 (119905) 119905

0le 119905 lt infin

(69)

with initial data 119909(1199050) = 119909

0and 120574(119905

0) If the assumption of

the existence-and-uniqueness theorem holds on every finitesubinterval [119905

0 119879] of [119905

0infin) then (69) has a unique solution

119909(119905) on the entire interval [1199050infin) Such a solution is called

a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R

+times S to R which is

given by

119871119881 (119909 119905 (1198941 1198942))

= 119881119905(119909 119905 (119894

1 1198942)) + 119881

119909(119909 119905 (119894

1 1198942)) 119891 (119909 119905 119894

1)

+ 05 trace [119892119879 (119909 119905 1198942) 119881119909119909

(119909 119905 (1198941 1198942)) 119892 (119909 119905 119894

2)]

+ sum

(11989511198952)isinS

1199021198941119894211989511198952

119881 (119909 119905 (1198951 1198952))

(70)

where 119881(119909 119905 (1198941 1198942)) isin 119862

21(R119899 times R

+times SR

+) and

119881119905(119909 119905 (119894

1 1198942)) =

120597119881 (119909 119905 (1198941 1198942))

120597119905

119881119909(119909 119905 (119894

1 1198942))

= (120597119881 (119909 119905 (119894

1 1198942))

1205971199091

120597119881 (119909 119905 (119894

1 1198942))

120597119909119899

)

119881119909119909

(119909 119905 (1198941 1198942)) = (

1205972119881 (119909 119905 (119894

1 1198942))

120597119909119894120597119909119895

)

119899times119899

(71)

Theorem 11 Assume that the local Lipschitz condition (57)holds Assume also that there is a function 119881(119909 119905 (119894

1 1198942)) isin

11986221

(R119899 times R+times SR

+) and a constant 120579 gt 0 such that

lim|119909|rarrinfin

( inf(119905(1198941 1198942))isinR+timesS

119881 (119909 119905 (1198941 1198942))) = infin (72)

119871119881 (119909 119905 (1198941 1198942)) le 120579 (1 + 119881 (119909 119905 (119894

1 1198942)))

forall (119909 119905 (1198941 1198942)) isin R

119899times R+times S

(73)

Then there exists a unique global solution 119909(119905) to (69)

Proof We need only to prove the theorem for any initialcondition 119909

0isin R119899 and (120574

1(1199050) 1205742(1199050)) isin S FromTheorem 10

we know that the local Lipschitz condition guarantees theexistence of the unique maximal solution 119909(119905) on [119905

0 120590infin

)where 120590

infinis the explosion time We need only to show 120590

infin=

infin as If this is not true then we can find a pair of positiveconstants 120576 and 119879 such that

119875 120590infin

le 119879 gt 2120576 (74)

For each integer 119896 ge 1 define the stopping time

120590119896= inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (75)

Since 120590119896

rarr 120590infinalmost surely we can find a sufficiently large

integer 1198960for

119875 120590119896le 119879 gt 120576 forall119896 ge 119896

0 (76)

Fix any 119896 ge 1198960 then for any 119905

0le 119905 le 119879 by virtue of the

generalized Ito formula (see eg [1])

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

= 119881 (1199090 1199050 1205740) + 119864int

119905and120590119896

1199050

119871119881 (119909 (119904) 119904 120574 (119904)) 119889119904

le 119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(77)

Making use of the Gronwall inequality gives

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(78)

Therefore

119864 (119868120590119896le119879

119881 (119909 (120590119896) 120590119896 120574 (120590119896)))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(79)

At the same time set

119892119896= inf 119881 (119909 119905 (119894

1 1198942)) |119909| ge 119896 119905 isin [119905

0 119879] (119894

1 1198942) isin S

(80)

8 Journal of Applied Mathematics

Then (72) means 119892119896

rarr infin It follows from (76) and (79) that

[119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

ge 119892119896119875 120590119896le 119879 ge 120576119892

119896

(81)

Letting 119896 rarr infin yields a contradiction that is to say120590infin

= infinThe proof is complete

3 119871119875-Estimates

In the previous section we have investigated the existenceand uniqueness of the solution to (8) In this section as abovelet 119909(119905) 119905

0le 119905 le 119879 be the unique solution of (8) with initial

conditions 119909(1199050) = 1199090and 120574(119905

0) and we will estimate the 119901th

moment of the solution

Theorem 12 Assume that there is a function 119881(119909 119905 (1198941 1198942)) isin

11986221

(R119899 times R+times SR

+) and positive constants 119901 120578 120579 such that

for all (119909 119905 (1198941 1198942)) isin R119899 times R

+times S

120578|119909|119901le 119881 (119909 119905 (119894

1 1198942)) (82)

119871119881 (119909 119905 (1198941 1198942)) le 120579119881 (119909 119905 (119894

1 1198942)) (83)

Assume also the initial condition 119909(1199050) = 119909

0and 120574(119905

0) obeys

119864119881(1199090 1199050 120574(1199050)) lt infin then one has

119864|119909 (119905)|119901

le119864119881 (119909

0 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

120578 forall119905 isin [119905

0 119879]

(84)

Proof For each integer 119896 ge 1 define the stopping time

120590119896= 119879 and inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (85)

Thus 120590119896

rarr 119879 as Using the generalized Itorsquos formula and(83) we obtain that for 119905 isin [119905

0 119879]

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050))

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(86)

Then the Gronwall inequality indicates

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

(87)

for all 119905 isin [1199050 119879] By virtue of condition (83) we obtain the

required assertion (84)

Corollary 13 Assume 119901 ge 2 and 1199090isin 119871119901

F1199050

(ΩR119899) Assumealso that there exists a constant 120579 gt 0 such that for all(119909 119905 (119894

1 1198942)) isin R119899 times R

+times S

119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 120579 (1 + |119909|

2) (88)

Then one has

119864|119909 (119905)|119901

le 2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119905 minus 119905

0) forall119905 isin [119905

0 119879]

(89)

Proof Define 119881(119909 119905 (1198941 1198942)) = (1 + |119909|

2)1199012 Making use of

(88) yields

119871119881 (119909 119905 (1198941 1198942))

= 119901(1 + |119909|2)(119901minus2)2

119909119879119891 (119909 119905 119894

1)

+119901

2(1 + |119909|

2)(119901minus2)21003816100381610038161003816119892 (119909 119905 119894

2)10038161003816100381610038162

+119901 (119901 minus 2)

2(1 + |119909|

2)(119901minus4)210038161003816100381610038161003816

119909119879119892 (119909 119905 119894

2)10038161003816100381610038161003816

2

le 119901(1 + |119909|2)(119901minus2)2

[119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162]

le 119901120579(1 + |119909|2)1199012

(90)

Then byTheorem 12 we get

119864(1 + |119909 (119905)|2)1199012

le 119864(1 +10038161003816100381610038161199090

10038161003816100381610038162)1199012

exp 119901120579 (119905 minus 1199050)

(91)

and the required assertion (89) followsIt is useful to point out that if the linear growth condition

(43) is satisfied then (88) is fulfilled with 120579 = radic119870 + 119870(119901 minus

1)2 Now we will show the other important properties of thesolution

Theorem 14 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume alsothat the linear growth condition (43) holds Then one has

119864|119909 (119905) minus 119909 (119904)|119901le 119862(119905 minus 119904)

1199012 (92)

where

119862 = 2119901minus2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119879 minus 119905

0)

times ([2 (119879 minus 1199050)]1199012

+ [119901 (119901 minus 1)]1199012

)

(93)

and 120579 = radic119870+119870(119901minus1)2 Particularly the 119901th moment of thesolution is continuous on [119905

0 119879]

Journal of Applied Mathematics 9

Proof Applying the elementary inequality |119886 + 119887|119901

le

2119901minus1

(|119886|119901+ |119887|119901) the Holder inequality and the linear growth

condition we can derive that

119864|119909 (119905) minus 119909 (119904)|119901

le 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119891 (119909 (119906) 119906 1205741(119906)) 119889119906

10038161003816100381610038161003816100381610038161003816

119901

+ 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119892 (119909 (119906) 119906 1205742(119906)) 119889119861 (119906)

10038161003816100381610038161003816100381610038161003816

119901

le [2 (119905 minus 119904)]119901minus1

119864int

119905

119904

1003816100381610038161003816119891 (119909 (119906) 119906 1205741(119906))

1003816100381610038161003816119901119889119906

+ 05[2119901 (119901 minus 1)]1199012

(119905 minus 119904)(119901minus2)2

119864

times int

119905

119904

1003816100381610038161003816119892 (119909 (119906) 119906 1205742 (119906))1003816100381610038161003816119901119889119906

le 1198621(119905 minus 119904)

(119901minus2)2int

119905

119904

119864(1 + |119909 (119906)|2)1199012

119889119906

(94)

where1198621= 2(119901minus2)2

1198701199012

([2(119879minus 1199050)]1199012

+ [119901(119901minus1)]1199012

) Using(91) yields

119864|119909 (119905) minus 119909 (119904)|119901

le 1198621(119905 minus 119904)

(119901minus2)2

times int

119905

119904

2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119906 minus 119905

0) 119889119906

le 11986212(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901)

times exp 119901120579 (119879 minus 1199050) (119905 minus 119904)

(1199012)

(95)

which is the desired inequality

Theorem 15 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume thatthere is a119870 gt 0 such that for all (119909 119905 (119894

1 1198942)) isin R119899times[119905

0 119879]timesS

119909119879 1003816100381610038161003816119891 (119909 119905 119894

1)1003816100381610038161003816 or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (96)

Then

119864( sup1199050le119905le119879

|119909 (119905)|119901)

le (1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 2119901 (10119901 + 1)119870 (119879 minus 119905

0)

(97)

Proof Making use of the generalized Itorsquos formula andcondition (96) we have

|119909 (119905)|119901

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905)

+ int

119905

1199050

119901|119909 (119904)|119901minus2

[119909119879(119904) 119891 (119909 (119904) 119904 120574

1(119904))

+119901 minus 1

2

1003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162] 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 05119901 (119901 + 1)119870

times int

119905

1199050

|119909 (119904)|119901minus2

(1 + |119909 (119904)|2) 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 119901 (119901 + 1)119870int

119905

1199050

(1 + |119909 (119904)|119901) 119889119904

(98)

where

119872(119905) = int

119905

1199050

119901|119909 (119904)|119901minus2

119909119879(119904) 119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904) (99)

Therefore for any 1199051isin [1199050 119879]

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 119864

100381610038161003816100381611990901003816100381610038161003816119901+ 119864( sup

1199050le119905le1199051

|119872 (119905)|)

+ 119901 (119901 + 1)119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(100)

At the same time applying the well-known Burkholder-Davis-Gundy inequality (see eg [2]) gives

119864( sup1199050le119905le1199051

|119872 (119905)|)

le 3119864(int

1199051

1199050

1199012|119909 (119904)|

2119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162119889119904)

05

le 3119864( sup1199050le119905le1199051

|119909 (119905)|119901

timesint

1199051

1199050

1199012|119909 (119904)|

119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742(119904))

10038161003816100381610038162119889119904)

05

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 45119864

times int

1199051

1199050

1199012119870|119909 (119904)|

119901minus2(1 + |119909 (119904)|

2) 119889119904

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 9119901

2119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(101)

10 Journal of Applied Mathematics

Substituting the above inequality into (100) gives

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 2119864

100381610038161003816100381611990901003816100381610038161003816119901

+ 2119901 (10119901 + 1)119870int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(102)

Thus

1 + 119864( sup1199050le119905le1199051

|119909 (119905)|119901)

le 1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901+ 2119901 (10119901 + 1)119870

times int

1199051

1199050

(1 + sup1199050le119905le119904

|119909 (119905)|119901)119889119904

(103)

Then the required assertion follows from the Gronwallinequality

Up to now we have discussed the 119871119901-estimates for the

solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies

119864|119909 (119905)|119901le [119864|119909 (119905)|

2]05119901

(104)

4 Example

Consider the following Black-Scholes model

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (105)

where 1205741(119905) is a right-continuous homogenous Markovian

chain taking values in finite state spaces S1= 1 2 and 120574

2(119905)

is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S

2= 1 2 3 120583(119894) = 119894 119894 = 1 2

](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution

5 Conclusions and Further Research

This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis

(H1)1015840 both 1205741(119905) and 120574

2(119905) are right-continuous homoge-

nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a

homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper

can be established similarly It is easy to see that if 1205741(119905) equiv

1205742(119905) and 120574

1(119905) is a right-continuous homogenous Markovian

chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574

2(119905) (8) will reduce to the classical SDEs with

single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions

Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress

Acknowledgments

The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)

References

[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989

[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006

[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004

[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002

[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009

[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007

[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009

[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

Then 120590119896

rarr 120590infin

as and

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590119896and 120590119896]] = 1 forall119896 ge 1 (66)

Letting 119896 rarr infin gives

119875 119909 (119905) = 119909 (119905) 119905 isin [[1199050 120590infin

and 120590infin

[[ = 1 (67)

In order to complete the proof we need only to show that120590infin

= 120590infin

as In fact for almost every 120596 isin 120590infin

lt 120590infin

wehave

1003816100381610038161003816119909 (120590infin

120596)1003816100381610038161003816

= lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = lim119896rarr+infin

1003816100381610038161003816119909 (120590119896 120596)

1003816100381610038161003816 = infin(68)

which contradicts the fact that 119909(119905 120596) is continuous on 119905 isin

[1199050 120590infin

(120596)) This implies 120590infin

ge 120590infin

as In the same way onecan show 120590

infinle 120590infinasThus wemust have 120590

infin= 120590infinasThis

completes the proof

In many situations we often consider an SDE on [1199050infin)

119889119909 (119905) = 119891 (119909 (119905) 119905 1205741(119905)) 119889119905

+ 119892 (119909 (119905) 119905 1205742(119905)) 119889119861 (119905) 119905

0le 119905 lt infin

(69)

with initial data 119909(1199050) = 119909

0and 120574(119905

0) If the assumption of

the existence-and-uniqueness theorem holds on every finitesubinterval [119905

0 119879] of [119905

0infin) then (69) has a unique solution

119909(119905) on the entire interval [1199050infin) Such a solution is called

a global solution To establish a more general result aboutglobal solution we need more notations To this end weintroduce an operator 119871119881 from R119899 times R

+times S to R which is

given by

119871119881 (119909 119905 (1198941 1198942))

= 119881119905(119909 119905 (119894

1 1198942)) + 119881

119909(119909 119905 (119894

1 1198942)) 119891 (119909 119905 119894

1)

+ 05 trace [119892119879 (119909 119905 1198942) 119881119909119909

(119909 119905 (1198941 1198942)) 119892 (119909 119905 119894

2)]

+ sum

(11989511198952)isinS

1199021198941119894211989511198952

119881 (119909 119905 (1198951 1198952))

(70)

where 119881(119909 119905 (1198941 1198942)) isin 119862

21(R119899 times R

+times SR

+) and

119881119905(119909 119905 (119894

1 1198942)) =

120597119881 (119909 119905 (1198941 1198942))

120597119905

119881119909(119909 119905 (119894

1 1198942))

= (120597119881 (119909 119905 (119894

1 1198942))

1205971199091

120597119881 (119909 119905 (119894

1 1198942))

120597119909119899

)

119881119909119909

(119909 119905 (1198941 1198942)) = (

1205972119881 (119909 119905 (119894

1 1198942))

120597119909119894120597119909119895

)

119899times119899

(71)

Theorem 11 Assume that the local Lipschitz condition (57)holds Assume also that there is a function 119881(119909 119905 (119894

1 1198942)) isin

11986221

(R119899 times R+times SR

+) and a constant 120579 gt 0 such that

lim|119909|rarrinfin

( inf(119905(1198941 1198942))isinR+timesS

119881 (119909 119905 (1198941 1198942))) = infin (72)

119871119881 (119909 119905 (1198941 1198942)) le 120579 (1 + 119881 (119909 119905 (119894

1 1198942)))

forall (119909 119905 (1198941 1198942)) isin R

119899times R+times S

(73)

Then there exists a unique global solution 119909(119905) to (69)

Proof We need only to prove the theorem for any initialcondition 119909

0isin R119899 and (120574

1(1199050) 1205742(1199050)) isin S FromTheorem 10

we know that the local Lipschitz condition guarantees theexistence of the unique maximal solution 119909(119905) on [119905

0 120590infin

)where 120590

infinis the explosion time We need only to show 120590

infin=

infin as If this is not true then we can find a pair of positiveconstants 120576 and 119879 such that

119875 120590infin

le 119879 gt 2120576 (74)

For each integer 119896 ge 1 define the stopping time

120590119896= inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (75)

Since 120590119896

rarr 120590infinalmost surely we can find a sufficiently large

integer 1198960for

119875 120590119896le 119879 gt 120576 forall119896 ge 119896

0 (76)

Fix any 119896 ge 1198960 then for any 119905

0le 119905 le 119879 by virtue of the

generalized Ito formula (see eg [1])

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

= 119881 (1199090 1199050 1205740) + 119864int

119905and120590119896

1199050

119871119881 (119909 (119904) 119904 120574 (119904)) 119889119904

le 119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(77)

Making use of the Gronwall inequality gives

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(78)

Therefore

119864 (119868120590119896le119879

119881 (119909 (120590119896) 120590119896 120574 (120590119896)))

le [119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

(79)

At the same time set

119892119896= inf 119881 (119909 119905 (119894

1 1198942)) |119909| ge 119896 119905 isin [119905

0 119879] (119894

1 1198942) isin S

(80)

8 Journal of Applied Mathematics

Then (72) means 119892119896

rarr infin It follows from (76) and (79) that

[119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

ge 119892119896119875 120590119896le 119879 ge 120576119892

119896

(81)

Letting 119896 rarr infin yields a contradiction that is to say120590infin

= infinThe proof is complete

3 119871119875-Estimates

In the previous section we have investigated the existenceand uniqueness of the solution to (8) In this section as abovelet 119909(119905) 119905

0le 119905 le 119879 be the unique solution of (8) with initial

conditions 119909(1199050) = 1199090and 120574(119905

0) and we will estimate the 119901th

moment of the solution

Theorem 12 Assume that there is a function 119881(119909 119905 (1198941 1198942)) isin

11986221

(R119899 times R+times SR

+) and positive constants 119901 120578 120579 such that

for all (119909 119905 (1198941 1198942)) isin R119899 times R

+times S

120578|119909|119901le 119881 (119909 119905 (119894

1 1198942)) (82)

119871119881 (119909 119905 (1198941 1198942)) le 120579119881 (119909 119905 (119894

1 1198942)) (83)

Assume also the initial condition 119909(1199050) = 119909

0and 120574(119905

0) obeys

119864119881(1199090 1199050 120574(1199050)) lt infin then one has

119864|119909 (119905)|119901

le119864119881 (119909

0 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

120578 forall119905 isin [119905

0 119879]

(84)

Proof For each integer 119896 ge 1 define the stopping time

120590119896= 119879 and inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (85)

Thus 120590119896

rarr 119879 as Using the generalized Itorsquos formula and(83) we obtain that for 119905 isin [119905

0 119879]

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050))

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(86)

Then the Gronwall inequality indicates

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

(87)

for all 119905 isin [1199050 119879] By virtue of condition (83) we obtain the

required assertion (84)

Corollary 13 Assume 119901 ge 2 and 1199090isin 119871119901

F1199050

(ΩR119899) Assumealso that there exists a constant 120579 gt 0 such that for all(119909 119905 (119894

1 1198942)) isin R119899 times R

+times S

119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 120579 (1 + |119909|

2) (88)

Then one has

119864|119909 (119905)|119901

le 2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119905 minus 119905

0) forall119905 isin [119905

0 119879]

(89)

Proof Define 119881(119909 119905 (1198941 1198942)) = (1 + |119909|

2)1199012 Making use of

(88) yields

119871119881 (119909 119905 (1198941 1198942))

= 119901(1 + |119909|2)(119901minus2)2

119909119879119891 (119909 119905 119894

1)

+119901

2(1 + |119909|

2)(119901minus2)21003816100381610038161003816119892 (119909 119905 119894

2)10038161003816100381610038162

+119901 (119901 minus 2)

2(1 + |119909|

2)(119901minus4)210038161003816100381610038161003816

119909119879119892 (119909 119905 119894

2)10038161003816100381610038161003816

2

le 119901(1 + |119909|2)(119901minus2)2

[119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162]

le 119901120579(1 + |119909|2)1199012

(90)

Then byTheorem 12 we get

119864(1 + |119909 (119905)|2)1199012

le 119864(1 +10038161003816100381610038161199090

10038161003816100381610038162)1199012

exp 119901120579 (119905 minus 1199050)

(91)

and the required assertion (89) followsIt is useful to point out that if the linear growth condition

(43) is satisfied then (88) is fulfilled with 120579 = radic119870 + 119870(119901 minus

1)2 Now we will show the other important properties of thesolution

Theorem 14 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume alsothat the linear growth condition (43) holds Then one has

119864|119909 (119905) minus 119909 (119904)|119901le 119862(119905 minus 119904)

1199012 (92)

where

119862 = 2119901minus2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119879 minus 119905

0)

times ([2 (119879 minus 1199050)]1199012

+ [119901 (119901 minus 1)]1199012

)

(93)

and 120579 = radic119870+119870(119901minus1)2 Particularly the 119901th moment of thesolution is continuous on [119905

0 119879]

Journal of Applied Mathematics 9

Proof Applying the elementary inequality |119886 + 119887|119901

le

2119901minus1

(|119886|119901+ |119887|119901) the Holder inequality and the linear growth

condition we can derive that

119864|119909 (119905) minus 119909 (119904)|119901

le 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119891 (119909 (119906) 119906 1205741(119906)) 119889119906

10038161003816100381610038161003816100381610038161003816

119901

+ 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119892 (119909 (119906) 119906 1205742(119906)) 119889119861 (119906)

10038161003816100381610038161003816100381610038161003816

119901

le [2 (119905 minus 119904)]119901minus1

119864int

119905

119904

1003816100381610038161003816119891 (119909 (119906) 119906 1205741(119906))

1003816100381610038161003816119901119889119906

+ 05[2119901 (119901 minus 1)]1199012

(119905 minus 119904)(119901minus2)2

119864

times int

119905

119904

1003816100381610038161003816119892 (119909 (119906) 119906 1205742 (119906))1003816100381610038161003816119901119889119906

le 1198621(119905 minus 119904)

(119901minus2)2int

119905

119904

119864(1 + |119909 (119906)|2)1199012

119889119906

(94)

where1198621= 2(119901minus2)2

1198701199012

([2(119879minus 1199050)]1199012

+ [119901(119901minus1)]1199012

) Using(91) yields

119864|119909 (119905) minus 119909 (119904)|119901

le 1198621(119905 minus 119904)

(119901minus2)2

times int

119905

119904

2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119906 minus 119905

0) 119889119906

le 11986212(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901)

times exp 119901120579 (119879 minus 1199050) (119905 minus 119904)

(1199012)

(95)

which is the desired inequality

Theorem 15 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume thatthere is a119870 gt 0 such that for all (119909 119905 (119894

1 1198942)) isin R119899times[119905

0 119879]timesS

119909119879 1003816100381610038161003816119891 (119909 119905 119894

1)1003816100381610038161003816 or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (96)

Then

119864( sup1199050le119905le119879

|119909 (119905)|119901)

le (1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 2119901 (10119901 + 1)119870 (119879 minus 119905

0)

(97)

Proof Making use of the generalized Itorsquos formula andcondition (96) we have

|119909 (119905)|119901

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905)

+ int

119905

1199050

119901|119909 (119904)|119901minus2

[119909119879(119904) 119891 (119909 (119904) 119904 120574

1(119904))

+119901 minus 1

2

1003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162] 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 05119901 (119901 + 1)119870

times int

119905

1199050

|119909 (119904)|119901minus2

(1 + |119909 (119904)|2) 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 119901 (119901 + 1)119870int

119905

1199050

(1 + |119909 (119904)|119901) 119889119904

(98)

where

119872(119905) = int

119905

1199050

119901|119909 (119904)|119901minus2

119909119879(119904) 119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904) (99)

Therefore for any 1199051isin [1199050 119879]

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 119864

100381610038161003816100381611990901003816100381610038161003816119901+ 119864( sup

1199050le119905le1199051

|119872 (119905)|)

+ 119901 (119901 + 1)119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(100)

At the same time applying the well-known Burkholder-Davis-Gundy inequality (see eg [2]) gives

119864( sup1199050le119905le1199051

|119872 (119905)|)

le 3119864(int

1199051

1199050

1199012|119909 (119904)|

2119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162119889119904)

05

le 3119864( sup1199050le119905le1199051

|119909 (119905)|119901

timesint

1199051

1199050

1199012|119909 (119904)|

119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742(119904))

10038161003816100381610038162119889119904)

05

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 45119864

times int

1199051

1199050

1199012119870|119909 (119904)|

119901minus2(1 + |119909 (119904)|

2) 119889119904

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 9119901

2119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(101)

10 Journal of Applied Mathematics

Substituting the above inequality into (100) gives

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 2119864

100381610038161003816100381611990901003816100381610038161003816119901

+ 2119901 (10119901 + 1)119870int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(102)

Thus

1 + 119864( sup1199050le119905le1199051

|119909 (119905)|119901)

le 1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901+ 2119901 (10119901 + 1)119870

times int

1199051

1199050

(1 + sup1199050le119905le119904

|119909 (119905)|119901)119889119904

(103)

Then the required assertion follows from the Gronwallinequality

Up to now we have discussed the 119871119901-estimates for the

solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies

119864|119909 (119905)|119901le [119864|119909 (119905)|

2]05119901

(104)

4 Example

Consider the following Black-Scholes model

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (105)

where 1205741(119905) is a right-continuous homogenous Markovian

chain taking values in finite state spaces S1= 1 2 and 120574

2(119905)

is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S

2= 1 2 3 120583(119894) = 119894 119894 = 1 2

](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution

5 Conclusions and Further Research

This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis

(H1)1015840 both 1205741(119905) and 120574

2(119905) are right-continuous homoge-

nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a

homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper

can be established similarly It is easy to see that if 1205741(119905) equiv

1205742(119905) and 120574

1(119905) is a right-continuous homogenous Markovian

chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574

2(119905) (8) will reduce to the classical SDEs with

single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions

Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress

Acknowledgments

The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)

References

[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989

[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006

[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004

[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002

[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009

[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007

[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009

[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

Then (72) means 119892119896

rarr infin It follows from (76) and (79) that

[119881 (1199090 1199050 1205740) + 120579 (119879 minus 119905

0)] exp 120579 (119879 minus 119905

0)

ge 119892119896119875 120590119896le 119879 ge 120576119892

119896

(81)

Letting 119896 rarr infin yields a contradiction that is to say120590infin

= infinThe proof is complete

3 119871119875-Estimates

In the previous section we have investigated the existenceand uniqueness of the solution to (8) In this section as abovelet 119909(119905) 119905

0le 119905 le 119879 be the unique solution of (8) with initial

conditions 119909(1199050) = 1199090and 120574(119905

0) and we will estimate the 119901th

moment of the solution

Theorem 12 Assume that there is a function 119881(119909 119905 (1198941 1198942)) isin

11986221

(R119899 times R+times SR

+) and positive constants 119901 120578 120579 such that

for all (119909 119905 (1198941 1198942)) isin R119899 times R

+times S

120578|119909|119901le 119881 (119909 119905 (119894

1 1198942)) (82)

119871119881 (119909 119905 (1198941 1198942)) le 120579119881 (119909 119905 (119894

1 1198942)) (83)

Assume also the initial condition 119909(1199050) = 119909

0and 120574(119905

0) obeys

119864119881(1199090 1199050 120574(1199050)) lt infin then one has

119864|119909 (119905)|119901

le119864119881 (119909

0 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

120578 forall119905 isin [119905

0 119879]

(84)

Proof For each integer 119896 ge 1 define the stopping time

120590119896= 119879 and inf 119905 ge 119905

0 |119909 (119905)| ge 119896 (85)

Thus 120590119896

rarr 119879 as Using the generalized Itorsquos formula and(83) we obtain that for 119905 isin [119905

0 119879]

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050))

+ 120579int

119905

1199050

119864119881 (119909 (119904 and 120590119896) 119904 and 120590

119896 120574 (119904 and 120590

119896)) 119889119904

(86)

Then the Gronwall inequality indicates

119864119881 (119909 (119905 and 120590119896) 119905 and 120590

119896 120574 (119905 and 120590

119896))

le 119864119881 (1199090 1199050 120574 (1199050)) exp 120579 (119905 minus 119905

0)

(87)

for all 119905 isin [1199050 119879] By virtue of condition (83) we obtain the

required assertion (84)

Corollary 13 Assume 119901 ge 2 and 1199090isin 119871119901

F1199050

(ΩR119899) Assumealso that there exists a constant 120579 gt 0 such that for all(119909 119905 (119894

1 1198942)) isin R119899 times R

+times S

119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 120579 (1 + |119909|

2) (88)

Then one has

119864|119909 (119905)|119901

le 2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119905 minus 119905

0) forall119905 isin [119905

0 119879]

(89)

Proof Define 119881(119909 119905 (1198941 1198942)) = (1 + |119909|

2)1199012 Making use of

(88) yields

119871119881 (119909 119905 (1198941 1198942))

= 119901(1 + |119909|2)(119901minus2)2

119909119879119891 (119909 119905 119894

1)

+119901

2(1 + |119909|

2)(119901minus2)21003816100381610038161003816119892 (119909 119905 119894

2)10038161003816100381610038162

+119901 (119901 minus 2)

2(1 + |119909|

2)(119901minus4)210038161003816100381610038161003816

119909119879119892 (119909 119905 119894

2)10038161003816100381610038161003816

2

le 119901(1 + |119909|2)(119901minus2)2

[119909119879119891 (119909 119905 119894

1) +

119901 minus 1

2

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162]

le 119901120579(1 + |119909|2)1199012

(90)

Then byTheorem 12 we get

119864(1 + |119909 (119905)|2)1199012

le 119864(1 +10038161003816100381610038161199090

10038161003816100381610038162)1199012

exp 119901120579 (119905 minus 1199050)

(91)

and the required assertion (89) followsIt is useful to point out that if the linear growth condition

(43) is satisfied then (88) is fulfilled with 120579 = radic119870 + 119870(119901 minus

1)2 Now we will show the other important properties of thesolution

Theorem 14 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume alsothat the linear growth condition (43) holds Then one has

119864|119909 (119905) minus 119909 (119904)|119901le 119862(119905 minus 119904)

1199012 (92)

where

119862 = 2119901minus2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119879 minus 119905

0)

times ([2 (119879 minus 1199050)]1199012

+ [119901 (119901 minus 1)]1199012

)

(93)

and 120579 = radic119870+119870(119901minus1)2 Particularly the 119901th moment of thesolution is continuous on [119905

0 119879]

Journal of Applied Mathematics 9

Proof Applying the elementary inequality |119886 + 119887|119901

le

2119901minus1

(|119886|119901+ |119887|119901) the Holder inequality and the linear growth

condition we can derive that

119864|119909 (119905) minus 119909 (119904)|119901

le 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119891 (119909 (119906) 119906 1205741(119906)) 119889119906

10038161003816100381610038161003816100381610038161003816

119901

+ 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119892 (119909 (119906) 119906 1205742(119906)) 119889119861 (119906)

10038161003816100381610038161003816100381610038161003816

119901

le [2 (119905 minus 119904)]119901minus1

119864int

119905

119904

1003816100381610038161003816119891 (119909 (119906) 119906 1205741(119906))

1003816100381610038161003816119901119889119906

+ 05[2119901 (119901 minus 1)]1199012

(119905 minus 119904)(119901minus2)2

119864

times int

119905

119904

1003816100381610038161003816119892 (119909 (119906) 119906 1205742 (119906))1003816100381610038161003816119901119889119906

le 1198621(119905 minus 119904)

(119901minus2)2int

119905

119904

119864(1 + |119909 (119906)|2)1199012

119889119906

(94)

where1198621= 2(119901minus2)2

1198701199012

([2(119879minus 1199050)]1199012

+ [119901(119901minus1)]1199012

) Using(91) yields

119864|119909 (119905) minus 119909 (119904)|119901

le 1198621(119905 minus 119904)

(119901minus2)2

times int

119905

119904

2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119906 minus 119905

0) 119889119906

le 11986212(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901)

times exp 119901120579 (119879 minus 1199050) (119905 minus 119904)

(1199012)

(95)

which is the desired inequality

Theorem 15 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume thatthere is a119870 gt 0 such that for all (119909 119905 (119894

1 1198942)) isin R119899times[119905

0 119879]timesS

119909119879 1003816100381610038161003816119891 (119909 119905 119894

1)1003816100381610038161003816 or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (96)

Then

119864( sup1199050le119905le119879

|119909 (119905)|119901)

le (1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 2119901 (10119901 + 1)119870 (119879 minus 119905

0)

(97)

Proof Making use of the generalized Itorsquos formula andcondition (96) we have

|119909 (119905)|119901

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905)

+ int

119905

1199050

119901|119909 (119904)|119901minus2

[119909119879(119904) 119891 (119909 (119904) 119904 120574

1(119904))

+119901 minus 1

2

1003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162] 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 05119901 (119901 + 1)119870

times int

119905

1199050

|119909 (119904)|119901minus2

(1 + |119909 (119904)|2) 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 119901 (119901 + 1)119870int

119905

1199050

(1 + |119909 (119904)|119901) 119889119904

(98)

where

119872(119905) = int

119905

1199050

119901|119909 (119904)|119901minus2

119909119879(119904) 119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904) (99)

Therefore for any 1199051isin [1199050 119879]

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 119864

100381610038161003816100381611990901003816100381610038161003816119901+ 119864( sup

1199050le119905le1199051

|119872 (119905)|)

+ 119901 (119901 + 1)119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(100)

At the same time applying the well-known Burkholder-Davis-Gundy inequality (see eg [2]) gives

119864( sup1199050le119905le1199051

|119872 (119905)|)

le 3119864(int

1199051

1199050

1199012|119909 (119904)|

2119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162119889119904)

05

le 3119864( sup1199050le119905le1199051

|119909 (119905)|119901

timesint

1199051

1199050

1199012|119909 (119904)|

119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742(119904))

10038161003816100381610038162119889119904)

05

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 45119864

times int

1199051

1199050

1199012119870|119909 (119904)|

119901minus2(1 + |119909 (119904)|

2) 119889119904

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 9119901

2119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(101)

10 Journal of Applied Mathematics

Substituting the above inequality into (100) gives

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 2119864

100381610038161003816100381611990901003816100381610038161003816119901

+ 2119901 (10119901 + 1)119870int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(102)

Thus

1 + 119864( sup1199050le119905le1199051

|119909 (119905)|119901)

le 1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901+ 2119901 (10119901 + 1)119870

times int

1199051

1199050

(1 + sup1199050le119905le119904

|119909 (119905)|119901)119889119904

(103)

Then the required assertion follows from the Gronwallinequality

Up to now we have discussed the 119871119901-estimates for the

solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies

119864|119909 (119905)|119901le [119864|119909 (119905)|

2]05119901

(104)

4 Example

Consider the following Black-Scholes model

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (105)

where 1205741(119905) is a right-continuous homogenous Markovian

chain taking values in finite state spaces S1= 1 2 and 120574

2(119905)

is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S

2= 1 2 3 120583(119894) = 119894 119894 = 1 2

](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution

5 Conclusions and Further Research

This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis

(H1)1015840 both 1205741(119905) and 120574

2(119905) are right-continuous homoge-

nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a

homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper

can be established similarly It is easy to see that if 1205741(119905) equiv

1205742(119905) and 120574

1(119905) is a right-continuous homogenous Markovian

chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574

2(119905) (8) will reduce to the classical SDEs with

single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions

Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress

Acknowledgments

The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)

References

[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989

[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006

[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004

[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002

[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009

[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007

[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009

[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

Proof Applying the elementary inequality |119886 + 119887|119901

le

2119901minus1

(|119886|119901+ |119887|119901) the Holder inequality and the linear growth

condition we can derive that

119864|119909 (119905) minus 119909 (119904)|119901

le 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119891 (119909 (119906) 119906 1205741(119906)) 119889119906

10038161003816100381610038161003816100381610038161003816

119901

+ 2119901minus1

119864

10038161003816100381610038161003816100381610038161003816int

119905

119904

119892 (119909 (119906) 119906 1205742(119906)) 119889119861 (119906)

10038161003816100381610038161003816100381610038161003816

119901

le [2 (119905 minus 119904)]119901minus1

119864int

119905

119904

1003816100381610038161003816119891 (119909 (119906) 119906 1205741(119906))

1003816100381610038161003816119901119889119906

+ 05[2119901 (119901 minus 1)]1199012

(119905 minus 119904)(119901minus2)2

119864

times int

119905

119904

1003816100381610038161003816119892 (119909 (119906) 119906 1205742 (119906))1003816100381610038161003816119901119889119906

le 1198621(119905 minus 119904)

(119901minus2)2int

119905

119904

119864(1 + |119909 (119906)|2)1199012

119889119906

(94)

where1198621= 2(119901minus2)2

1198701199012

([2(119879minus 1199050)]1199012

+ [119901(119901minus1)]1199012

) Using(91) yields

119864|119909 (119905) minus 119909 (119904)|119901

le 1198621(119905 minus 119904)

(119901minus2)2

times int

119905

119904

2(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 119901120579 (119906 minus 119905

0) 119889119906

le 11986212(119901minus2)2

(1 + 11986410038161003816100381610038161199090

1003816100381610038161003816119901)

times exp 119901120579 (119879 minus 1199050) (119905 minus 119904)

(1199012)

(95)

which is the desired inequality

Theorem 15 Let 119901 ge 2 and 1199090

isin 119871119901

F1199050

(ΩR119899) Assume thatthere is a119870 gt 0 such that for all (119909 119905 (119894

1 1198942)) isin R119899times[119905

0 119879]timesS

119909119879 1003816100381610038161003816119891 (119909 119905 119894

1)1003816100381610038161003816 or

1003816100381610038161003816119892 (119909 119905 1198942)10038161003816100381610038162le 119870(1 + |119909|)

2 (96)

Then

119864( sup1199050le119905le119879

|119909 (119905)|119901)

le (1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901) exp 2119901 (10119901 + 1)119870 (119879 minus 119905

0)

(97)

Proof Making use of the generalized Itorsquos formula andcondition (96) we have

|119909 (119905)|119901

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905)

+ int

119905

1199050

119901|119909 (119904)|119901minus2

[119909119879(119904) 119891 (119909 (119904) 119904 120574

1(119904))

+119901 minus 1

2

1003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162] 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 05119901 (119901 + 1)119870

times int

119905

1199050

|119909 (119904)|119901minus2

(1 + |119909 (119904)|2) 119889119904

le10038161003816100381610038161199090

1003816100381610038161003816119901+ 119872(119905) + 119901 (119901 + 1)119870int

119905

1199050

(1 + |119909 (119904)|119901) 119889119904

(98)

where

119872(119905) = int

119905

1199050

119901|119909 (119904)|119901minus2

119909119879(119904) 119892 (119909 (119904) 119904 1205742 (119904)) 119889119861 (119904) (99)

Therefore for any 1199051isin [1199050 119879]

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 119864

100381610038161003816100381611990901003816100381610038161003816119901+ 119864( sup

1199050le119905le1199051

|119872 (119905)|)

+ 119901 (119901 + 1)119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(100)

At the same time applying the well-known Burkholder-Davis-Gundy inequality (see eg [2]) gives

119864( sup1199050le119905le1199051

|119872 (119905)|)

le 3119864(int

1199051

1199050

1199012|119909 (119904)|

2119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742 (119904))10038161003816100381610038162119889119904)

05

le 3119864( sup1199050le119905le1199051

|119909 (119905)|119901

timesint

1199051

1199050

1199012|119909 (119904)|

119901minus21003816100381610038161003816119892 (119909 (119904) 119904 1205742(119904))

10038161003816100381610038162119889119904)

05

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 45119864

times int

1199051

1199050

1199012119870|119909 (119904)|

119901minus2(1 + |119909 (119904)|

2) 119889119904

le 05119864( sup1199050le119905le1199051

|119909 (119905)|119901) + 9119901

2119870119864int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(101)

10 Journal of Applied Mathematics

Substituting the above inequality into (100) gives

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 2119864

100381610038161003816100381611990901003816100381610038161003816119901

+ 2119901 (10119901 + 1)119870int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(102)

Thus

1 + 119864( sup1199050le119905le1199051

|119909 (119905)|119901)

le 1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901+ 2119901 (10119901 + 1)119870

times int

1199051

1199050

(1 + sup1199050le119905le119904

|119909 (119905)|119901)119889119904

(103)

Then the required assertion follows from the Gronwallinequality

Up to now we have discussed the 119871119901-estimates for the

solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies

119864|119909 (119905)|119901le [119864|119909 (119905)|

2]05119901

(104)

4 Example

Consider the following Black-Scholes model

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (105)

where 1205741(119905) is a right-continuous homogenous Markovian

chain taking values in finite state spaces S1= 1 2 and 120574

2(119905)

is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S

2= 1 2 3 120583(119894) = 119894 119894 = 1 2

](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution

5 Conclusions and Further Research

This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis

(H1)1015840 both 1205741(119905) and 120574

2(119905) are right-continuous homoge-

nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a

homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper

can be established similarly It is easy to see that if 1205741(119905) equiv

1205742(119905) and 120574

1(119905) is a right-continuous homogenous Markovian

chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574

2(119905) (8) will reduce to the classical SDEs with

single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions

Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress

Acknowledgments

The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)

References

[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989

[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006

[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004

[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002

[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009

[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007

[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009

[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

Substituting the above inequality into (100) gives

119864( sup1199050le119905le1199051

|119909 (119905)|119901) le 2119864

100381610038161003816100381611990901003816100381610038161003816119901

+ 2119901 (10119901 + 1)119870int

1199051

1199050

(1 + |119909 (119904)|119901) 119889119904

(102)

Thus

1 + 119864( sup1199050le119905le1199051

|119909 (119905)|119901)

le 1 + 211986410038161003816100381610038161199090

1003816100381610038161003816119901+ 2119901 (10119901 + 1)119870

times int

1199051

1199050

(1 + sup1199050le119905le119904

|119909 (119905)|119901)119889119904

(103)

Then the required assertion follows from the Gronwallinequality

Up to now we have discussed the 119871119901-estimates for the

solution in the case when 119901 ge 2 As for 0 lt 119901 lt 2 the similarresults can be given without any difficulty as long as we notethat the Holder inequality implies

119864|119909 (119905)|119901le [119864|119909 (119905)|

2]05119901

(104)

4 Example

Consider the following Black-Scholes model

119889119883 (119905) = 120583 (1205741(119905))119883 (119905) 119889119905 + ] (120574

2(119905))119883 (119905) 119889119861 (119905) (105)

where 1205741(119905) is a right-continuous homogenous Markovian

chain taking values in finite state spaces S1= 1 2 and 120574

2(119905)

is a right-continuous homogenous Markovian chain takingvalues in finite state spaces S

2= 1 2 3 120583(119894) = 119894 119894 = 1 2

](119895) = 119895 + 1 119895 = 1 2 3 Taking 119870 = 16 119870 = 9 then (42)and (43) hold Therefore by Theorem 7 (105) has a uniquesolution

5 Conclusions and Further Research

This paper is devoted to studying the existence and unique-ness of solution of SDEs with multi-Markovian switchingsand estimating the119901thmoment of the solutionWe have usedtwo continuous-time Markovian chains to model the SDEsThis area is becoming increasingly useful in engineeringeconomics communication theory active networking andso forth The sufficient criteria for existence and unique-ness of solution local solution and maximal local solutionwere established Those results indicate that (8) keeps manyproperties that (89) owns At the same time although thehypothesis (H1) is used in this paper wewant to point out thatthis hypothesis is not essential In fact (H1) can be replacedby the following generalized hypothesis

(H1)1015840 both 1205741(119905) and 120574

2(119905) are right-continuous homoge-

nous Markovian chains such that 120574(119905) = (1205741(119905) 1205742(119905)) is a

homogenous vector chainUnder hypothesis (H1)1015840 the results given in this paper

can be established similarly It is easy to see that if 1205741(119905) equiv

1205742(119905) and 120574

1(119905) is a right-continuous homogenous Markovian

chain then (H1)1015840 is fulfilled immediately At the same timeif 1205741(119905) equiv 120574

2(119905) (8) will reduce to the classical SDEs with

single Markovian chain that is to say the classical theoryabout SDEs with single Markovian chain is a special caseof our theory On the other hand many theorems in thispaper will play important roles in further study For exampleTheorem 15 will be useful when one studies the approximatesolutions

Some important and interesting questions can be furtherinvestigated using the results in this paper For exampleapproximate solutions boundedness and stability stochas-tic functional differential equations with vector Markovianswitching and their applications In particular the stability of(8) is one of the most important and interesting topics andthose investigations are in progress

Acknowledgments

The authors thank the editor and referees for their veryimportant and helpful comments and suggestions Theauthors also thank Dr C Zhang for helping them to improvethe English exposition This research is supported by theNSFC of China (nos 11171081 and 11171056)

References

[1] A V SkorokhodAsymptoticMethods in theTheory of StochasticDifferential Equations vol 78 of Translations of MathematicalMonographs American Mathematical Society Providence RIUSA 1989

[2] X Mao and C Yuan Stochastic Differential Euations withMarkovian Switching Imperial College Press London UK2006

[3] G Yin and X Y Zhou ldquoMarkowitzrsquos mean-variance portfolioselection with regime switching from discrete-time models totheir continuous-time limitsrdquo IEEE Transactions on AutomaticControl vol 49 no 3 pp 349ndash360 2004

[4] J Buffington and R J Elliott ldquoAmerican options with regimeswitchingrdquo International Journal of Theoretical and AppliedFinance vol 5 no 5 pp 497ndash514 2002

[5] C Zhu and G Yin ldquoOn competitive Lotka-Volterra model inrandom environmentsrdquo Journal of Mathematical Analysis andApplications vol 357 no 1 pp 154ndash170 2009

[6] Q Luo and X Mao ldquoStochastic population dynamics underregime switchingrdquo Journal of Mathematical Analysis and Appli-cations vol 334 no 1 pp 69ndash84 2007

[7] X Li D Jiang and X Mao ldquoPopulation dynamical behaviorof Lotka-Volterra system under regime switchingrdquo Journal ofComputational and Applied Mathematics vol 232 no 2 pp427ndash448 2009

[8] X Li A Gray D Jiang and X Mao ldquoSufficient and neces-sary conditions of stochastic permanence and extinction forstochastic logistic populations under regime switchingrdquo Journal

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

ofMathematical Analysis and Applications vol 376 no 1 pp 11ndash28 2011

[9] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011

[10] G Hu and K Wang ldquoStability in distribution of competi-tive Lotka-Volterra system with Markovian switchingrdquo AppliedMathematical Modelling vol 35 no 7 pp 3189ndash3200 2011

[11] M Liu and K Wang ldquoAsymptotic properties and simulationsof a stochastic logistic model under regime switching IIrdquoMathematical andComputerModelling vol 55 no 3-4 pp 405ndash418 2012

[12] ZWu H Huang and LWang ldquoStochastic delay logistic modelunder regime switchingrdquo Abstract and Applied Analysis vol2012 Article ID 241702 26 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of