Research ArticleStatistical Inference for Stochastic DifferentialEquations with Small Noises

Liang Shen12 and Qingsong Xu1

1 School of Mathematics and Statistics Central South University Changsha Hunan 410075 China2 School of Science Linyi University Linyi Shandong 276005 China

Correspondence should be addressed to Qingsong Xu csuqingsongxu126com

Received 13 November 2013 Accepted 11 February 2014 Published 13 March 2014

Academic Editor Zhi-Bo Huang

Copyright copy 2014 L Shen and Q Xu This 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 proposes the least squares method to estimate the drift parameter for the stochastic differential equations driven bysmall noises which is more general than pure jump 120572-stable noises The asymptotic property of this least squares estimator isstudied under some regularity conditions The asymptotic distribution of the estimator is shown to be the convolution of a stabledistribution and a normal distribution which is completely different from the classical cases

1 Introduction

Stochastic differential equations (SDEs) are being extensivelyused as a model to describe some phenomena which aresubject to random influences it has found many applicationsin biology [1] medicine [2] econometrics [3 4] finance[5] geophysics [6] and oceanography [7] Then statisticalinference for these differential equations was of great interestand became a challenging theoretical problem For a morerecent comprehensive discussion we refer to [8 9]

The asymptotic theory of parametric estimation for dif-fusion processes with small white noise based on continuoustime observations is well developed and it has been studiedby many authors (see eg [10ndash14]) There have been manyapplications of small noise in mathematical finance see forexample [15ndash18]

In parametric inference due to the impossibility ofobserving diffusions continuously throughout a time intervalit is more practical and interesting to consider asymptoticestimation for diffusion processes with small noise based ondiscrete observations There are many approaches to driftestimation for discretely observed diffusions (see eg [19ndash23]) Long [24] has started the study on parameter estimationfor a class of stochastic differential equations driven by smallstable noise 119885

119905 119905 ge 0 However there has been no study on

parametric inference for stochastic processes with small Levynoises yet

In this paper we are interested in the study of parameterestimation for the following stochastic differential equationsdriven by more general Levy noise 119871

119905 119905 ge 0 based

on discrete observations We will employ the least squaresmethod to obtain an asymptotically consistent estimator

Let (ΩF F119905ge0

P) be a basic complete filtered prob-ability space satisfying the usual conditions that is thefiltration is continuous on the right and F

0contains all

P-null sets In this paper we consider a class of stochasticdifferential equations as follows

119889119883119905

= 120579119891 (119883119905) 119889119905 + 120576119892 (119883

119905minus

) 119889119871119905 119905 isin [0 1]

119871119905

= 119886119861119905+ 119887119885119905

119883 (0) = 1199090

(1)

where 119891 R rarr R and 119892 R rarr R are known functionsand 119886 119887 are known constants Let 119861

119905 119905 ge 0 be a standard

Brownian motion and let 119885119905 119905 ge 0 be a standard 120572-stable

Levy motion independent of 119861119905 119905 ge 0 with 119885

1sim 119878120572(1 120573 0)

for 120573 isin [0 1] 1 lt 120572 lt 2Let 119883 = 119883

119905 119905 ge 0 be a real-valued stationary process

satisfying the stochastic differential equation (1) and weassume that this process is observed at regularly spaced timepoints 119905

119894= 119894119899 119894 = 1 2 119899 Assume 119883

0

119905is the solution of

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 473681 6 pageshttpdxdoiorg1011552014473681

2 Abstract and Applied Analysis

the underlying ordinary differential equation (ODE) with thetrue value of the drift parameter 120579

0

1198891198830

119905= 1205790119891 (1198830

119905) 119889119905 119883

0

0= 1199090 (2)

Then we get

119883119905119894

minus 119883119905119894minus1

= int

119905119894

119905119894minus1

1205790119891 (119883119904) 119889119904 + 120576 int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119871119904 (3)

2 Preliminaries

In this paper we denote 119862 as a generic constant whose valuemay vary from place to place

The following regularity conditions are assumed to hold

(A1) The functions 119891(119909) and 119892(119909) satisfy the Lipschitzconditions that is there exists a constant 119871 gt 0 suchthat

1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816+

1003816100381610038161003816119892 (119909) minus 119892 (119910)

1003816100381610038161003816

le 1198711003816100381610038161003816119909 minus 119910

1003816100381610038161003816 119909 119910 isin R

(4)

(A2) There exist constants 119872 gt 0 and 119903 ge 0 satisfying

the growth condition

119892minus2

(119909) le 119872 (1 + |119909|119903) 119909 isin R (5)

(A3) There exists a positive constant 119873 gt 0 such that

0 lt |119892(119909)| le 119873 lt infin

(A4) For 119862

119903= 2119903minus1

or 1 119903 gt 0

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

le 119862119903(

100381610038161003816100381610038161198830

119905119894minus1

10038161003816100381610038161003816

119903

+

10038161003816100381610038161003816119883119905119894minus1

minus 1198830

119905119894minus1

10038161003816100381610038161003816

119903

) (6)

The LSE of 120579119899120576

is defined as

120579119899120576

= argmin120579

120588119899120576

(120579) (7)

where the contrast function

120588119899120576

(120579) =

119899

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

119883119905119894

minus 119883119905119894minus1

minus 120579119891 (119883119905119894minus1

) Δ119905119894minus1

120576119892 (119883119905119894minus1

)

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

(8)

Then the 120579119899120576

can be represented explicitly as follows

120579119899120576

=

sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) (119883119905119894

minus 119883119905119894minus1

)

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

(9)

Based on (3) and (9) there is a special decomposition for 120579119899120576

120579119899120576

=

1205790

sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119891 (119883119904) 119889119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

+

120576 sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119871119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

=1205790+

1205790sum119899

119894=1119892minus2

(119883119905119894minus1

)119891 (119883119905119894minus1

)int

119905119894

119905119894minus1

(119891 (119883119904)minus119891 (119883

119905119894minus1

)) 119889119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

+

120576 sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119871119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

=1205790+

1205790sum119899

119894=1119892minus2

(119883119905119894minus1

)119891 (119883119905119894minus1

)int

119905119894

119905119894minus1

(119891 (119883119904)minus119891 (119883

119905119894minus1

)) 119889119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

+

119887120576 sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119885119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

+

119886120576 sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

= 1205790

+

Φ2

(119899 120576)

Φ1

(119899 120576)

+

Φ3

(119899 120576)

Φ1

(119899 120576)

+

Φ4

(119899 120576)

Φ1

(119899 120576)

(10)

Now we give an explicit expression for 120576minus1

(120579119899120576

minus 1205790) By using

(10) we have

120576minus1

(120579119899120576

minus 1205790) =

120576minus1

Φ2

(119899 120576)

Φ1

(119899 120576)

+

120576minus1

Φ3

(119899 120576)

Φ1

(119899 120576)

+

120576minus1

Φ4

(119899 120576)

Φ1

(119899 120576)

=

Ψ2

(119899 120576)

Φ1

(119899 120576)

+

Ψ3

(119899 120576)

Φ1

(119899 120576)

+

Ψ4

(119899 120576)

Φ1

(119899 120576)

(11)

One of the important tools we will employ is the under-lying lemma (see (35) in the Lemma 32 of [24])

Lemma 1 Under conditions (A1)-(A2) one has

10038161003816100381610038161003816119883119905minus 1198830

119905

10038161003816100381610038161003816

le 120576119890119871|1205790

|119905

10038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119883119904minus

) 119889119885119904

10038161003816100381610038161003816100381610038161003816

(12)

sup0le119905le1

10038161003816100381610038161003816119883119905minus 1198830

119905

10038161003816100381610038161003816997888rarr1198750 as 120576 997888rarr 0 (13)

Abstract and Applied Analysis 3

3 Asymptotic Property of the LeastSquares Estimator

Theorem 2 Under the conditions (A1)ndash(A4) as 119899 rarr

infin 120576 rarr 0 119899120576 rarr infin and 119899120576120572(120572minus1)

rarr infin one has

120576minus1

(120579119899120576

minus 1205790)

997904rArr 119886

(int

1

0119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904)

12

int

1

0119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904

119873

+ 119887 (((int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

+119889119904)

1120572

1198801

minus (int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

minus119889119904)

1120572

1198802)

times (int

1

0

119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904)

minus1

)

(14)

where 1198801and 119880

2are independent random variables with 120572-

stable distribution 119878120572(1 120573 0) and 119873 is an independent random

variable with standard normal distribution

The theoremwill be proved by establishing several propo-sitionsWewill consider the asymptotic behaviors ofΦ

1(119899 120576)

Ψ119894(119899 120576) 119894 = 2 3 4 respectively

Proposition 3 Under conditions (A1)ndash(A4) and 119899 rarr infin

120576 rarr 0 one has

Φ1

(119899 120576) 997888rarr119875

int

1

0

119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904 (15)

Proof Under conditions (A1)ndash(A3) Proposition 3 can be

proved by using condition (A4) (see the proof of Proposition

33 in [24])

Proposition 4 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 and 119899120576 rarr infin one has

Ψ2

(119899 120576) 997888rarr1198750 (16)

Proof For 119905119894minus1

le 119905 le 119905119894 119894 = 1 2 119899

119883119905

= 119883119905119894minus1

+ int

119905

119905119894minus1

1205790119891 (119883119904) 119889119904 + 120576 int

119905

119905119894minus1

119892 (119883119904minus

) 119889119871119904 (17)

It follows that10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

le int

119905

119905119894minus1

10038161003816100381610038161205790

1003816100381610038161003816(

10038161003816100381610038161003816119891 (119883119904) minus 119891 (119883

119905119894minus1

)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816) 119889119904

+ 120576

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119871119904

100381610038161003816100381610038161003816100381610038161003816

le10038161003816100381610038161205790

1003816100381610038161003816119872 int

119905

119905119894minus1

10038161003816100381610038161003816119891 (119883119904) minus 119891 (119883

119905119894minus1

)

10038161003816100381610038161003816+ 119899minus1 1003816

1003816100381610038161205790

1003816100381610038161003816

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

+ 119886120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

+ 119887120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

(18)

Using Gronwall inequality we get10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

le 119890|1205790

|119872(119905minus119905119894minus1

)[

10038161003816100381610038161205790

1003816100381610038161003816

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

119899

+ 119886120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

+119887120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

]

(19)

which yields

sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

le 119890|1205790

|119872119899[

10038161003816100381610038161205790

1003816100381610038161003816

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

119899

+ 119886120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

+119887120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

]

(20)

thus under conditions (A1) and (A

3)

1003816100381610038161003816Φ2

(119899 120576)1003816100381610038161003816

le10038161003816100381610038161205790

1003816100381610038161003816

119899

sum

119894=1

119872 (1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

(119891 (119883119904) minus 119891 (119883

119905119894minus1

)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

le

11987211987010038161003816100381610038161205790

1003816100381610038161003816

119899

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

4 Abstract and Applied Analysis

le

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890|1205790

|119872119899

1198992

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

2

+

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890(|1205790

|119872)119899

119899

119887120576

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

+

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890|1205790

|119872119899

119899

119886120576

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

= Φ21

(119899 120576) + Φ22

(119899 120576) + Φ23

(119899 120576)

(21)

Then

1003816100381610038161003816Ψ2

(119899 120576)1003816100381610038161003816

le 120576minus1

Φ21

(119899 120576) + 120576minus1

Φ22

(119899 120576)

+ 120576minus1

Φ23

(119899 120576)

= Ψ21

(119899 120576) + Ψ22

(119899 120576) + Ψ23

(119899 120576)

(22)

Using (13) in Lemma 1 conditions (A1) and (A

4) we get

Ψ21

(119899 120576) rarr1198750 as 119899 rarr infin 120576 rarr 0 and 119899120576 rarr infin (see (326)

in [24]) By using the same techniques under condition (A2)

we can prove thatΨ2119895

(119899 120576) rarr1198750 119895 = 2 3 as 119899 rarr infin 120576 rarr 0

respectively

Proposition 5 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 and 119899120576120572(120572minus1)

rarr infin one has

Ψ3

(119899 120576)

997904rArr 119887(int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

+119889119904)

1120572

1198801

minus 119887(int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

minus119889119904)

1120572

1198802

(23)

Proof Under conditions (A1)ndash(A3) Proposition 5 can be

proved by using condition (A4) (see the proof of Proposition

44 in [24])

Proposition 6 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 one has

Ψ4

(119899 120576) 997904rArr 119886(int

1

0

10038161003816100381610038161003816119892minus2

(1198830

119904)

100381610038161003816100381610038161198912

(1198830

119904) 119889119904)

12

119873 (24)

Proof Note that

Ψ4

(119899 120576) = 119886

119899

sum

119894=1

119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

= 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

) int

119905119894

119905119894minus1

119892 (1198830

119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

)

times int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

)) 119889119861119904

+ 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) (119891 (119883119905119894minus1

) minus 119891 (1198830

119905119894minus1

))

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

(119892minus2

(119883119905119894minus1

) minus 119892minus2

(1198830

119905119894minus1

)) 119891 (1198830

119905119894minus1

)

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

(119892minus2

(119883119905119894minus1

) minus 119892minus2

(1198830

119905119894minus1

))

times (119891 (119883119905119894minus1

) minus 119891 (1198830

119905119894minus1

))

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

=

5

sum

119895=1

Ψ4119895

(119899 120576)

(25)

For Ψ41

(119899 120576) let 119884119894

= int

119905119894

119905119894minus1

119892(119883119904minus

)119889119861119904 119895 = 1 119899 Then it is

easy to see that 119884119894

sim 119873(0 int

119905119894

119905119894minus1

1198922(119883119904minus

)119889119904) and 1198841 119884

119899are

independent normal random variablesIt follows that

Ψ41

(119899 120576)

= 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

) 119884119894

sim 119873 (0 1198862

119899

sum

119894=1

119892minus4

(1198830

119905119894minus1

) 1198912

(1198830

119905119894minus1

) int

119905119894

119905119894minus1

1198922

(119883119904minus

) 119889119904)

997904rArr 119886(int

1

0

119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904)

12

119873

(26)

as 119899 rarr infin 120576 rarr 0

Abstract and Applied Analysis 5

For Ψ42

(119899 120576) using Markov inequality and Itorsquos isometryproperty for any given 120578 gt 0

Ψ42

(119899 120576)

le

1

120578

E[119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

)) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

]

le

119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

))

2

119889119904]

12

le

119871119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [int

119905119894

119905119894minus1

(119883119904minus

minus 1198830

119904minus

)

2

119889119904]

12

le

119871119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [ sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119904minus

minus 1198830

119904minus

10038161003816100381610038161003816119899minus12

]

(27)

By using (13) Ψ42

(119899 120576) rarr 0 as 119899 rarr infin 120576 rarr 0Applying similar techniques to Ψ

4119895(119899 120576) 119895 = 3 4 5 we

get Ψ4119895

(119899 120576) rarr 0 119895 = 3 4 5 as 119899 rarr infin 120576 rarr 0

Now we can proveTheorem 2

Proof By using Propositions 3 4 5 6 and Slutskyrsquos theoremwe can get the conclusion

4 Example

We consider the following nonlinear SDE driven by generalLevy noises

119889119883119905

= 120579119883119905119889119905 +

120576

1 + 1198832

119905minus

119889119871119905 119905 isin [0 1] 119883

0= 1199090

(28)

where 119891(119909) = 119909 119892(119909) = 1(1 + 1199092) 1199090and 120576 are known

constants and 120579 = 0 is an unknown parameterFor simplicity let 119909

0gt 0 120576 = 0 we get the ODE

1198891198830

119905= 12057901198830

119905119889119905 119905 isin [0 1] 119883

0

0= 1199090

(29)

and the solution

1198830

119905= 11990901198901205790

119905 (30)

Then the asymptotic distribution is

119886(int

1

0

(1 + 1199092

011989021205790

119904)

2

1199092

011989021205790

119904119889119904)

minus12

119873

+ 119887

(int

1

0(1 + 119909

2

011989021205790

119904)

120572

119909120572

01198901205721205790

119904119889119904)

1120572

int

1

0(1 + 119909

2

011989021205790

119904)2

1199092

011989021205790

119904119889119904

119878120572

(1 120573 0)

(31)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R I Jennrich and P B Bright ldquoFitting systems of linear dif-ferential equations using computer generated exact derivativesrdquoTechnometrics vol 18 no 4 pp 385ndash392 1976

[2] R H Jones ldquoFitting multivariate models to unequally spaceddatardquo in Time Series Analysis of Irregularly Observed Data pp158ndash188 1984

[3] A R Bergstrom Statistical Inference in Continuous TimeEconomic Models vol 99 North-Holland Amsterdam TheNetherlands 1976

[4] A R Bergstrom ldquoThe history of continuous-time econometricmodelsrdquo Econometric Theory vol 4 no 3 pp 365ndash383 1988

[5] F Black and M Scholes ldquoThe pricing of options and corporateliabilitiesrdquoThe Journal of Political Economy vol 81 pp 637ndash6541973

[6] M Arato Linear Stochastic Systems with Constant CoefficientsSpringer Berlin Germany 1982

[7] R J Adler and P Meuller Stochastic Modelling on PhysicalOceanography vol 39 Springer New York NY USA 1996

[8] B P Rao B L P Rao I Statisticien B L P Rao B L P Rao andI Statistician Statistical Inference for di Usion Type ProcessesArnold London UK 1999

[9] Y A Kutoyants Statistical Inference for Ergodic Diffusion Pro-cesses Springer London UK 2004

[10] Yu A Kutoyants Parameter Estimation for Stochastic Processesvol 6 Heldermann Berlin Germany 1984

[11] Yu Kutoyants Identification of Dynamical Systems with SmallNoise Kluwer Academic Dordrecht The Netherlands 1994

[12] N Yoshida ldquoAsymptotic expansions of maximum likelihoodestimators for small diffusions via the theory of Malliavin-Watanaberdquo Probability Theory and Related Fields vol 92 no 3pp 275ndash311 1992

[13] N Yoshida ldquoConditional expansions and their applicationsrdquoStochastic Processes and their Applications vol 107 no 1 pp 53ndash81 2003

[14] M Uchida and N Yoshida ldquoInformation criteria for smalldiffusions via the theory of Malliavin-Watanaberdquo StatisticalInference for Stochastic Processes vol 7 no 1 pp 35ndash67 2004

[15] N Yoshida ldquoAsymptotic expansion of Bayes estimators for smalldiffusionsrdquo Probability Theory and Related Fields vol 95 no 4pp 429ndash450 1993

[16] A Takahashi ldquoAn asymptotic expansion approach to pricingfinancial contingent claimsrdquoAsia-Pacific FinancialMarkets vol6 no 2 pp 115ndash151 1999

6 Abstract and Applied Analysis

[17] N Kunitomo and A Takahashi ldquoThe asymptotic expansionapproach to the valuation of interest rate contingent claimsrdquoMathematical Finance vol 11 no 1 pp 117ndash151 2001

[18] A Takahashi and N Yoshida ldquoAn asymptotic expansionscheme for optimal investment problemsrdquo Statistical Inferencefor Stochastic Processes vol 7 no 2 pp 153ndash188 2004

[19] V Genon-Catalot ldquoMaximum contrast estimation for diffusionprocesses fromdiscrete observationsrdquo Statistics vol 21 no 1 pp99ndash116 1990

[20] A Gloter and M Soslashrensen ldquoEstimation for stochastic differ-ential equations with a small diffusion coefficientrdquo StochasticProcesses and their Applications vol 119 no 3 pp 679ndash6992009

[21] C F Laredo ldquoA sufficient condition for asymptotic sufficiencyof incomplete observations of a diffusion processrdquo The Annalsof Statistics vol 18 no 3 pp 1158ndash1171 1990

[22] M Uchida ldquoApproximate martingale estimating functions forstochastic differential equations with small noisesrdquo StochasticProcesses and their Applications vol 118 no 9 pp 1706ndash17212008

[23] M Soslashrensen and M Uchida ldquoSmall-diffusion asymptotics fordiscretely sampled stochastic differential equationsrdquo Bernoullivol 9 no 6 pp 1051ndash1069 2003

[24] H Long ldquoParameter estimation for a class of stochastic dif-ferential equations driven by small stable noises from discreteobservationsrdquo Acta Mathematica Scientia B vol 30 no 3 pp645ndash663 2010

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

2 Abstract and Applied Analysis

the underlying ordinary differential equation (ODE) with thetrue value of the drift parameter 120579

0

1198891198830

119905= 1205790119891 (1198830

119905) 119889119905 119883

0

0= 1199090 (2)

Then we get

119883119905119894

minus 119883119905119894minus1

= int

119905119894

119905119894minus1

1205790119891 (119883119904) 119889119904 + 120576 int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119871119904 (3)

2 Preliminaries

In this paper we denote 119862 as a generic constant whose valuemay vary from place to place

The following regularity conditions are assumed to hold

(A1) The functions 119891(119909) and 119892(119909) satisfy the Lipschitzconditions that is there exists a constant 119871 gt 0 suchthat

1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816+

1003816100381610038161003816119892 (119909) minus 119892 (119910)

1003816100381610038161003816

le 1198711003816100381610038161003816119909 minus 119910

1003816100381610038161003816 119909 119910 isin R

(4)

(A2) There exist constants 119872 gt 0 and 119903 ge 0 satisfying

the growth condition

119892minus2

(119909) le 119872 (1 + |119909|119903) 119909 isin R (5)

(A3) There exists a positive constant 119873 gt 0 such that

0 lt |119892(119909)| le 119873 lt infin

(A4) For 119862

119903= 2119903minus1

or 1 119903 gt 0

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

le 119862119903(

100381610038161003816100381610038161198830

119905119894minus1

10038161003816100381610038161003816

119903

+

10038161003816100381610038161003816119883119905119894minus1

minus 1198830

119905119894minus1

10038161003816100381610038161003816

119903

) (6)

The LSE of 120579119899120576

is defined as

120579119899120576

= argmin120579

120588119899120576

(120579) (7)

where the contrast function

120588119899120576

(120579) =

119899

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

119883119905119894

minus 119883119905119894minus1

minus 120579119891 (119883119905119894minus1

) Δ119905119894minus1

120576119892 (119883119905119894minus1

)

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

(8)

Then the 120579119899120576

can be represented explicitly as follows

120579119899120576

=

sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) (119883119905119894

minus 119883119905119894minus1

)

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

(9)

Based on (3) and (9) there is a special decomposition for 120579119899120576

120579119899120576

=

1205790

sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119891 (119883119904) 119889119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

+

120576 sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119871119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

=1205790+

1205790sum119899

119894=1119892minus2

(119883119905119894minus1

)119891 (119883119905119894minus1

)int

119905119894

119905119894minus1

(119891 (119883119904)minus119891 (119883

119905119894minus1

)) 119889119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

+

120576 sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119871119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

=1205790+

1205790sum119899

119894=1119892minus2

(119883119905119894minus1

)119891 (119883119905119894minus1

)int

119905119894

119905119894minus1

(119891 (119883119904)minus119891 (119883

119905119894minus1

)) 119889119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

+

119887120576 sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119885119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

+

119886120576 sum119899

119894=1119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

119899minus1

sum119899

119894=1119892minus2

(119883119905119894minus1

) 1198912

(119883119905119894minus1

)

= 1205790

+

Φ2

(119899 120576)

Φ1

(119899 120576)

+

Φ3

(119899 120576)

Φ1

(119899 120576)

+

Φ4

(119899 120576)

Φ1

(119899 120576)

(10)

Now we give an explicit expression for 120576minus1

(120579119899120576

minus 1205790) By using

(10) we have

120576minus1

(120579119899120576

minus 1205790) =

120576minus1

Φ2

(119899 120576)

Φ1

(119899 120576)

+

120576minus1

Φ3

(119899 120576)

Φ1

(119899 120576)

+

120576minus1

Φ4

(119899 120576)

Φ1

(119899 120576)

=

Ψ2

(119899 120576)

Φ1

(119899 120576)

+

Ψ3

(119899 120576)

Φ1

(119899 120576)

+

Ψ4

(119899 120576)

Φ1

(119899 120576)

(11)

One of the important tools we will employ is the under-lying lemma (see (35) in the Lemma 32 of [24])

Lemma 1 Under conditions (A1)-(A2) one has

10038161003816100381610038161003816119883119905minus 1198830

119905

10038161003816100381610038161003816

le 120576119890119871|1205790

|119905

10038161003816100381610038161003816100381610038161003816

int

119905

0

119892 (119883119904minus

) 119889119885119904

10038161003816100381610038161003816100381610038161003816

(12)

sup0le119905le1

10038161003816100381610038161003816119883119905minus 1198830

119905

10038161003816100381610038161003816997888rarr1198750 as 120576 997888rarr 0 (13)

Abstract and Applied Analysis 3

3 Asymptotic Property of the LeastSquares Estimator

Theorem 2 Under the conditions (A1)ndash(A4) as 119899 rarr

infin 120576 rarr 0 119899120576 rarr infin and 119899120576120572(120572minus1)

rarr infin one has

120576minus1

(120579119899120576

minus 1205790)

997904rArr 119886

(int

1

0119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904)

12

int

1

0119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904

119873

+ 119887 (((int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

+119889119904)

1120572

1198801

minus (int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

minus119889119904)

1120572

1198802)

times (int

1

0

119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904)

minus1

)

(14)

where 1198801and 119880

2are independent random variables with 120572-

stable distribution 119878120572(1 120573 0) and 119873 is an independent random

variable with standard normal distribution

The theoremwill be proved by establishing several propo-sitionsWewill consider the asymptotic behaviors ofΦ

1(119899 120576)

Ψ119894(119899 120576) 119894 = 2 3 4 respectively

Proposition 3 Under conditions (A1)ndash(A4) and 119899 rarr infin

120576 rarr 0 one has

Φ1

(119899 120576) 997888rarr119875

int

1

0

119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904 (15)

Proof Under conditions (A1)ndash(A3) Proposition 3 can be

proved by using condition (A4) (see the proof of Proposition

33 in [24])

Proposition 4 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 and 119899120576 rarr infin one has

Ψ2

(119899 120576) 997888rarr1198750 (16)

Proof For 119905119894minus1

le 119905 le 119905119894 119894 = 1 2 119899

119883119905

= 119883119905119894minus1

+ int

119905

119905119894minus1

1205790119891 (119883119904) 119889119904 + 120576 int

119905

119905119894minus1

119892 (119883119904minus

) 119889119871119904 (17)

It follows that10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

le int

119905

119905119894minus1

10038161003816100381610038161205790

1003816100381610038161003816(

10038161003816100381610038161003816119891 (119883119904) minus 119891 (119883

119905119894minus1

)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816) 119889119904

+ 120576

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119871119904

100381610038161003816100381610038161003816100381610038161003816

le10038161003816100381610038161205790

1003816100381610038161003816119872 int

119905

119905119894minus1

10038161003816100381610038161003816119891 (119883119904) minus 119891 (119883

119905119894minus1

)

10038161003816100381610038161003816+ 119899minus1 1003816

1003816100381610038161205790

1003816100381610038161003816

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

+ 119886120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

+ 119887120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

(18)

Using Gronwall inequality we get10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

le 119890|1205790

|119872(119905minus119905119894minus1

)[

10038161003816100381610038161205790

1003816100381610038161003816

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

119899

+ 119886120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

+119887120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

]

(19)

which yields

sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

le 119890|1205790

|119872119899[

10038161003816100381610038161205790

1003816100381610038161003816

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

119899

+ 119886120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

+119887120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

]

(20)

thus under conditions (A1) and (A

3)

1003816100381610038161003816Φ2

(119899 120576)1003816100381610038161003816

le10038161003816100381610038161205790

1003816100381610038161003816

119899

sum

119894=1

119872 (1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

(119891 (119883119904) minus 119891 (119883

119905119894minus1

)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

le

11987211987010038161003816100381610038161205790

1003816100381610038161003816

119899

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

4 Abstract and Applied Analysis

le

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890|1205790

|119872119899

1198992

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

2

+

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890(|1205790

|119872)119899

119899

119887120576

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

+

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890|1205790

|119872119899

119899

119886120576

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

= Φ21

(119899 120576) + Φ22

(119899 120576) + Φ23

(119899 120576)

(21)

Then

1003816100381610038161003816Ψ2

(119899 120576)1003816100381610038161003816

le 120576minus1

Φ21

(119899 120576) + 120576minus1

Φ22

(119899 120576)

+ 120576minus1

Φ23

(119899 120576)

= Ψ21

(119899 120576) + Ψ22

(119899 120576) + Ψ23

(119899 120576)

(22)

Using (13) in Lemma 1 conditions (A1) and (A

4) we get

Ψ21

(119899 120576) rarr1198750 as 119899 rarr infin 120576 rarr 0 and 119899120576 rarr infin (see (326)

in [24]) By using the same techniques under condition (A2)

we can prove thatΨ2119895

(119899 120576) rarr1198750 119895 = 2 3 as 119899 rarr infin 120576 rarr 0

respectively

Proposition 5 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 and 119899120576120572(120572minus1)

rarr infin one has

Ψ3

(119899 120576)

997904rArr 119887(int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

+119889119904)

1120572

1198801

minus 119887(int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

minus119889119904)

1120572

1198802

(23)

Proof Under conditions (A1)ndash(A3) Proposition 5 can be

proved by using condition (A4) (see the proof of Proposition

44 in [24])

Proposition 6 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 one has

Ψ4

(119899 120576) 997904rArr 119886(int

1

0

10038161003816100381610038161003816119892minus2

(1198830

119904)

100381610038161003816100381610038161198912

(1198830

119904) 119889119904)

12

119873 (24)

Proof Note that

Ψ4

(119899 120576) = 119886

119899

sum

119894=1

119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

= 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

) int

119905119894

119905119894minus1

119892 (1198830

119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

)

times int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

)) 119889119861119904

+ 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) (119891 (119883119905119894minus1

) minus 119891 (1198830

119905119894minus1

))

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

(119892minus2

(119883119905119894minus1

) minus 119892minus2

(1198830

119905119894minus1

)) 119891 (1198830

119905119894minus1

)

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

(119892minus2

(119883119905119894minus1

) minus 119892minus2

(1198830

119905119894minus1

))

times (119891 (119883119905119894minus1

) minus 119891 (1198830

119905119894minus1

))

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

=

5

sum

119895=1

Ψ4119895

(119899 120576)

(25)

For Ψ41

(119899 120576) let 119884119894

= int

119905119894

119905119894minus1

119892(119883119904minus

)119889119861119904 119895 = 1 119899 Then it is

easy to see that 119884119894

sim 119873(0 int

119905119894

119905119894minus1

1198922(119883119904minus

)119889119904) and 1198841 119884

119899are

independent normal random variablesIt follows that

Ψ41

(119899 120576)

= 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

) 119884119894

sim 119873 (0 1198862

119899

sum

119894=1

119892minus4

(1198830

119905119894minus1

) 1198912

(1198830

119905119894minus1

) int

119905119894

119905119894minus1

1198922

(119883119904minus

) 119889119904)

997904rArr 119886(int

1

0

119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904)

12

119873

(26)

as 119899 rarr infin 120576 rarr 0

Abstract and Applied Analysis 5

For Ψ42

(119899 120576) using Markov inequality and Itorsquos isometryproperty for any given 120578 gt 0

Ψ42

(119899 120576)

le

1

120578

E[119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

)) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

]

le

119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

))

2

119889119904]

12

le

119871119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [int

119905119894

119905119894minus1

(119883119904minus

minus 1198830

119904minus

)

2

119889119904]

12

le

119871119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [ sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119904minus

minus 1198830

119904minus

10038161003816100381610038161003816119899minus12

]

(27)

By using (13) Ψ42

(119899 120576) rarr 0 as 119899 rarr infin 120576 rarr 0Applying similar techniques to Ψ

4119895(119899 120576) 119895 = 3 4 5 we

get Ψ4119895

(119899 120576) rarr 0 119895 = 3 4 5 as 119899 rarr infin 120576 rarr 0

Now we can proveTheorem 2

Proof By using Propositions 3 4 5 6 and Slutskyrsquos theoremwe can get the conclusion

4 Example

We consider the following nonlinear SDE driven by generalLevy noises

119889119883119905

= 120579119883119905119889119905 +

120576

1 + 1198832

119905minus

119889119871119905 119905 isin [0 1] 119883

0= 1199090

(28)

where 119891(119909) = 119909 119892(119909) = 1(1 + 1199092) 1199090and 120576 are known

constants and 120579 = 0 is an unknown parameterFor simplicity let 119909

0gt 0 120576 = 0 we get the ODE

1198891198830

119905= 12057901198830

119905119889119905 119905 isin [0 1] 119883

0

0= 1199090

(29)

and the solution

1198830

119905= 11990901198901205790

119905 (30)

Then the asymptotic distribution is

119886(int

1

0

(1 + 1199092

011989021205790

119904)

2

1199092

011989021205790

119904119889119904)

minus12

119873

+ 119887

(int

1

0(1 + 119909

2

011989021205790

119904)

120572

119909120572

01198901205721205790

119904119889119904)

1120572

int

1

0(1 + 119909

2

011989021205790

119904)2

1199092

011989021205790

119904119889119904

119878120572

(1 120573 0)

(31)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R I Jennrich and P B Bright ldquoFitting systems of linear dif-ferential equations using computer generated exact derivativesrdquoTechnometrics vol 18 no 4 pp 385ndash392 1976

[2] R H Jones ldquoFitting multivariate models to unequally spaceddatardquo in Time Series Analysis of Irregularly Observed Data pp158ndash188 1984

[3] A R Bergstrom Statistical Inference in Continuous TimeEconomic Models vol 99 North-Holland Amsterdam TheNetherlands 1976

[4] A R Bergstrom ldquoThe history of continuous-time econometricmodelsrdquo Econometric Theory vol 4 no 3 pp 365ndash383 1988

[5] F Black and M Scholes ldquoThe pricing of options and corporateliabilitiesrdquoThe Journal of Political Economy vol 81 pp 637ndash6541973

[6] M Arato Linear Stochastic Systems with Constant CoefficientsSpringer Berlin Germany 1982

[7] R J Adler and P Meuller Stochastic Modelling on PhysicalOceanography vol 39 Springer New York NY USA 1996

[8] B P Rao B L P Rao I Statisticien B L P Rao B L P Rao andI Statistician Statistical Inference for di Usion Type ProcessesArnold London UK 1999

[9] Y A Kutoyants Statistical Inference for Ergodic Diffusion Pro-cesses Springer London UK 2004

[10] Yu A Kutoyants Parameter Estimation for Stochastic Processesvol 6 Heldermann Berlin Germany 1984

[11] Yu Kutoyants Identification of Dynamical Systems with SmallNoise Kluwer Academic Dordrecht The Netherlands 1994

[12] N Yoshida ldquoAsymptotic expansions of maximum likelihoodestimators for small diffusions via the theory of Malliavin-Watanaberdquo Probability Theory and Related Fields vol 92 no 3pp 275ndash311 1992

[13] N Yoshida ldquoConditional expansions and their applicationsrdquoStochastic Processes and their Applications vol 107 no 1 pp 53ndash81 2003

[14] M Uchida and N Yoshida ldquoInformation criteria for smalldiffusions via the theory of Malliavin-Watanaberdquo StatisticalInference for Stochastic Processes vol 7 no 1 pp 35ndash67 2004

[15] N Yoshida ldquoAsymptotic expansion of Bayes estimators for smalldiffusionsrdquo Probability Theory and Related Fields vol 95 no 4pp 429ndash450 1993

[16] A Takahashi ldquoAn asymptotic expansion approach to pricingfinancial contingent claimsrdquoAsia-Pacific FinancialMarkets vol6 no 2 pp 115ndash151 1999

6 Abstract and Applied Analysis

[17] N Kunitomo and A Takahashi ldquoThe asymptotic expansionapproach to the valuation of interest rate contingent claimsrdquoMathematical Finance vol 11 no 1 pp 117ndash151 2001

[18] A Takahashi and N Yoshida ldquoAn asymptotic expansionscheme for optimal investment problemsrdquo Statistical Inferencefor Stochastic Processes vol 7 no 2 pp 153ndash188 2004

[19] V Genon-Catalot ldquoMaximum contrast estimation for diffusionprocesses fromdiscrete observationsrdquo Statistics vol 21 no 1 pp99ndash116 1990

[20] A Gloter and M Soslashrensen ldquoEstimation for stochastic differ-ential equations with a small diffusion coefficientrdquo StochasticProcesses and their Applications vol 119 no 3 pp 679ndash6992009

[21] C F Laredo ldquoA sufficient condition for asymptotic sufficiencyof incomplete observations of a diffusion processrdquo The Annalsof Statistics vol 18 no 3 pp 1158ndash1171 1990

[22] M Uchida ldquoApproximate martingale estimating functions forstochastic differential equations with small noisesrdquo StochasticProcesses and their Applications vol 118 no 9 pp 1706ndash17212008

[23] M Soslashrensen and M Uchida ldquoSmall-diffusion asymptotics fordiscretely sampled stochastic differential equationsrdquo Bernoullivol 9 no 6 pp 1051ndash1069 2003

[24] H Long ldquoParameter estimation for a class of stochastic dif-ferential equations driven by small stable noises from discreteobservationsrdquo Acta Mathematica Scientia B vol 30 no 3 pp645ndash663 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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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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

Abstract and Applied Analysis 3

3 Asymptotic Property of the LeastSquares Estimator

Theorem 2 Under the conditions (A1)ndash(A4) as 119899 rarr

infin 120576 rarr 0 119899120576 rarr infin and 119899120576120572(120572minus1)

rarr infin one has

120576minus1

(120579119899120576

minus 1205790)

997904rArr 119886

(int

1

0119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904)

12

int

1

0119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904

119873

+ 119887 (((int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

+119889119904)

1120572

1198801

minus (int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

minus119889119904)

1120572

1198802)

times (int

1

0

119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904)

minus1

)

(14)

where 1198801and 119880

2are independent random variables with 120572-

stable distribution 119878120572(1 120573 0) and 119873 is an independent random

variable with standard normal distribution

The theoremwill be proved by establishing several propo-sitionsWewill consider the asymptotic behaviors ofΦ

1(119899 120576)

Ψ119894(119899 120576) 119894 = 2 3 4 respectively

Proposition 3 Under conditions (A1)ndash(A4) and 119899 rarr infin

120576 rarr 0 one has

Φ1

(119899 120576) 997888rarr119875

int

1

0

119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904 (15)

Proof Under conditions (A1)ndash(A3) Proposition 3 can be

proved by using condition (A4) (see the proof of Proposition

33 in [24])

Proposition 4 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 and 119899120576 rarr infin one has

Ψ2

(119899 120576) 997888rarr1198750 (16)

Proof For 119905119894minus1

le 119905 le 119905119894 119894 = 1 2 119899

119883119905

= 119883119905119894minus1

+ int

119905

119905119894minus1

1205790119891 (119883119904) 119889119904 + 120576 int

119905

119905119894minus1

119892 (119883119904minus

) 119889119871119904 (17)

It follows that10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

le int

119905

119905119894minus1

10038161003816100381610038161205790

1003816100381610038161003816(

10038161003816100381610038161003816119891 (119883119904) minus 119891 (119883

119905119894minus1

)

10038161003816100381610038161003816+

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816) 119889119904

+ 120576

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119871119904

100381610038161003816100381610038161003816100381610038161003816

le10038161003816100381610038161205790

1003816100381610038161003816119872 int

119905

119905119894minus1

10038161003816100381610038161003816119891 (119883119904) minus 119891 (119883

119905119894minus1

)

10038161003816100381610038161003816+ 119899minus1 1003816

1003816100381610038161205790

1003816100381610038161003816

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

+ 119886120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

+ 119887120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

(18)

Using Gronwall inequality we get10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

le 119890|1205790

|119872(119905minus119905119894minus1

)[

10038161003816100381610038161205790

1003816100381610038161003816

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

119899

+ 119886120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

+119887120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

]

(19)

which yields

sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

le 119890|1205790

|119872119899[

10038161003816100381610038161205790

1003816100381610038161003816

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

119899

+ 119886120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

+119887120576 sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

]

(20)

thus under conditions (A1) and (A

3)

1003816100381610038161003816Φ2

(119899 120576)1003816100381610038161003816

le10038161003816100381610038161205790

1003816100381610038161003816

119899

sum

119894=1

119872 (1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

(119891 (119883119904) minus 119891 (119883

119905119894minus1

)) 119889119904

100381610038161003816100381610038161003816100381610038161003816

le

11987211987010038161003816100381610038161205790

1003816100381610038161003816

119899

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119905minus 119883119905119894minus1

10038161003816100381610038161003816

4 Abstract and Applied Analysis

le

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890|1205790

|119872119899

1198992

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

2

+

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890(|1205790

|119872)119899

119899

119887120576

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

+

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890|1205790

|119872119899

119899

119886120576

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

= Φ21

(119899 120576) + Φ22

(119899 120576) + Φ23

(119899 120576)

(21)

Then

1003816100381610038161003816Ψ2

(119899 120576)1003816100381610038161003816

le 120576minus1

Φ21

(119899 120576) + 120576minus1

Φ22

(119899 120576)

+ 120576minus1

Φ23

(119899 120576)

= Ψ21

(119899 120576) + Ψ22

(119899 120576) + Ψ23

(119899 120576)

(22)

Using (13) in Lemma 1 conditions (A1) and (A

4) we get

Ψ21

(119899 120576) rarr1198750 as 119899 rarr infin 120576 rarr 0 and 119899120576 rarr infin (see (326)

in [24]) By using the same techniques under condition (A2)

we can prove thatΨ2119895

(119899 120576) rarr1198750 119895 = 2 3 as 119899 rarr infin 120576 rarr 0

respectively

Proposition 5 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 and 119899120576120572(120572minus1)

rarr infin one has

Ψ3

(119899 120576)

997904rArr 119887(int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

+119889119904)

1120572

1198801

minus 119887(int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

minus119889119904)

1120572

1198802

(23)

Proof Under conditions (A1)ndash(A3) Proposition 5 can be

proved by using condition (A4) (see the proof of Proposition

44 in [24])

Proposition 6 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 one has

Ψ4

(119899 120576) 997904rArr 119886(int

1

0

10038161003816100381610038161003816119892minus2

(1198830

119904)

100381610038161003816100381610038161198912

(1198830

119904) 119889119904)

12

119873 (24)

Proof Note that

Ψ4

(119899 120576) = 119886

119899

sum

119894=1

119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

= 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

) int

119905119894

119905119894minus1

119892 (1198830

119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

)

times int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

)) 119889119861119904

+ 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) (119891 (119883119905119894minus1

) minus 119891 (1198830

119905119894minus1

))

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

(119892minus2

(119883119905119894minus1

) minus 119892minus2

(1198830

119905119894minus1

)) 119891 (1198830

119905119894minus1

)

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

(119892minus2

(119883119905119894minus1

) minus 119892minus2

(1198830

119905119894minus1

))

times (119891 (119883119905119894minus1

) minus 119891 (1198830

119905119894minus1

))

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

=

5

sum

119895=1

Ψ4119895

(119899 120576)

(25)

For Ψ41

(119899 120576) let 119884119894

= int

119905119894

119905119894minus1

119892(119883119904minus

)119889119861119904 119895 = 1 119899 Then it is

easy to see that 119884119894

sim 119873(0 int

119905119894

119905119894minus1

1198922(119883119904minus

)119889119904) and 1198841 119884

119899are

independent normal random variablesIt follows that

Ψ41

(119899 120576)

= 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

) 119884119894

sim 119873 (0 1198862

119899

sum

119894=1

119892minus4

(1198830

119905119894minus1

) 1198912

(1198830

119905119894minus1

) int

119905119894

119905119894minus1

1198922

(119883119904minus

) 119889119904)

997904rArr 119886(int

1

0

119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904)

12

119873

(26)

as 119899 rarr infin 120576 rarr 0

Abstract and Applied Analysis 5

For Ψ42

(119899 120576) using Markov inequality and Itorsquos isometryproperty for any given 120578 gt 0

Ψ42

(119899 120576)

le

1

120578

E[119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

)) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

]

le

119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

))

2

119889119904]

12

le

119871119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [int

119905119894

119905119894minus1

(119883119904minus

minus 1198830

119904minus

)

2

119889119904]

12

le

119871119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [ sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119904minus

minus 1198830

119904minus

10038161003816100381610038161003816119899minus12

]

(27)

By using (13) Ψ42

(119899 120576) rarr 0 as 119899 rarr infin 120576 rarr 0Applying similar techniques to Ψ

4119895(119899 120576) 119895 = 3 4 5 we

get Ψ4119895

(119899 120576) rarr 0 119895 = 3 4 5 as 119899 rarr infin 120576 rarr 0

Now we can proveTheorem 2

Proof By using Propositions 3 4 5 6 and Slutskyrsquos theoremwe can get the conclusion

4 Example

We consider the following nonlinear SDE driven by generalLevy noises

119889119883119905

= 120579119883119905119889119905 +

120576

1 + 1198832

119905minus

119889119871119905 119905 isin [0 1] 119883

0= 1199090

(28)

where 119891(119909) = 119909 119892(119909) = 1(1 + 1199092) 1199090and 120576 are known

constants and 120579 = 0 is an unknown parameterFor simplicity let 119909

0gt 0 120576 = 0 we get the ODE

1198891198830

119905= 12057901198830

119905119889119905 119905 isin [0 1] 119883

0

0= 1199090

(29)

and the solution

1198830

119905= 11990901198901205790

119905 (30)

Then the asymptotic distribution is

119886(int

1

0

(1 + 1199092

011989021205790

119904)

2

1199092

011989021205790

119904119889119904)

minus12

119873

+ 119887

(int

1

0(1 + 119909

2

011989021205790

119904)

120572

119909120572

01198901205721205790

119904119889119904)

1120572

int

1

0(1 + 119909

2

011989021205790

119904)2

1199092

011989021205790

119904119889119904

119878120572

(1 120573 0)

(31)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R I Jennrich and P B Bright ldquoFitting systems of linear dif-ferential equations using computer generated exact derivativesrdquoTechnometrics vol 18 no 4 pp 385ndash392 1976

[2] R H Jones ldquoFitting multivariate models to unequally spaceddatardquo in Time Series Analysis of Irregularly Observed Data pp158ndash188 1984

[3] A R Bergstrom Statistical Inference in Continuous TimeEconomic Models vol 99 North-Holland Amsterdam TheNetherlands 1976

[4] A R Bergstrom ldquoThe history of continuous-time econometricmodelsrdquo Econometric Theory vol 4 no 3 pp 365ndash383 1988

[5] F Black and M Scholes ldquoThe pricing of options and corporateliabilitiesrdquoThe Journal of Political Economy vol 81 pp 637ndash6541973

[6] M Arato Linear Stochastic Systems with Constant CoefficientsSpringer Berlin Germany 1982

[7] R J Adler and P Meuller Stochastic Modelling on PhysicalOceanography vol 39 Springer New York NY USA 1996

[8] B P Rao B L P Rao I Statisticien B L P Rao B L P Rao andI Statistician Statistical Inference for di Usion Type ProcessesArnold London UK 1999

[9] Y A Kutoyants Statistical Inference for Ergodic Diffusion Pro-cesses Springer London UK 2004

[10] Yu A Kutoyants Parameter Estimation for Stochastic Processesvol 6 Heldermann Berlin Germany 1984

[11] Yu Kutoyants Identification of Dynamical Systems with SmallNoise Kluwer Academic Dordrecht The Netherlands 1994

[12] N Yoshida ldquoAsymptotic expansions of maximum likelihoodestimators for small diffusions via the theory of Malliavin-Watanaberdquo Probability Theory and Related Fields vol 92 no 3pp 275ndash311 1992

[13] N Yoshida ldquoConditional expansions and their applicationsrdquoStochastic Processes and their Applications vol 107 no 1 pp 53ndash81 2003

[14] M Uchida and N Yoshida ldquoInformation criteria for smalldiffusions via the theory of Malliavin-Watanaberdquo StatisticalInference for Stochastic Processes vol 7 no 1 pp 35ndash67 2004

[15] N Yoshida ldquoAsymptotic expansion of Bayes estimators for smalldiffusionsrdquo Probability Theory and Related Fields vol 95 no 4pp 429ndash450 1993

[16] A Takahashi ldquoAn asymptotic expansion approach to pricingfinancial contingent claimsrdquoAsia-Pacific FinancialMarkets vol6 no 2 pp 115ndash151 1999

6 Abstract and Applied Analysis

[17] N Kunitomo and A Takahashi ldquoThe asymptotic expansionapproach to the valuation of interest rate contingent claimsrdquoMathematical Finance vol 11 no 1 pp 117ndash151 2001

[18] A Takahashi and N Yoshida ldquoAn asymptotic expansionscheme for optimal investment problemsrdquo Statistical Inferencefor Stochastic Processes vol 7 no 2 pp 153ndash188 2004

[19] V Genon-Catalot ldquoMaximum contrast estimation for diffusionprocesses fromdiscrete observationsrdquo Statistics vol 21 no 1 pp99ndash116 1990

[20] A Gloter and M Soslashrensen ldquoEstimation for stochastic differ-ential equations with a small diffusion coefficientrdquo StochasticProcesses and their Applications vol 119 no 3 pp 679ndash6992009

[21] C F Laredo ldquoA sufficient condition for asymptotic sufficiencyof incomplete observations of a diffusion processrdquo The Annalsof Statistics vol 18 no 3 pp 1158ndash1171 1990

[22] M Uchida ldquoApproximate martingale estimating functions forstochastic differential equations with small noisesrdquo StochasticProcesses and their Applications vol 118 no 9 pp 1706ndash17212008

[23] M Soslashrensen and M Uchida ldquoSmall-diffusion asymptotics fordiscretely sampled stochastic differential equationsrdquo Bernoullivol 9 no 6 pp 1051ndash1069 2003

[24] H Long ldquoParameter estimation for a class of stochastic dif-ferential equations driven by small stable noises from discreteobservationsrdquo Acta Mathematica Scientia B vol 30 no 3 pp645ndash663 2010

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

4 Abstract and Applied Analysis

le

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890|1205790

|119872119899

1198992

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

2

+

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890(|1205790

|119872)119899

119899

119887120576

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119885119904

100381610038161003816100381610038161003816100381610038161003816

+

11987211987010038161003816100381610038161205790

1003816100381610038161003816

2

119890|1205790

|119872119899

119899

119886120576

119899

sum

119894=1

(1 +

10038161003816100381610038161003816119883119905119894minus1

10038161003816100381610038161003816

119903

)

10038161003816100381610038161003816119891 (119883119905119894minus1

)

10038161003816100381610038161003816

times sup119905119894minus1

le119905le119905119894

100381610038161003816100381610038161003816100381610038161003816

int

119905

119905119894minus1

119892 (119883119904minus

) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

= Φ21

(119899 120576) + Φ22

(119899 120576) + Φ23

(119899 120576)

(21)

Then

1003816100381610038161003816Ψ2

(119899 120576)1003816100381610038161003816

le 120576minus1

Φ21

(119899 120576) + 120576minus1

Φ22

(119899 120576)

+ 120576minus1

Φ23

(119899 120576)

= Ψ21

(119899 120576) + Ψ22

(119899 120576) + Ψ23

(119899 120576)

(22)

Using (13) in Lemma 1 conditions (A1) and (A

4) we get

Ψ21

(119899 120576) rarr1198750 as 119899 rarr infin 120576 rarr 0 and 119899120576 rarr infin (see (326)

in [24]) By using the same techniques under condition (A2)

we can prove thatΨ2119895

(119899 120576) rarr1198750 119895 = 2 3 as 119899 rarr infin 120576 rarr 0

respectively

Proposition 5 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 and 119899120576120572(120572minus1)

rarr infin one has

Ψ3

(119899 120576)

997904rArr 119887(int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

+119889119904)

1120572

1198801

minus 119887(int

1

0

10038161003816100381610038161003816119892 (1198830

119904)

10038161003816100381610038161003816

minus2120572

(119891 (1198830

119904) 119892 (119883

0

119904))

120572

minus119889119904)

1120572

1198802

(23)

Proof Under conditions (A1)ndash(A3) Proposition 5 can be

proved by using condition (A4) (see the proof of Proposition

44 in [24])

Proposition 6 Under conditions (A1)ndash(A4) as 119899 rarr infin

120576 rarr 0 one has

Ψ4

(119899 120576) 997904rArr 119886(int

1

0

10038161003816100381610038161003816119892minus2

(1198830

119904)

100381610038161003816100381610038161198912

(1198830

119904) 119889119904)

12

119873 (24)

Proof Note that

Ψ4

(119899 120576) = 119886

119899

sum

119894=1

119892minus2

(119883119905119894minus1

) 119891 (119883119905119894minus1

) int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

= 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

) int

119905119894

119905119894minus1

119892 (1198830

119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

)

times int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

)) 119889119861119904

+ 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) (119891 (119883119905119894minus1

) minus 119891 (1198830

119905119894minus1

))

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

(119892minus2

(119883119905119894minus1

) minus 119892minus2

(1198830

119905119894minus1

)) 119891 (1198830

119905119894minus1

)

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

+ 119886

119899

sum

119894=1

(119892minus2

(119883119905119894minus1

) minus 119892minus2

(1198830

119905119894minus1

))

times (119891 (119883119905119894minus1

) minus 119891 (1198830

119905119894minus1

))

times int

119905119894

119905119894minus1

119892 (119883119904minus

) 119889119861119904

=

5

sum

119895=1

Ψ4119895

(119899 120576)

(25)

For Ψ41

(119899 120576) let 119884119894

= int

119905119894

119905119894minus1

119892(119883119904minus

)119889119861119904 119895 = 1 119899 Then it is

easy to see that 119884119894

sim 119873(0 int

119905119894

119905119894minus1

1198922(119883119904minus

)119889119904) and 1198841 119884

119899are

independent normal random variablesIt follows that

Ψ41

(119899 120576)

= 119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

) 119891 (1198830

119905119894minus1

) 119884119894

sim 119873 (0 1198862

119899

sum

119894=1

119892minus4

(1198830

119905119894minus1

) 1198912

(1198830

119905119894minus1

) int

119905119894

119905119894minus1

1198922

(119883119904minus

) 119889119904)

997904rArr 119886(int

1

0

119892minus2

(1198830

119904) 1198912

(1198830

119904) 119889119904)

12

119873

(26)

as 119899 rarr infin 120576 rarr 0

Abstract and Applied Analysis 5

For Ψ42

(119899 120576) using Markov inequality and Itorsquos isometryproperty for any given 120578 gt 0

Ψ42

(119899 120576)

le

1

120578

E[119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

)) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

]

le

119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

))

2

119889119904]

12

le

119871119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [int

119905119894

119905119894minus1

(119883119904minus

minus 1198830

119904minus

)

2

119889119904]

12

le

119871119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [ sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119904minus

minus 1198830

119904minus

10038161003816100381610038161003816119899minus12

]

(27)

By using (13) Ψ42

(119899 120576) rarr 0 as 119899 rarr infin 120576 rarr 0Applying similar techniques to Ψ

4119895(119899 120576) 119895 = 3 4 5 we

get Ψ4119895

(119899 120576) rarr 0 119895 = 3 4 5 as 119899 rarr infin 120576 rarr 0

Now we can proveTheorem 2

Proof By using Propositions 3 4 5 6 and Slutskyrsquos theoremwe can get the conclusion

4 Example

We consider the following nonlinear SDE driven by generalLevy noises

119889119883119905

= 120579119883119905119889119905 +

120576

1 + 1198832

119905minus

119889119871119905 119905 isin [0 1] 119883

0= 1199090

(28)

where 119891(119909) = 119909 119892(119909) = 1(1 + 1199092) 1199090and 120576 are known

constants and 120579 = 0 is an unknown parameterFor simplicity let 119909

0gt 0 120576 = 0 we get the ODE

1198891198830

119905= 12057901198830

119905119889119905 119905 isin [0 1] 119883

0

0= 1199090

(29)

and the solution

1198830

119905= 11990901198901205790

119905 (30)

Then the asymptotic distribution is

119886(int

1

0

(1 + 1199092

011989021205790

119904)

2

1199092

011989021205790

119904119889119904)

minus12

119873

+ 119887

(int

1

0(1 + 119909

2

011989021205790

119904)

120572

119909120572

01198901205721205790

119904119889119904)

1120572

int

1

0(1 + 119909

2

011989021205790

119904)2

1199092

011989021205790

119904119889119904

119878120572

(1 120573 0)

(31)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R I Jennrich and P B Bright ldquoFitting systems of linear dif-ferential equations using computer generated exact derivativesrdquoTechnometrics vol 18 no 4 pp 385ndash392 1976

[2] R H Jones ldquoFitting multivariate models to unequally spaceddatardquo in Time Series Analysis of Irregularly Observed Data pp158ndash188 1984

[3] A R Bergstrom Statistical Inference in Continuous TimeEconomic Models vol 99 North-Holland Amsterdam TheNetherlands 1976

[4] A R Bergstrom ldquoThe history of continuous-time econometricmodelsrdquo Econometric Theory vol 4 no 3 pp 365ndash383 1988

[5] F Black and M Scholes ldquoThe pricing of options and corporateliabilitiesrdquoThe Journal of Political Economy vol 81 pp 637ndash6541973

[6] M Arato Linear Stochastic Systems with Constant CoefficientsSpringer Berlin Germany 1982

[7] R J Adler and P Meuller Stochastic Modelling on PhysicalOceanography vol 39 Springer New York NY USA 1996

[8] B P Rao B L P Rao I Statisticien B L P Rao B L P Rao andI Statistician Statistical Inference for di Usion Type ProcessesArnold London UK 1999

[9] Y A Kutoyants Statistical Inference for Ergodic Diffusion Pro-cesses Springer London UK 2004

[10] Yu A Kutoyants Parameter Estimation for Stochastic Processesvol 6 Heldermann Berlin Germany 1984

[11] Yu Kutoyants Identification of Dynamical Systems with SmallNoise Kluwer Academic Dordrecht The Netherlands 1994

[12] N Yoshida ldquoAsymptotic expansions of maximum likelihoodestimators for small diffusions via the theory of Malliavin-Watanaberdquo Probability Theory and Related Fields vol 92 no 3pp 275ndash311 1992

[13] N Yoshida ldquoConditional expansions and their applicationsrdquoStochastic Processes and their Applications vol 107 no 1 pp 53ndash81 2003

[14] M Uchida and N Yoshida ldquoInformation criteria for smalldiffusions via the theory of Malliavin-Watanaberdquo StatisticalInference for Stochastic Processes vol 7 no 1 pp 35ndash67 2004

[15] N Yoshida ldquoAsymptotic expansion of Bayes estimators for smalldiffusionsrdquo Probability Theory and Related Fields vol 95 no 4pp 429ndash450 1993

[16] A Takahashi ldquoAn asymptotic expansion approach to pricingfinancial contingent claimsrdquoAsia-Pacific FinancialMarkets vol6 no 2 pp 115ndash151 1999

6 Abstract and Applied Analysis

[17] N Kunitomo and A Takahashi ldquoThe asymptotic expansionapproach to the valuation of interest rate contingent claimsrdquoMathematical Finance vol 11 no 1 pp 117ndash151 2001

[18] A Takahashi and N Yoshida ldquoAn asymptotic expansionscheme for optimal investment problemsrdquo Statistical Inferencefor Stochastic Processes vol 7 no 2 pp 153ndash188 2004

[19] V Genon-Catalot ldquoMaximum contrast estimation for diffusionprocesses fromdiscrete observationsrdquo Statistics vol 21 no 1 pp99ndash116 1990

[20] A Gloter and M Soslashrensen ldquoEstimation for stochastic differ-ential equations with a small diffusion coefficientrdquo StochasticProcesses and their Applications vol 119 no 3 pp 679ndash6992009

[21] C F Laredo ldquoA sufficient condition for asymptotic sufficiencyof incomplete observations of a diffusion processrdquo The Annalsof Statistics vol 18 no 3 pp 1158ndash1171 1990

[22] M Uchida ldquoApproximate martingale estimating functions forstochastic differential equations with small noisesrdquo StochasticProcesses and their Applications vol 118 no 9 pp 1706ndash17212008

[23] M Soslashrensen and M Uchida ldquoSmall-diffusion asymptotics fordiscretely sampled stochastic differential equationsrdquo Bernoullivol 9 no 6 pp 1051ndash1069 2003

[24] H Long ldquoParameter estimation for a class of stochastic dif-ferential equations driven by small stable noises from discreteobservationsrdquo Acta Mathematica Scientia B vol 30 no 3 pp645ndash663 2010

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

Abstract and Applied Analysis 5

For Ψ42

(119899 120576) using Markov inequality and Itorsquos isometryproperty for any given 120578 gt 0

Ψ42

(119899 120576)

le

1

120578

E[119886

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

)) 119889119861119904

100381610038161003816100381610038161003816100381610038161003816

]

le

119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [int

119905119894

119905119894minus1

(119892 (119883119904minus

) minus 119892 (1198830

119904minus

))

2

119889119904]

12

le

119871119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [int

119905119894

119905119894minus1

(119883119904minus

minus 1198830

119904minus

)

2

119889119904]

12

le

119871119886

120578

119899

sum

119894=1

119892minus2

(1198830

119905119894minus1

)

10038161003816100381610038161003816119891 (1198830

119905119894minus1

)

10038161003816100381610038161003816

times [ sup119905119894minus1

le119905le119905119894

10038161003816100381610038161003816119883119904minus

minus 1198830

119904minus

10038161003816100381610038161003816119899minus12

]

(27)

By using (13) Ψ42

(119899 120576) rarr 0 as 119899 rarr infin 120576 rarr 0Applying similar techniques to Ψ

4119895(119899 120576) 119895 = 3 4 5 we

get Ψ4119895

(119899 120576) rarr 0 119895 = 3 4 5 as 119899 rarr infin 120576 rarr 0

Now we can proveTheorem 2

Proof By using Propositions 3 4 5 6 and Slutskyrsquos theoremwe can get the conclusion

4 Example

We consider the following nonlinear SDE driven by generalLevy noises

119889119883119905

= 120579119883119905119889119905 +

120576

1 + 1198832

119905minus

119889119871119905 119905 isin [0 1] 119883

0= 1199090

(28)

where 119891(119909) = 119909 119892(119909) = 1(1 + 1199092) 1199090and 120576 are known

constants and 120579 = 0 is an unknown parameterFor simplicity let 119909

0gt 0 120576 = 0 we get the ODE

1198891198830

119905= 12057901198830

119905119889119905 119905 isin [0 1] 119883

0

0= 1199090

(29)

and the solution

1198830

119905= 11990901198901205790

119905 (30)

Then the asymptotic distribution is

119886(int

1

0

(1 + 1199092

011989021205790

119904)

2

1199092

011989021205790

119904119889119904)

minus12

119873

+ 119887

(int

1

0(1 + 119909

2

011989021205790

119904)

120572

119909120572

01198901205721205790

119904119889119904)

1120572

int

1

0(1 + 119909

2

011989021205790

119904)2

1199092

011989021205790

119904119889119904

119878120572

(1 120573 0)

(31)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R I Jennrich and P B Bright ldquoFitting systems of linear dif-ferential equations using computer generated exact derivativesrdquoTechnometrics vol 18 no 4 pp 385ndash392 1976

[2] R H Jones ldquoFitting multivariate models to unequally spaceddatardquo in Time Series Analysis of Irregularly Observed Data pp158ndash188 1984

[3] A R Bergstrom Statistical Inference in Continuous TimeEconomic Models vol 99 North-Holland Amsterdam TheNetherlands 1976

[4] A R Bergstrom ldquoThe history of continuous-time econometricmodelsrdquo Econometric Theory vol 4 no 3 pp 365ndash383 1988

[5] F Black and M Scholes ldquoThe pricing of options and corporateliabilitiesrdquoThe Journal of Political Economy vol 81 pp 637ndash6541973

[6] M Arato Linear Stochastic Systems with Constant CoefficientsSpringer Berlin Germany 1982

[7] R J Adler and P Meuller Stochastic Modelling on PhysicalOceanography vol 39 Springer New York NY USA 1996

[8] B P Rao B L P Rao I Statisticien B L P Rao B L P Rao andI Statistician Statistical Inference for di Usion Type ProcessesArnold London UK 1999

[9] Y A Kutoyants Statistical Inference for Ergodic Diffusion Pro-cesses Springer London UK 2004

[10] Yu A Kutoyants Parameter Estimation for Stochastic Processesvol 6 Heldermann Berlin Germany 1984

[11] Yu Kutoyants Identification of Dynamical Systems with SmallNoise Kluwer Academic Dordrecht The Netherlands 1994

[12] N Yoshida ldquoAsymptotic expansions of maximum likelihoodestimators for small diffusions via the theory of Malliavin-Watanaberdquo Probability Theory and Related Fields vol 92 no 3pp 275ndash311 1992

[13] N Yoshida ldquoConditional expansions and their applicationsrdquoStochastic Processes and their Applications vol 107 no 1 pp 53ndash81 2003

[14] M Uchida and N Yoshida ldquoInformation criteria for smalldiffusions via the theory of Malliavin-Watanaberdquo StatisticalInference for Stochastic Processes vol 7 no 1 pp 35ndash67 2004

[15] N Yoshida ldquoAsymptotic expansion of Bayes estimators for smalldiffusionsrdquo Probability Theory and Related Fields vol 95 no 4pp 429ndash450 1993

[16] A Takahashi ldquoAn asymptotic expansion approach to pricingfinancial contingent claimsrdquoAsia-Pacific FinancialMarkets vol6 no 2 pp 115ndash151 1999

6 Abstract and Applied Analysis

[17] N Kunitomo and A Takahashi ldquoThe asymptotic expansionapproach to the valuation of interest rate contingent claimsrdquoMathematical Finance vol 11 no 1 pp 117ndash151 2001

[18] A Takahashi and N Yoshida ldquoAn asymptotic expansionscheme for optimal investment problemsrdquo Statistical Inferencefor Stochastic Processes vol 7 no 2 pp 153ndash188 2004

[19] V Genon-Catalot ldquoMaximum contrast estimation for diffusionprocesses fromdiscrete observationsrdquo Statistics vol 21 no 1 pp99ndash116 1990

[20] A Gloter and M Soslashrensen ldquoEstimation for stochastic differ-ential equations with a small diffusion coefficientrdquo StochasticProcesses and their Applications vol 119 no 3 pp 679ndash6992009

[21] C F Laredo ldquoA sufficient condition for asymptotic sufficiencyof incomplete observations of a diffusion processrdquo The Annalsof Statistics vol 18 no 3 pp 1158ndash1171 1990

[22] M Uchida ldquoApproximate martingale estimating functions forstochastic differential equations with small noisesrdquo StochasticProcesses and their Applications vol 118 no 9 pp 1706ndash17212008

[23] M Soslashrensen and M Uchida ldquoSmall-diffusion asymptotics fordiscretely sampled stochastic differential equationsrdquo Bernoullivol 9 no 6 pp 1051ndash1069 2003

[24] H Long ldquoParameter estimation for a class of stochastic dif-ferential equations driven by small stable noises from discreteobservationsrdquo Acta Mathematica Scientia B vol 30 no 3 pp645ndash663 2010

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

6 Abstract and Applied Analysis

[17] N Kunitomo and A Takahashi ldquoThe asymptotic expansionapproach to the valuation of interest rate contingent claimsrdquoMathematical Finance vol 11 no 1 pp 117ndash151 2001

[18] A Takahashi and N Yoshida ldquoAn asymptotic expansionscheme for optimal investment problemsrdquo Statistical Inferencefor Stochastic Processes vol 7 no 2 pp 153ndash188 2004

[19] V Genon-Catalot ldquoMaximum contrast estimation for diffusionprocesses fromdiscrete observationsrdquo Statistics vol 21 no 1 pp99ndash116 1990

[20] A Gloter and M Soslashrensen ldquoEstimation for stochastic differ-ential equations with a small diffusion coefficientrdquo StochasticProcesses and their Applications vol 119 no 3 pp 679ndash6992009

[21] C F Laredo ldquoA sufficient condition for asymptotic sufficiencyof incomplete observations of a diffusion processrdquo The Annalsof Statistics vol 18 no 3 pp 1158ndash1171 1990

[22] M Uchida ldquoApproximate martingale estimating functions forstochastic differential equations with small noisesrdquo StochasticProcesses and their Applications vol 118 no 9 pp 1706ndash17212008

[23] M Soslashrensen and M Uchida ldquoSmall-diffusion asymptotics fordiscretely sampled stochastic differential equationsrdquo Bernoullivol 9 no 6 pp 1051ndash1069 2003

[24] H Long ldquoParameter estimation for a class of stochastic dif-ferential equations driven by small stable noises from discreteobservationsrdquo Acta Mathematica Scientia B vol 30 no 3 pp645ndash663 2010

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