linear quadratic stochastic optimal control of forward...

Research ArticleLinear Quadratic Stochastic Optimal Control ofForward Backward Stochastic Control System Associated withLeacutevy Process

Hong Huang12 XiangrongWang1 Ting Hou3 and Lu Xu4

1 Institute of Financial Engineering College of Mathematics and Systems Science Shandong University of Science and TechnologyQingdao 266590 China2Institute of Financial Engineering Shandong Womenrsquos University Jinan 250300 China3College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China4School of Statistics and Management Shanghai University of Finance and Economics Shanghai 200433 China

Correspondence should be addressed to Xiangrong Wang xrwang2000126com

Received 5 April 2017 Accepted 17 August 2017 Published 25 September 2017

Academic Editor Zhongwei Lin

Copyright copy 2017 Hong Huang et al 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 analyzes one kind of linear quadratic (LQ) stochastic control problem of forward backward stochastic control systemassociated with Levy process We obtain the explicit form of the optimal control then prove it to be unique and get the linearfeedback regulator by introducing one kind of generalized Riccati equation Finally we discuss the solvability of the generalizedRiccati equation and its existence and uniqueness of the solutions are proved in a special case

1 Introduction

LQ stochastic optimal control is a kind of special optimalcontrol problem which not only can be used to model manylinear optimal problems practically but also can reasonablybe used to approach and solve many nonlinear problemsIn 1962 Kushner [1] firstly established a forward randomstochastic LQ model with a dynamic programming methodand Wonham [2] firstly studied a LQ stochastic optimalcontrol problem by introducing a Riccati equation in 1968Then a lot of works have been done for forward or backwardstochastic LQ control problems the corresponding Riccatiequation and its application in finance such as Li and Zhang[3] Ma and Hou [4] Liu et al [5] Wang et al [6] andShen and Wang [7] In 2003 Wang et al [8] discussed aspecial kind of forward backward stochastic LQ problemand got the existence and uniqueness of the optimal controlfor the control system Subsequently Wu [9] extended thisconclusion to the fully coupled forward backward stochasticLQ problem

Theoptimal control problemwith random jumpswas firstconsidered by Boel and Varaiya [10] in this case the control

system is often described by Brownian motion and Poissonprocesses On the basis of proving the existence and unique-ness of solutions of a kind of forward backward stochasticdifferential equation with Poisson jumps (FBSDEP) Wu andWang [11] got the explicit form of the optimal control forLQ stochastic control problem where the state variable wasdescribed by a stochastic differential equation with a Poissonprocess (SDEP) In 2009 Shi and Wu [12] extended Wu andWangrsquos results in [11] to a fully coupled LQ stochastic controlproblem of forward backward stochastic control system withPoisson jumps Moreover Lin and Zhang [13] considered the119867infin control problem for linear stochastic systems driven byboth Brownian motion and Poisson jumps In 2016 Li et al[14] studied a stochastic differential equations driven by G-Brownianmotion and got the existence and uniqueness of thesolution for these equations

In 2000 Nualart and Schoutens [15] introduced a classof Levy processes with exponential moments satisfyingsome conditions Using these exponential moments andthe standard orthogonalization process they constructeda series of orthogonal normal martingales called Teugelsmartingale And they also proved amartingale representation

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 2541687 11 pageshttpsdoiorg10115520172541687

2 Mathematical Problems in Engineering

theorem associated with Teugels martingale In the next yearNualart and Schoutens [16] considered a backward stochasticdifferential equation (BSDE) driven by Teugels martingaleand proved the existence and uniqueness theory of this BSDEIn 2003 Bahlali et al [17] studied a BSDE driven by Teugelsmartingale and an independent Brownian motion theygot the existence uniqueness and comparison of solutionsfor these equations having a Lipschitz or locally Lipschitzcoefficient El Otmani [18] considered a kind of generalizedBSDE (GBSDE) associated with Teugels martingale andBrownian motion associated with a pure jump-independentLevy process They got the existence and uniqueness theoryof this GBSDE when the coefficient verifies some conditionsof LipschitzMore results about BSDE associatedwithTeugelsmartingale can be found in the theses of El Otmani [19] Renand Fan [20] Tang and Zhang [21] and Huang and Wang[22] On the basis of these results in 2008 Mitsui and Tabata[23] studied a LQ regulation stochastic control problemwith Levy process and obtained the optimal control for thenonhomogeneous case In [24] Tang andWu considered thefollowing LQ stochastic control problem in a given finitehorizon [119904 119879] with Levy process

119889119909 (119905) = [119860 (120596 119905) 119909 (119905) + 119861 (120596 119905) 119906 (119905)] 119889119905+ infinsum119894=1

[119862119894 (120596 119905) 119909 (119905minus) + 119863119894 (120596 119905) 119906 (119905)] 119889119867119894 (119905)119909 (119904) = 120585

(1)

and the cost function was

119869 (119904 120585 119906 (sdot)) = 119864119904 [int119879119904(119909120591 (119905) 119876 (120596 119905) 119909 (119905)

+ 119906120591 (119905) 119877 (120596 119905) 119906 (119905)) 119889119905 + 119909120591 (119879)119867 (120596) 119909 (119879)] (2)

They show that the solvability of one kind of generalizedRiccati equation is sufficient to the well-posedness of this LQproblem and proved the existence of the optimal control

In this paper we consider one kind of LQ stochasticcontrol problem where the controlled system is driven by afully coupled linear forward backward stochastic differentialequation associated with Levy process (FBSDEL)

119889119909119905 = [119860 (120596 119905) 119909119905 + 119861 (120596 119905) 119906119905 minus 119871 (120596 119905) 119910119905] 119889119905+ [119862 (120596 119905) 119909119905 + 119863 (120596 119905) 119906119905] 119889119861119905+ infinsum119894=1

[119864119894 (120596 119905) 119909119905minus

+ 119865119894 (120596 119905) 119906119905] 119889119867119894119905minus 119889119910119905 = [119860120591 (120596 119905) 119910119905 + 119862120591 (120596 119905) 119911119905 + infinsum

119894=1

119864119894120591 (120596 119905) 119903119894119905+ 119877 (120596 119905) 119909119905]119889119905 minus 119911119905119889119861119905 minus infinsum

119894=1

119903119894119905119889119867119894119905

1199090 = 119886119910119879 = 119876 (120596) 119909119879(3)

where (119909119905 119910119905 119911119905 119903119894119905) areF119905-adapted stochastic processes tak-ing values in 119877119899 times 119877119899 times 119877119899 times 1198972(119877119899) and 119906(sdot) is F119905-adaptedstochastic process called admissible control process Assumethe control process set 119880 = 119877119896 and define the admissiblecontrol set as follows119880119886119889 = 119906 (sdot) isin 1198722 (0 119879 119877119896) 119906119905 isin 119880 0 le 119905

le 119879 ae as (4)

The cost functional we considered is

119869 (119906) = 12119864 [int1198790 (⟨119877 (120596 119905) 119909119905 119909119905⟩ + ⟨119873 (120596 119905) 119906119905 119906119905⟩+ ⟨119871 (120596 119905) 119910119905 119910119905⟩) 119889119905 + ⟨119876 (120596) 119909119879 119909119879⟩] (5)

And the optimal control problem is to find 119906119905 isin 119880119886119889 suchthat 119869 (119906 (sdot)) = inf

119906(sdot)119869 (119906 (sdot)) (6)

Note that (3) is a fully coupled FBSDEL In 2012Pereira and Shamarova [25] firstly considered this kind ofFBSDEL obtained a solution to this FBSDEL via a partialintegrodifferential equation and proved the uniquenessUnder some monotonicity assumptions Baghery et al [26]proved the existence and uniqueness of solutions of fullycoupled FBSDEL and then obtained the existence of an open-loop Nash equilibrium point for nonzero sum stochasticdifferential games by using this result Based on [25] Wangand Huang [27] got the maximum principle for forwardbackward stochastic control system driven by Levy processthen they discussed a kind of LQ stochastic control problemof forward backward stochastic control system and got anecessary condition for the optimal control

We extend the result of Shi and Wu [12] to the fullycoupled linear forward backward stochastic control systemdriven by Brownian motion and an independent Teugelsmartingale Since Teugels martingale is more complex thanthe Poisson process we also needmore general formula aboutcadlag semimartingale The rest of this paper is organized asfollows In Section 2 we provide a list of notations and resultsof the existence and uniqueness of solutions of fully coupledFBSDEL In Section 3 we prove the existence and uniquenessof the optimal control of LQ stochastic control problem(6) and give the linear feedback regulator for the optimalcontrol by the solution of a kind of generalizedmatrix-valuedRiccati equation when assuming the coefficient matrices aredeterministic In Section 4 the solvability of this kind ofmatrix-valued Riccati equation is discussed

2 Preliminaries and Notations

Let (ΩF119905 119875) be a complete probability space satisfying theusual conditionsF119905 is a right continuous increasing family of

Mathematical Problems in Engineering 3

complete sub-120590-algebra which is generated by the followingtwo mutually independent processes a one-dimensionalstandard Brownian motion 1198611199050le119905le119879 and an 119877-valued Levyprocess 119871 1199050le119905le119879 with a standard Levy measure ] satisfy

(i) int119877(1 and 1199092)](119889119909) lt infin

(ii) int(minus120576120576)119888

119890120582|119909|](119889119909) lt infin for every 120576 gt 0 and for some120582 gt 0Naluart and Schoutens denoted Teugels martingale asso-

ciated with the Levy process 119871 1199050le119905le119879 by 119867119894119905infin119894=1 and 119867119894119905 isgiven by

119867119894119905 = 119888119894119894119884119894119905 + 119888119894119894minus1119884119894minus1119905 + 119888119894119894minus2119884119894minus2119905 + sdot sdot sdot + 11988811989411198841119905 (7)

where 119884119894119905 = 119871119894119905 minus 119864[119871119894119905] is the compensated power-jumpprocess of order 119894 and 119871119894119905 is power-jump processes

119871119894119905 = 119871 119905 119894 = 1sum0lt119904le119905

(Δ119871 119904)119894 119894 ge 2 (8)

Coefficients 119888119894119896 correspond to orthonormalization of thepolynomials 1 119909 1199092 with respect the measure 120583(119889119909) =120592(119889119909) + 12059021205750(119889119909) Please refer to Naluart and Schoutens [15]for more details about Teugels martingale

Introduce the following notations adopted in this paper

⟨119860 119861⟩ = tr(119860119861119879) the inner product in 119877119899times119898 forall119860 119861 isin119877119899times119898|120572| = radic⟨120572 120572⟩ the norm in 119877119899 forall120572 isin 1198771198991198712(Ω119867) the space of 119867-valued F119879-measurablerandom variable 120585 satisfies 119864|120585|2 lt infin1198722(0 119879119867) the space of 119867-valued F119905-measurableprocess 120601(sdot) = 120601(119905 120596) (119905 120596) isin [0 119879] times Ω satisfies119864int1198790|120601119905|2119889119905 lt infin1198972(119867) the space of 119867-valued 119891119894119894ge1 satisfiessuminfin119894=1 |119891119894|2 lt infin1198972(0 119879119867) the space of 1198972(119867)-valuedF119905-measurable

processes satisfies 119864int1198790suminfin119894=1 |119891119894119905 |2119889119905 lt infin1198782(0 119879119867) the space of 119867-valued F119905-measurable

cadlag process 119891(sdot) = 119891(119905 120596) (119905 120596) isin [0 119879] times Ωsatisfies 119864 sup0le119905le119879|119891119905|2119889119905 lt infin

For notational brevity we set

1198722 (0 119879) = 1198722 (0 119879 119877119899) times 1198722 (0 119879 119877119899)times 1198722 (0 119879 119877119899) times 1198972 (0 119879 119877119899) (9)

Next consider the following fully coupled FBSDEL

119889119909119905 = 119887 (119905 119909119905 119910119905 119911119905 119903119905) 119889119905 + 119889sum119894=1

120590119894 (119905 119909119905 119910119905 119911119905 119903119905) 119889119861119894119905+ infinsum119894=1

119892119894 (119905 119909119905minus 119910119905minus 119911119905 119903119905) 119889119867119894119905minus119889119910119905 = 119891 (119905 119909119905 119910119905 119911119905 119903119905) 119889119905 minus 119889sum

119894=1

119911119894119905119889119861119894119905 minus infinsum119894=1

1199031198941199051198891198671198941199051199090 = 119886119910119879 = Φ (119909119879)

(10)

where 119887 Ω times [0 119879] times 119877119899 times 119877119898 times 119877119898times119889 times 1198972(119877119898) rarr 119877119899 120590 Ωtimes [0 119879] times119877119899 times119877119898 times119877119898times119889 times 1198972(119877119898) rarr 119877119899times119889 119892 Ωtimes [0 119879] times119877119899 times119877119898 times119877119898times119889 times 1198972(119877119898) rarr 1198972(119877119899) 119891 Ωtimes [0 119879] times119877119899 times119877119898 times119877119898times119889 times 1198972(119877119898) rarr 119877119898For a given119898 times 119899 full rank matrix 119866 set

120582 = (119909119910119911)

119860 (119905 120582 119903) = (minus119866120591119891 (119905 120582 119903)119866119887 (119905 120582 119903)119866120590 (119905 120582 119903) ) (11)

Assumption 1 (i) 119887 120590 119892 and 119891 are uniformly Lipschitzcontinuous with respect to (119909 119910 119911 119903)

(ii) For each (120596 119905) isin Ωtimes[0 119879] 119897(120596 119905 0 0 0 0) isin 1198722(0 119879)and 119892(120596 119905 0 0 0 0) isin 1198672(1198972) where 119897 = 119887 120590 119891 respectively

(iii) Φ(sdot) is uniformly Lipschitz continuous with respectto 119909 and forall119909 Φ(119909) isin 1198712(Ω 119865119879 119875)Assumption 2

⟨119860 (119905 1205821 1199031) minus 119860 (119905 1205822 1199032) 1205821 minus 1205822⟩ + infinsum119894=1

⟨119866119892119894 119903119894⟩le minus1205731 |119866119909|2minus 1205732(100381610038161003816100381611986612059111991010038161003816100381610038162 + 100381610038161003816100381611986612059110038161003816100381610038162 + infinsum

119894=1

1003817100381710038171003817100381710038171198661205911199031198941003817100381710038171003817100381710038172) ⟨Φ (1199091) minus Φ (1199092) 119866 (1199091 minus 1199092)⟩ ge 1205831 |119866119909|2

(12)

where 1205821 = (1199091 1199101 1199111) 1205822 = (1199092 1199102 1199112) 119909 = 1199091 minus 1199092 119910 =1199101minus1199102 = 1199111minus1199112 119892119894 = 119892119894(119905 1205821 1199031)minus119892119894(119905 1205822 1199032) 119903119894 = 1199031198941minus1199031198942and 12057311205732 1205831 are nonnegative constants with 1205731 + 1205732 gt 0 and1205732 + 1205831 gt 0 Moreover we have 1205731 gt 0 1205831 gt 0 (resp 1205732 gt 0)when119898 gt 119899 (resp 119899 gt 119898)

Lemma 3 (existence and uniqueness theorem of FBSDEL[25]) Under Assumptions 1 and 2 FBSDEL (10) admits aunique solution in1198722(0 119879)


In the following sections we also need the more generalItorsquos formula about a cadlag semimartingales

Lemma 4 (Itorsquos formula [27]) Let 119883 = 119883119905 119905 isin [0 119879] becadlag semimartingales denote [119883] = [119883]119905 119905 isin [0 119879] as thequadratic variation process119865 is aC2 real valued function then119865(119883) is also a semimartingales and the following Itorsquos formulaholds

119865 (119883119905) = 119865 (1198830) + int11990501198651015840 (119883119904minus) 119889119883119904

+ 12 int1199050 11986510158401015840 (119883119904) 119889 [119883]C119904+ sum0lt119904le119905

119865 (119883119904) minus 119865 (119883119904minus) minus 1198651015840 (119883119904minus) Δ119883119904 (13)

where [119883]C is the continuous part of [119883]In particular when 119865(119883) = 1198832 and 119865(119883) = 119883119905119884119905 where119883119884 are two cadlag semimartingales we get

1198832119905 = 11988320 + int11990502119883119904minus119889119883119904 + int119905

0119889 [119883]119904

119883119905119884119905 = 11988301198840 + int1199050119883119904minus119889119884119904 + int119905

0119884119904minus119889119883119904

+ int1199050119889 [119883 119884]119904

(14)

Here [119883 119884] is the quadratic covariation of 1198831198843 Linear Quadratic Stochastic OptimalControl Problem

Let us consider the LQ stochastic optimal control problem(6) First of all we give some necessary explanations for thecoefficients in the system119860(120596 119905) 119862(120596 119905) 119864119894(120596 119905) (119894 = 1 2 3 ) isin 119877119899times119899119861(120596 119905) 119863(120596 119905) and 119865119894(120596 119905) (119894 = 1 2 3 ) isin 119877119899times119896 areall bounded progressively measurable matrix-valued pro-cesses 119877(120596 119905) 119871(120596 119905) isin 119877119899times119899 are nonnegative symmetricbounded progressively measurable matrix-valued processesand 119873(120596 119905) is a positive bounded 119896 times 119896 progressivelymeasurable matrix-valued process the inverse is 119873minus1(120596 119905)which is also bounded 119876(120596) is a F119905-adapted nonnegativesymmetric bounded matrix-valued random variable

For a given admissible control 119906(sdot) isin 119880119886119889 underassumptions of the coefficients above we can verify thatFBSDEL (3) satisfies Assumptions 1 and 2 Therefore thereexists a unique solution (119909119906119905 119910119906119905 119911119906119905 119903119906119905 ) isin 1198722(0 119879) satisfyingthe control system (3) from Lemma 3

Then we get the explicit form of the optimal control 119906119905 forthe LQ stochastic optimal control problem (6)

Theorem 5 There exists a unique optimal control 119906119905 for LQstochastic optimal control problem (6) and 119906119905 is given by thefollowing equation

119906119905 = minus119873minus1 (120596 119905)sdot (119861120591 (120596 119905) 119910119905 + 119863120591 (120596 119905) 119911119905 + infinsum

119894=1

119865119894120591 (120596 119905) 119903119894119905) (15)

Proof As we know for a given admissible control 119906119905 thecontrol system (15) has a unique solution (119909119905 119910119905 119911119905 119903119905) isin1198722(0 119879)Existence For any admissible control V119905 assume the corre-sponding trajectory is (119909V119905 119910V119905 119911V119905 119903V119905 ) isin 1198722(0 119879) then

119869 (V119905) minus 119869 (119906119905) = 12sdot 119864 [int119879

0(⟨119877 (120596 119905) (119909V119905 minus 119909119905) 119909V119905 minus 119909119905⟩

+ ⟨2119877 (120596 119905) 119909119905 119909V119905 minus 119909119905⟩+ ⟨119873 (120596 119905) (V119905 minus 119906119905) V119905 minus 119906119905⟩+ ⟨2119873 (120596 119905) 119906119905 V119905 minus 119906119905⟩+ ⟨119871 (120596 119905) (119910V119905 minus 119910119905) 119910V119905 minus 119910119905⟩+ ⟨2119871 (120596 119905) 119910119905 119910V119905 minus 119910119905⟩) 119889119905+ ⟨119876 (120596) (119909V119879 minus 119909119879) 119909V119879 minus 119909119879⟩ + ⟨2119876 (120596) 119909119879 119909V119879minus 119909119879⟩]

(16)

Applying Itorsquos formula to ⟨119909V119905 minus 119909119905 119910119905⟩ we have119864 ⟨119909V119879 minus 119909119879 119910119879⟩ = 119864int119879

0(⟨minus119877 (120596 119905) 119909119905 119909V119905 minus 119909119905⟩

+ ⟨119861120591 (120596 119905) 119910119905 V119905 minus 119906119905⟩ minus ⟨119871 (120596 119905) 119910119905 119910V119905 minus 119910119905⟩+ ⟨119863120591 (120596 119905) 119911119905 V119905 minus 119906119905⟩+ infinsum119894=1

⟨119865119894120591 (120596 119905) 119903119894119905 V119905 minus 119906119905⟩)119889119905(17)

Since 119877(120596 119905) 119871(120596 119905) and 119876(120596) are nonnegative and 119873(120596 119905)is positive we can get

119869 (V119905) minus 119869 (119906119905)ge 119864int1198790[⟨(119861120591 (120596 119905) 119910119905 + 119863120591 (120596 119905) 119911119905 + infinsum

119894=1

119865119894120591 (120596 119905) 119903119894119905)


V119905 minus 119906119905⟩+ ⟨119873 (120596 119905) 119906119905 V119905 minus 119906119905⟩] = 119864int1198790⟨(119861120591 (120596 119905) 119910119905

+ 119863120591 (120596 119905) 119911119905 + infinsum119894=1

119865119894120591 (120596 119905) 119903119894119905) V119905 minus 119906119905⟩minus⟨119873(120596 119905)sdot 119873minus1 (120596 119905) (119861120591 (120596 119905) 119910119905 + 119863120591 (120596 119905) 119911119905 + infinsum

119894=1

119865119894120591 (120596 119905) 119903119894119905) V119905 minus 119906119905⟩ = 0

(18)Then the admissible control 119906119905 defined by (15) is the

optimal control of LQ stochastic control problem (6)

Unique Assume admissible control 1199061119905 is an optimal controlthe corresponding trajectories are (1199091119905 1199101119905 1199111119905 1199031119905 ) and 1199062119905 isanother optimal control the corresponding trajectories are(1199092119905 1199102119905 1199112119905 1199032119905 ) So the trajectories corresponding to (1199061119905+1199062119905 )2are

(1199091119905 + 11990921199052 1199101119905 + 11991021199052 1199111119905 + 11991121199052 1199031119905 + 11990321199052 ) (19)

and the trajectories corresponding to (1199061119905 minus 1199062119905 )2 are(1199091119905 minus 11990921199052 1199101119905 minus 11991021199052 1199111119905 minus 11991121199052 1199031119905 minus 11990321199052 ) (20)

Since 1199061119905 and 1199062119905 are both optimal controls 119873(120596 119905) ispositive and 119877(120596 119905) 119871(120596 119905) 119876(120596) are nonnegative we have119869 (1199061119905 ) = 119869 (1199062119905 ) = 120572 ge 02120572 = 119869 (1199061119905 ) + 119869 (1199062119905 ) = 2119869(1199061119905 + 11990621199052 )

+ 119864int1198790(⟨119877 (120596 119905) 1199091119905 minus 11990921199052 1199091119905 minus 11990921199052 ⟩

+⟨119873(120596 119905) 1199061119905 minus 11990621199052 1199061119905 minus 11990621199052 ⟩+⟨119871 (120596 119905) 1199101119905 minus 11991021199052 1199101119905 minus 11991021199052 ⟩)119889119905+ 119864⟨119876 (120596) 1199091119879 minus 11990921198792 1199091119879 minus 11990921198792 ⟩ ge 2119869(1199061119905 + 11990621199052 )+ 119864int1198790⟨119873(120596 119905) 1199061119905 minus 11990621199052 1199061119905 minus 11990621199052 ⟩119889119905 ge 2120572

+ 1205752119864int1198790 100381610038161003816100381610038161199061119905 minus 1199062119905 100381610038161003816100381610038162 119889119905

(21)

Here 120575 is a constant and 120575 gt 0 then119864int1198790

100381610038161003816100381610038161199061119905 minus 1199062119905 100381610038161003816100381610038162 119889119905 le 0 (22)

hence 1199061119905 = 1199062119905 in1198722(0 119879 119877119896)

Assume 119860(120596 119905) 119861(120596 119905) 119862(120596 119905) 119863(120596 119905) 119864(120596 119905) 119865(120596 119905)119877(120596 119905)119873(120596 119905) 119871(120596 119905) and 119876(120596) are all deterministicmatrices denoted as 119860 119905 119861119905 119862119905 119863119905 119864119905 119865119905 119877119905 119873119905 119871 119905 and 119876for convenience Introducing the following generalized 119899 times 119899matrix-valued Riccati equation (23) 119905 isin [0 119879] 119894 = 1 2

minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1

(119864119894119905)120591 119884119894119905minus 119870119905 (119871120591119905 + 119861119905119873minus1119905 119861120591119905 )119870119905 minus 119870119905119861119905119873minus1119905 119863120591119905119872119905minus infinsum119894=1

119870119905119861119905119873minus1119905 (119865119894119905)120591 119884119894119905 + 119877119905119872119905 = 119870119905119862119905 minus 119870119905119863119905119873minus1119905 119861120591119905119870119905 minus 119870119905119863119905119873minus1119905 119863119905119872119905

minus infinsum119894=1

119870119905119863119905119873minus1119905 (119865119894119905)120591 119884119894119905119884119894119905 = 119870119905119864119894119905 minus 119870119905119865119894119905119873minus1119905 119861120591119905119870119905 minus 119870119905119865119894119905119873minus1119905 119863120591119905119872119905


119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(23)

Then we can get the following conclusions

Theorem 6 Suppose the generalized matrix-valued Riccatiequation (23) has solution (119870119905119872119905 119884119894119905 ) for all 119905 isin [0 119879] thenthe optimal linear feedback regulator for LQ stochastic optimalcontrol problem (6) is

119906119905 = minus119873minus1119905 [119861120591119905119870119905 + 119863120591119905119872119905 + infinsum119894=1

(119865119894119905)120591 119884119894119905]119909119905 (24)

and the optimal value function is

119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (25)

Proof If (119870119905119872119905 119884119894119905 ) is the solution of the matrix-valuedRiccati equation (23) then we can check that the solution of(6) (119909119905 119910119905 119911119905 119903119894119905) satisfies

119910119905 = 119870119905119909119905119911119905 = 119872119905119909119905119903119894119905 = 119884119894119905119909119905(26)

As we have proved that the optimal control has the formof (15) take (26) into (15) then the optimal control can bewritten by


(119865119894119905)120591 119884119894119905]119909119905 (27)


For the optimal value function using Itorsquos formula to⟨119909119905 119910119905⟩ then119864int1198790⟨119877119905119909119905 119909119905⟩ 119889119905 + 119864int119879

0⟨119871 119905119910119905 119910119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩

minus ⟨119870119886 119886⟩= 119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905(28)

On the other hand from the relationship of 119906 and(119909119905 119910119905 119911119905 119903119894119905) we can verify that

119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905 = minus119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 (29)

and then

119864int1198790⟨119877119905119909119905 119909119905⟩ 119889119905 + 119864int119879

0⟨119871 119905119910119905 119910119905⟩ 119889119905

+ 119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩ = ⟨119870119886 119886⟩ (30)

By the definition of cost function 119869(sdot) (5) we prove that theoptimal value function is

119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (31)

Now consider a special case of stochastic LQ controlproblem when 119871(120596 119905) = 0 and the control system is reducedto 119889119909119905 = (119860 (120596 119905) 119909119905 + 119861 (120596 119905) 119906119905) 119889119905+ (119862 (120596 119905) 119909119905 + 119863 (120596 119905) 119906119905) 119889119861119905

+ infinsum119894=1

(119864119894 (120596 119905) 119909119905minus

+ 119865119894 (120596 119905) 119906119905) 1198891198671198941199051199090 = 119886

(32)

The cost functional now is

119869 (119906) = 12119864 [int1198790 ⟨119877 (120596 119905) 119909119905 119909119905⟩ + ⟨119873 (120596 119905) 119906119905 119906119905⟩+ ⟨119876 (120596) 119909119879 119909119879⟩] (33)

Remark 7 Comparing the LQ stochastic optimal controlsystem (32) and control system (1) which was considered in[22] by Tang and Wu we know that control system (1) is aspecial case of control system (32) when 119862(120596 119905) = 119863(120596 119905) =0

We can get the following Corollary 8 easily from Theo-rem 5

Corollary 8 There exists a unique optimal control for LQstochastic optimal control problem (32)-(33) and

119906119905 = minus119873minus1 (120596 119905)sdot [119861120591 (120596 119905) 119910119905 + 119863120591 (120596 119905) 119911119905 + infinsum

119894=1

119865119894120591 (120596 119905) 119903119894119905] (34)

where the (119910119905 119911119905 119903119905) is the solution of the following BSDEdrivenby Levy process

minus 119889119910119905 = [119860120591 (120596 119905) 119910119905 + 119862120591 (120596 119905) 119911119905 + infinsum119894=1


119894=1

119903119894119905119889119867119894119905119910119879 = 119876 (120596) 119909119879

(35)

Assume 119860(120596 119905) 119861(120596 119905) 119862(120596 119905) 119863(120596 119905) 119864(120596 119905) 119865(120596 119905)119877(120596 119905) 119873(120596 119905) and 119876(120596) are all deterministic then Riccatiequation (23) changes tominus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905

+ infinsum119894=1

(119864119894119905)120591 119884119894119905 minus 119870119905119861119905119873minus1119905 119861120591119905119870119905minus 119870119905119861119905119873minus1119905 119863120591119905119872119905 minus infinsum

119894=1

119870119905119861119905119873minus1119905 (119865119894119905)120591 119884119894119905+ 119877119905119872119905 = 119870119905119862119905 minus 119870119905119863119905119873minus1119905 119861120591119905119870119905 minus 119870119905119863119905119873minus1119905 119863119905119872119905minus infinsum119894=1

119870119905119863119905119873minus1119905 (119865119894119905)120591 119884119894119905119884119894119905 = 119870119905119864119894119905 minus 119870119905119865i

119905119873minus1119905 119861120591119905119870119905 minus 119870119905119865119894119905119873minus1119905 119863120591119905119872119905minus infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(36)

Then fromTheorem 6 we can get Corollary 9

Corollary 9 For LQ stochastic optimal control problem (32)-(33) if for all 119905 isin [0 119879] there exist matrices (119870119905119872119905 119884119894119905 )satisfying (36) then the optimal linear feedback regulator is


(119865119894119905)120591 119884119894119905]119909119905 (37)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (38)


4 Solvability of the GeneralizedRiccati Equation

From the discussion of the previous section we can see thatthe key to get the optimal linear feedback regulator for LQstochastic optimal control problem is the solvability of thegeneralized Riccati equation (23) But (23) is so complicatedthat we cannot prove its existence and uniqueness at thismoment Using technique introduced by Shi and Wu [12]we only discuss a special case 119863119905 = 0 in this case Riccatiequation (23) becomes

minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1

(119864119894119905)120591 119884119894119905minus 119870119905 (119871120591119905 + 119861119905119873minus1119905 119861120591119905 )119870119905minus infinsum119894=1

119870119905119861119905119873minus1119905 (119865119894119905)120591 119884119894119905 + 119877119905119884119894119905 = 119870119905119864119894119905 minus 119870119905119865119894119905119873minus1119905 119861120591119905119870119905 minus infinsum

119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905 119872119905 = 119870119905119862119905119870119879 = 119876 119894 = 1 2 3

(39)

Equivalently consider the following equation

minus 119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119870119905119862119905 minus 119870119905 (119871120591119905 + 119861119905119873minus1119905 119861120591119905 )sdot 119870119905 + 119877119905 + infinsum

119894=1

(119864119894119905)120591 [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1sdot [119870119905119864119894119905 minus 119870119905119865119894119905119873minus1119905 119861120591119905119870119905] minus infinsum

119894=1

119870119905119861119905119873minus1119905 (119865119894119905)120591sdot [119868119899 + infinsum

119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1119870119905119864119894119905+ infinsum119894=1

119870119905119861119905119873minus1119905 (119865119894119905)120591 [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1sdot 119870119905119865119894119905119873minus1119905 119861120591119905119870119905119870119879 = 119876

119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(40)

Compare (39) and (40) we can find that if we can prove119870119905 the solution of (40) then119872119905 = 119870119905119862119905119884119894119905 = [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 [119870119905119864119894119905 minus 119870119905119865119894119905119873minus1119905 119861120591119905119870119905] (41)

is the solution of the Riccati equation (39)

In the following we will focus on the existence anduniqueness of solutions of (40) Firstly let 119878119899+ denote the spaceof all 119899times119899 nonnegative symmetric matrices and119862([0 119879] 119878119899+)is a Banach space of 119878119899+-valued continuous functions on [0 119879]We have the following uniqueness result

Theorem 10 The Riccati equation (40) admits at most onesolution 119870119905 isin 119862[0 119879 119878119899+]Proof Suppose 119905 isin 119862[0 119879 119878119899+] satisfying 119868119899 +suminfin119894=1 119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0 is another solution of (40) Let119905 = 119870119905 minus 119905 then

minus 119870119905= 119860120591119905119905 + 119905119860 119905 + 119862120591119905 119905119862119905minus 119905 (119871120591119905 + 119861119905119873minus1119905 119861120591119905 )119870119905minus 119905 (119871120591119905 + 119861119905119873minus1119905 119861120591119905 ) 119905 + 1198681 + 1198682 + 1198683 + 1198684

119879 = 0119868119899 + infinsum119894=1

119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0 (119894 = 1 2 3 )

(42)

where

1198681 = infinsum119894=1

[(119864119894119905)120591 [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119864119894119905]minus infinsum119894=1

[(119864119894119905)120591 [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119865119894119905119873minus1119905 (119865119894119905)120591sdot [119868119899 + 119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119864119894119905]

1198682 = infinsum119894=1

[119905119861119905119873minus1119905 (119865119894119905)120591 [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1sdot 119870119905119865119894119905119873minus1119905 119861120591119905119870119905] + infinsum

119894=1

[119905119861119905119873minus1119905 (119865119894119905)120591sdot [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119865119894119905119873minus1119905 119861120591119905119870119905]+ infinsum119894=1

[119905119861119905119873minus1119905 (119865119894119905)120591 [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1sdot 119905119865119894119905119873minus1119905 119861120591119905 119905] minus infinsum

119894=1

[119905119861119905119873minus1119905 (119865119894119905)120591sdot [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119865119894119905119873minus1119905 (119865119894119905)120591sdot [119868119899 + 119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119865119894119905119873minus1119905 119861120591119905 119905]


1198683 = minusinfinsum119894=1

[(119864119894119905)120591 [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1sdot 119905119865119894119905119873minus1119905 119861120591119905119870119905] minus infinsum

119894=1

[(119864119894119905)120591sdot [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119865119894119905119873minus1119905 119861120591119905 119905]minus infinsum119894=1

[119905119861119905119873minus1119905 (119865119894119905)120591 [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1sdot 119870119905119864119894119905] minus infinsum

119894=1

[119905119861119905119873minus1119905 (119865119894119905)120591sdot [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119864119894119905] + infinsum

119894=1

[(119864119894119905)120591sdot [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119865119894119905119873minus1119905 (119865119894119905)120591sdot [119868119899 + 119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119865119894119905119873minus1119905 119861120591119905 119905]+ infinsum119894=1

[119905119861119905119873minus1119905 (119865119894119905)120591 [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1sdot 119905119865119894119905119873minus1119905 (119865119894119905)120591 [119868119899 + 119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119864119894119905]



119894=1

[119905119861119905119873minus1119905 (119865119894119905)120591sdot [119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 119905119864119894119905]+ infinsum119894=1


(43)[119868119899 + 119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 and [119868119899 + 119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1 areuniformly bounded as they are continuously in [0 119879] applyGronwallrsquos inequality we can get for all 119905 isin [0 119879] 119905 = 0Then we prove the uniqueness of solution

For the existence part first of all if we let

Φ119905 = Λ (119870) = [119868119899 + infinsum119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870 (44)

then from the conventional Riccati equation theory forall Φ119905 isin 119862([0 119879] 119878119899+) the following conventional Riccatiequation

minus 119905 = [119860 119905 minus infinsum119894=1

119861119905119873minus1119905 (119865119894119905)120591Φ119905119864119894119905]120591119870119905 + 119870119905 [119860 119905


119861119905119873minus1119905 (119865119894119905)120591Φ119905119864119894119905] minus 119870119905 [119871120591119905 + 119861119905119873minus1119905 119861120591119905minus infinsum119894=1

119861119905119873minus1119905 (119865119894119905)120591Φ119905119865119894119905119873minus1119905 119861120591119905]119870119905 + 119862120591119905119870119905119862119905+ infinsum119894=1

(119864119894119905)120591Φ119905119864119894119905 + 119877119905119870119879 = 119876119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(45)

has a unique solution119870(sdot) isin 119862([0 119879] 119878119899+) when[119871120591119905 + 119861119905119873minus1119905 119861120591119905 minus infinsum

119894=1

[119861119905119873minus1119905 (119865119894119905)120591Φ119865119894119905119873minus1119905 119861119905]]isin 119862 ([0 119879] 119878119899+)

(46)

Let 119878119899119904 be the subspace of 119878119899+ which is formed by thesymmetric matrices satisfying (46) Obviously as 119870119905 equiv 0 isin119878119899119904 the definition of 119878119899119904 is reasonable Define a mapping Ψ 119862([0 119879] 119878119899119904 ) rarr 119862([0 119879] 119878119899+) we can get Lemma 11 about Φand ΨLemma 11 The operators Φ = Λ(119870) are monotonouslyincreasing when 119870 gt 0 and the operator Ψ is continuous andmonotonously increasing

Proof When 119870 gt 0 from the definition of Λ(119870) we haveΛ (119870) = [119868119899 + infinsum

119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870= [119870minus1(119868119899 + infinsum

119894=1

119870119865119894119873minus1 (119865119894)120591)]minus1

= [119870minus1 + infinsum119894=1

119865119894119873minus1 (119865119894)120591]minus1 (47)

So if 1198701 ge 1198702 then Λ(1198701) ge Λ(1198702) that is Λ(119870) ismonotonously increasing when 119870 gt 0

As 119870 = Ψ(Φ) set 119870 = Ψ(Φ) then the conventionalRiccati equation (45) can be rewritten

minus 119905= 119860120591119905119870119905 + 119870119905119860 119905 minus 119870119905 [119871120591119905 + 119861119905119873minus1119905 119861120591119905 ]119870119905 + 119877119905


+ 119862120591119905119870119905119862119905+ infinsum119894=1

[[119864119894119905 minus 119865119894119905119873minus1119905 119861120591119905119870119905]120591Φ119905 [119864119894119905 minus 119865119894119905119873minus1119905 119861120591119905119870119905]]119870119879 = 119876 (119894 = 1 2 3 )

(48)

From the conclusion of Λ(119870) above in this lemma andLemma 82 in [28] if Φ ge Φ then 119870 ge 119870 the operator Ψ ismonotonously increasing On the other hand by Gronwallrsquosinequality we know that if Φ rarr Φ then 119870 minus 119870 rarr 0 so theoperator Ψ is also continuous

For (45) it is easy to know that if there exists Φ(sdot) isin119862([0 119879] 119878119899+) satisfyingΦ = [119868119899 + infinsum

119894=1

Ψ (Φ) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φ) (49)

then Riccati equation (40) admits a unique solution So thefollowing task is to find the suitable Φ(sdot) isin 119862([0 119879] 119878119899+)satisfying (49) We need the following lemma

Lemma 12 If there exist Φ+Φminus isin 119862([0 119879] 119878119899119904 ) which satisfyΦ+ ge [119868119899 + infinsum

119894=1

Ψ (Φ+) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φ+)ge [119868119899 + infinsum

119894=1

Ψ (Φminus) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φminus) ge Φminus(50)

then Riccati equation (40) admits a solution 119870(sdot) isin 119862([0 119879]119878119899+)Proof For given Φ+ Φminus which satisfied (49) define thesequences Φ+119895 Φminus119895 119870+119895 119870minus119895 as followsΦ+0 = Φ+ isin 119878119899119904 Φminus0 = Φminus isin 119878119899119904 119870+0 = Ψ (Φ+0 ) 119870minus0 = Ψ (Φminus0 ) Φ+119895+1 = [119868119899 + infinsum

119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870+119895 Φminus119895+1 = [119868119899 + infinsum

119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870minus119895 119870+119895+1 = Ψ (Φ+119895+1) 119870minus119895+1 = Ψ (Φminus119895+1) 119895 = 1 2 3

(51)

From (50) and Lemma 11 by induction we obtain119870+0 ge 119870+119895 ge 119870+119895+1 ge 119870minus119895+1 ge 119870minus119895 ge 119870minus0 ge 0Φ+0 ge Φ+119895 ge Φ+119895+1 ge Φminus119895+1 ge Φminus119895 ge Φminus0 ge 0 (52)

and Φ+119895 Φminus119895 isin 119878119899119904 we havelim119895rarrinfin

Φ+119895 = Φ+ isin 119878119899119904 lim119895rarrinfin

119870+119895 = 119870+ isin 119878119899+119870+ = lim

119895rarrinfin119870+119895 = lim

119895rarrinfinΨ(Φ+119895 ) = Ψ( lim

119895rarrinfinΦ+119895)

= Ψ (Φ+119895 ) (53)

So 119870+ is a solution of (45) corresponding to Φ = Φ+ thenΦ+ = [119868119899 + infinsum

119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870+ (54)

where 119870+ is a solution of Riccati equation (40) By the samestep we can get

lim119895rarrinfin

Φminus119895 = Φminus isin 119878119899119904 lim119895rarrinfin

119870minus119895 = 119870minus isin 119878119899+ (55)

and 119870minus is also a solution of Riccati equation (40) FromTheorem 10119870+ = 119870minus

From Lemma 12 in order to get the existence of solutionfor Riccati equation (40) we only need to find Φ+ and Φminussatisfying (50) Obviously we can let Φminus = 0 and for theexistence ofΦ+ we need the following Assumption 13

Assumption 13 There exists Φ(sdot) isin 119878119899119904 such thatinfinsum119894=1

(119865119894)120591 (119905) Φ (119905) 119865119894 (119905) = 119873 (119905) infinsum119894=1

[119868119899 + 119870 (119905) 119865119894 (119905)119873minus1 (119865119894)120591 (119905)]minus1119870 (119905) le Φ (119905) (56)

where 119870(119905) is the unique solution of the following equation

minus119905 = [119860 119905 minus infinsum119894=1

119861119905119873minus1119905 (119865119894119905)120591Φ (119905) 119864119894119905]120591119870119905+ 119870119905 [119860 119905 minus infinsum

119894=1

119861119905119873minus1119905 (119865119894119905)120591Φ (119905) 119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591Φ (119905) 119864119894119905+ 119877119905119870119879 = 119876

(57)


It is easy to known that when matrix 119865119905 is invertible and119896 = 119899 Assumption 13 is satisfiedThen we get the main resultof this section

Theorem 14 Let 119863 = 0 and Assumption 13 holds theRiccati equation (40) has a unique solution (119870119872 119884) isin1198621([0 119879] 119878119899+) times 119871infin([0 119879] 119877119899times119899) times 119871infin([0 119879] 119877119899times119899)

At last we give a simple example of the Riccati equationwhich has a unique solution

Example 15 Assume the dimensions of the state and controlare the same that is 119896 = 119899 when 119863 = 0 119865119894 = 119868119899 (119894 =1 2 3 ) in Riccati equation (39) letΦ(sdot) = 119873(sdot) then checkAssumption 13

In fact Φ(sdot) = 119873(sdot) ge 0 so suminfin119894=1[Φ119905119865119894119905 +119870119905119865119894119905119873minus1119905 (119865119894119905)120591Φ119905119865119894119905] ge suminfin119894=1119870119905119865119894119905 here 119870119905 is the solution of


119861119905119864119894119905]120591119870119905 + 119870119905 [119860 119905 minus infinsum119894=1

119861119905119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591119873(119905) 119864119894119905+ 119877119905119870119879 = 119876

(58)

Φ119905 + suminfin119894=1119870119905119865119894119905119873minus1119905 (119865119894119905)120591Φ119905 ge 119870119905 and then [119868119899 +suminfin119894=1119870119905119865119894119905119873minus1119905 (119865119894119905)120591]minus1119870119905 le Φ119905 FromTheorem 14 the Riccatiequation (39) has a unique solution when 119896 = 119899 119863 = 0 119865119894 =119868119899 (119894 = 1 2 3 )5 Conclusion

In this paper we discussed one kind of LQ stochastic controlproblem with Levy process as noise source where the controlsystem is described by a linear FBSDEL Explicit form ofoptimal control is obtained and it can be proved to be uniqueWhen assuming that all the coefficientmatrices in this controlproblem are deterministic it has been shown that the linearfeedback regulator for this LQ problem has a close relation tothe solutions of a kind of generalizedRiccati equation Finallywe discuss the solvability of the generalized Riccati equationand prove the existence and uniqueness of the solution for itin a special case

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 11271007 no 61673013) theNatural Science Foundation of Shandong Province(no ZR2016JZ022) the SDUST Research Fund (no

2014JQJH103) and the Graduate Student Innovation Fund ofSDUST (no SDKDYC170345)

References

[1] H Kushner ldquoOptimal Stochastic Controlrdquo IRE Transactions onAutomatic Control vol 7 no 5 pp 120ndash122 1962

[2] W M Wonham ldquoOn a matrix Riccati equation of stochasticcontrolrdquo SIAM Journal on Control and Optimization vol 6 pp681ndash697 1968

[3] G Li and W Zhang ldquoStudy on indefinite stochastic linearquadratic optimal control with inequality constraintrdquo Journalof Applied Mathematics vol 2013 Article ID 805829 9 pages2013

[4] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActa Mathematicae Applicatae Sinica English Series vol 29 no2 pp 303ndash314 2013

[5] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014

[6] X Wang X Zhang and P Zhao ldquoBinary nonlinearization forAKNS-KN coupling systemrdquo Abstract and Applied AnalysisArticle ID 253102 12 pages 2014

[7] C H Shen and X R Wang ldquoNonlinear analysis on the patternstructures of connection between final marketsrdquo Journal ofManagement sciences in China vol 18 no 2 pp 66ndash75 2015

[8] X-RWang Z-YGao andZWu ldquoForward-backward stochas-tic differential equation and the linear quadratic stochasticoptimal controlrdquo Acta Automatica Sinica vol 29 no 1 pp 32ndash37 2003

[9] Z Wu ldquoForward-backward stochastic differential equationslinear quadratic stochastic optimal control and nonzero sumdifferential gamesrdquo Journal of Systems Science and Complexityvol 2 pp 179ndash192 18

[10] R Boel and P Varaiya ldquoOptimal control of jump processesrdquoSIAM Journal on Control and Optimization vol 15 no 1 pp92ndash119 1977

[11] ZWu and XWang ldquoFBSDEwith Poisson process and its appli-cation to linear quadratic stochastic optimal control problemwith random jumpsrdquo Acta Automatica Sinica vol 29 no 6 pp821ndash826 2003

[12] J T Shi and Z Wu ldquoOne kind of fully coupled linear quadraticstochastic control problemwith random jumpsrdquoActa Automat-ica Sinica Zidonghua Xuebao vol 35 no 1 pp 92ndash97 2009

[13] X Lin and R Zhang ldquoHinfin control for stochastic systems withPoisson jumpsrdquo Journal of Systems Science amp Complexity vol24 no 4 pp 683ndash700 2011

[14] X Li X Lin and Y Lin ldquoLyapunov-type conditions andstochastic differential equations driven by G-Brownianmotionrdquo Journal of Mathematical Analysis and Applicationsvol 439 no 1 pp 235ndash255 2016

[15] D Nualart and W Schoutens ldquoChaotic and predictable rep-resentations for Levy processesrdquo Stochastic Processes and theirApplications vol 90 no 1 pp 109ndash122 2000

[16] D Nualart and W Schoutens ldquoBackward stochastic differen-tial equations and Feynman-Kac formula for Levy processeswith applications in financerdquo Bernoulli Official Journal of theBernoulli Society forMathematical Statistics and Probability vol7 no 5 pp 761ndash776 2001


[17] K Bahlali M Eddahbi and E Essaky ldquoBSDE associated withLevy processes and application to PDIErdquo Journal of AppliedMathematics and Stochastic Analysis vol 16 no 1 pp 1ndash17 2003

[18] M El Otmani ldquoGeneralized BSDE driven by a Levy processrdquoJournal of Applied Mathematics and Stochastic Analysis Art ID85407 25 pages 2006

[19] M El Otmani ldquoBackward stochastic differential equationsassociated with Levy processes and partial integro-differentialequationsrdquo Communications on Stochastic Analysis vol 2 no 2pp 277ndash288 2008

[20] Y Ren and X Fan ldquoReflected backward stochastic differentialequations driven by a lvy processrdquo ANZIAM Journal vol 50no 4 pp 486ndash500 2009

[21] M Tang and Q Zhang ldquoOptimal variational principle for back-ward stochastic control systems associatedwith Levy processesrdquoScience China Mathematics vol 55 no 4 pp 745ndash761 2012

[22] H Huang and X Wang ldquoLQ stochastic optimal control offorward-backward stochastic control system driven by Levyprocessrdquo in Proceedings of the 2016 IEEE Advanced InformationManagement Communicates Electronic and Automation Con-trol Conference IMCEC 2016 pp 1939ndash1943 chn October 2016

[23] K-i Mitsui and Y Tabata ldquoA stochastic linear-quadraticproblem with Levy processes and its application to financerdquoStochastic Processes and their Applications vol 118 no 1 pp 120ndash152 2008

[24] H Tang and Z Wu ldquoStochastic differential equations andstochastic linear quadratic optimal control problem with Levyprocessesrdquo Journal of Systems Science amp Complexity vol 22 no1 pp 122ndash136 2009

[25] R S Pereira and E Shamarova ldquoForward backward SDEs drivenby Levy processes and application to option pricingrdquo RandomOperators and Stochastic Equations vol 2 no 1 pp 1ndash20 2012

[26] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differential gamesrdquoRandom Operators and Stochastic Equations vol 22 no 3 pp151ndash161 2014

[27] X Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering Article ID 702802 ArtID 702802 12 pages 2015

[28] S Peng ldquoProblem of eigenvalues of stochastic Hamiltoniansystems with boundary conditionsrdquo Stochastic Processes andtheir Applications vol 88 no 2 pp 259ndash290 2000

Submit your manuscripts athttpswwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


theorem associated with Teugels martingale In the next yearNualart and Schoutens [16] considered a backward stochasticdifferential equation (BSDE) driven by Teugels martingaleand proved the existence and uniqueness theory of this BSDEIn 2003 Bahlali et al [17] studied a BSDE driven by Teugelsmartingale and an independent Brownian motion theygot the existence uniqueness and comparison of solutionsfor these equations having a Lipschitz or locally Lipschitzcoefficient El Otmani [18] considered a kind of generalizedBSDE (GBSDE) associated with Teugels martingale andBrownian motion associated with a pure jump-independentLevy process They got the existence and uniqueness theoryof this GBSDE when the coefficient verifies some conditionsof LipschitzMore results about BSDE associatedwithTeugelsmartingale can be found in the theses of El Otmani [19] Renand Fan [20] Tang and Zhang [21] and Huang and Wang[22] On the basis of these results in 2008 Mitsui and Tabata[23] studied a LQ regulation stochastic control problemwith Levy process and obtained the optimal control for thenonhomogeneous case In [24] Tang andWu considered thefollowing LQ stochastic control problem in a given finitehorizon [119904 119879] with Levy process

119889119909 (119905) = [119860 (120596 119905) 119909 (119905) + 119861 (120596 119905) 119906 (119905)] 119889119905+ infinsum119894=1

[119862119894 (120596 119905) 119909 (119905minus) + 119863119894 (120596 119905) 119906 (119905)] 119889119867119894 (119905)119909 (119904) = 120585

(1)

and the cost function was

119869 (119904 120585 119906 (sdot)) = 119864119904 [int119879119904(119909120591 (119905) 119876 (120596 119905) 119909 (119905)

+ 119906120591 (119905) 119877 (120596 119905) 119906 (119905)) 119889119905 + 119909120591 (119879)119867 (120596) 119909 (119879)] (2)

They show that the solvability of one kind of generalizedRiccati equation is sufficient to the well-posedness of this LQproblem and proved the existence of the optimal control

In this paper we consider one kind of LQ stochasticcontrol problem where the controlled system is driven by afully coupled linear forward backward stochastic differentialequation associated with Levy process (FBSDEL)

119889119909119905 = [119860 (120596 119905) 119909119905 + 119861 (120596 119905) 119906119905 minus 119871 (120596 119905) 119910119905] 119889119905+ [119862 (120596 119905) 119909119905 + 119863 (120596 119905) 119906119905] 119889119861119905+ infinsum119894=1

[119864119894 (120596 119905) 119909119905minus

+ 119865119894 (120596 119905) 119906119905] 119889119867119894119905minus 119889119910119905 = [119860120591 (120596 119905) 119910119905 + 119862120591 (120596 119905) 119911119905 + infinsum

119894=1


119894=1

119903119894119905119889119867119894119905

1199090 = 119886119910119879 = 119876 (120596) 119909119879(3)

where (119909119905 119910119905 119911119905 119903119894119905) areF119905-adapted stochastic processes tak-ing values in 119877119899 times 119877119899 times 119877119899 times 1198972(119877119899) and 119906(sdot) is F119905-adaptedstochastic process called admissible control process Assumethe control process set 119880 = 119877119896 and define the admissiblecontrol set as follows119880119886119889 = 119906 (sdot) isin 1198722 (0 119879 119877119896) 119906119905 isin 119880 0 le 119905

le 119879 ae as (4)

The cost functional we considered is

119869 (119906) = 12119864 [int1198790 (⟨119877 (120596 119905) 119909119905 119909119905⟩ + ⟨119873 (120596 119905) 119906119905 119906119905⟩+ ⟨119871 (120596 119905) 119910119905 119910119905⟩) 119889119905 + ⟨119876 (120596) 119909119879 119909119879⟩] (5)

And the optimal control problem is to find 119906119905 isin 119880119886119889 suchthat 119869 (119906 (sdot)) = inf

119906(sdot)119869 (119906 (sdot)) (6)

Note that (3) is a fully coupled FBSDEL In 2012Pereira and Shamarova [25] firstly considered this kind ofFBSDEL obtained a solution to this FBSDEL via a partialintegrodifferential equation and proved the uniquenessUnder some monotonicity assumptions Baghery et al [26]proved the existence and uniqueness of solutions of fullycoupled FBSDEL and then obtained the existence of an open-loop Nash equilibrium point for nonzero sum stochasticdifferential games by using this result Based on [25] Wangand Huang [27] got the maximum principle for forwardbackward stochastic control system driven by Levy processthen they discussed a kind of LQ stochastic control problemof forward backward stochastic control system and got anecessary condition for the optimal control

We extend the result of Shi and Wu [12] to the fullycoupled linear forward backward stochastic control systemdriven by Brownian motion and an independent Teugelsmartingale Since Teugels martingale is more complex thanthe Poisson process we also needmore general formula aboutcadlag semimartingale The rest of this paper is organized asfollows In Section 2 we provide a list of notations and resultsof the existence and uniqueness of solutions of fully coupledFBSDEL In Section 3 we prove the existence and uniquenessof the optimal control of LQ stochastic control problem(6) and give the linear feedback regulator for the optimalcontrol by the solution of a kind of generalizedmatrix-valuedRiccati equation when assuming the coefficient matrices aredeterministic In Section 4 the solvability of this kind ofmatrix-valued Riccati equation is discussed

2 Preliminaries and Notations

Let (ΩF119905 119875) be a complete probability space satisfying theusual conditionsF119905 is a right continuous increasing family of



(i) int119877(1 and 1199092)](119889119909) lt infin

(ii) int(minus120576120576)119888





119871119894119905 = 119871 119905 119894 = 1sum0lt119904le119905

(Δ119871 119904)119894 119894 ge 2 (8)









119889119909119905 = 119887 (119905 119909119905 119910119905 119911119905 119903119905) 119889119905 + 119889sum119894=1

120590119894 (119905 119909119905 119910119905 119911119905 119903119905) 119889119861119894119905+ infinsum119894=1


119894=1

119911119894119905119889119861119894119905 minus infinsum119894=1

1199031198941199051198891198671198941199051199090 = 119886119910119879 = Φ (119909119879)

(10)


120582 = (119909119910119911)

119860 (119905 120582 119903) = (minus119866120591119891 (119905 120582 119903)119866119887 (119905 120582 119903)119866120590 (119905 120582 119903) ) (11)






119894=1

1003817100381710038171003817100381710038171198661205911199031198941003817100381710038171003817100381710038172) ⟨Φ (1199091) minus Φ (1199092) 119866 (1199091 minus 1199092)⟩ ge 1205831 |119866119909|2

(12)






119865 (119883119905) = 119865 (1198830) + int11990501198651015840 (119883119904minus) 119889119883119904

+ 12 int1199050 11986510158401015840 (119883119904) 119889 [119883]C119904+ sum0lt119904le119905



1198832119905 = 11988320 + int11990502119883119904minus119889119883119904 + int119905

0119889 [119883]119904

119883119905119884119905 = 11988301198840 + int1199050119883119904minus119889119884119904 + int119905

0119884119904minus119889119883119904

+ int1199050119889 [119883 119884]119904

(14)







119894=1

119865119894120591 (120596 119905) 119903119894119905) (15)


119869 (V119905) minus 119869 (119906119905) = 12sdot 119864 [int119879

0(⟨119877 (120596 119905) (119909V119905 minus 119909119905) 119909V119905 minus 119909119905⟩


(16)


0(⟨minus119877 (120596 119905) 119909119905 119909V119905 minus 119909119905⟩


⟨119865119894120591 (120596 119905) 119903119894119905 V119905 minus 119906119905⟩)119889119905(17)



119894=1

119865119894120591 (120596 119905) 119903119894119905)



+ 119863120591 (120596 119905) 119911119905 + infinsum119894=1


119894=1

119865119894120591 (120596 119905) 119903119894119905) V119905 minus 119906119905⟩ = 0




(1199091119905 + 11990921199052 1199101119905 + 11991021199052 1199111119905 + 11991121199052 1199031119905 + 11990321199052 ) (19)



+ 119864int1198790(⟨119877 (120596 119905) 1199091119905 minus 11990921199052 1199091119905 minus 11990921199052 ⟩


+ 1205752119864int1198790 100381610038161003816100381610038161199061119905 minus 1199062119905 100381610038161003816100381610038162 119889119905

(21)


100381610038161003816100381610038161199061119905 minus 1199062119905 100381610038161003816100381610038162 119889119905 le 0 (22)

hence 1199061119905 = 1199062119905 in1198722(0 119879 119877119896)


minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1






119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(23)




(119865119894119905)120591 119884119894119905]119909119905 (24)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (25)


119910119905 = 119870119905119909119905119911119905 = 119872119905119909119905119903119894119905 = 119884119894119905119909119905(26)



(119865119894119905)120591 119884119894119905]119909119905 (27)



0⟨119871 119905119910119905 119910119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩

minus ⟨119870119886 119886⟩= 119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905(28)


119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905 = minus119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 (29)

and then

119864int1198790⟨119877119905119909119905 119909119905⟩ 119889119905 + 119864int119879

0⟨119871 119905119910119905 119910119905⟩ 119889119905

+ 119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩ = ⟨119870119886 119886⟩ (30)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (31)


+ infinsum119894=1

(119864119894 (120596 119905) 119909119905minus

+ 119865119894 (120596 119905) 119906119905) 1198891198671198941199051199090 = 119886

(32)


119869 (119906) = 12119864 [int1198790 ⟨119877 (120596 119905) 119909119905 119909119905⟩ + ⟨119873 (120596 119905) 119906119905 119906119905⟩+ ⟨119876 (120596) 119909119879 119909119879⟩] (33)





119894=1

119865119894120591 (120596 119905) 119903119894119905] (34)


minus 119889119910119905 = [119860120591 (120596 119905) 119910119905 + 119862120591 (120596 119905) 119911119905 + infinsum119894=1


119894=1

119903119894119905119889119867119894119905119910119879 = 119876 (120596) 119909119879

(35)


+ infinsum119894=1


119894=1


119870119905119863119905119873minus1119905 (119865119894119905)120591 119884119894119905119884119894119905 = 119870119905119864119894119905 minus 119870119905119865i


119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(36)




(119865119894119905)120591 119884119894119905]119909119905 (37)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (38)




minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1



119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905 119872119905 = 119870119905119862119905119870119879 = 119876 119894 = 1 2 3

(39)



119894=1


119894=1


119894=1



119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(40)






119879 = 0119868119899 + infinsum119894=1

119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0 (119894 = 1 2 3 )

(42)

where

1198681 = infinsum119894=1



1198682 = infinsum119894=1


119894=1



119894=1





119894=1



119894=1


119894=1





119894=1





Φ119905 = Λ (119870) = [119868119899 + infinsum119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870 (44)



119861119905119873minus1119905 (119865119894119905)120591Φ119905119864119894119905]120591119870119905 + 119870119905 [119860 119905




(119864119894119905)120591Φ119905119864119894119905 + 119877119905119870119879 = 119876119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(45)


119894=1


(46)



119894=1


119894=1

119870119865119894119873minus1 (119865119894)120591)]minus1

= [119870minus1 + infinsum119894=1

119865119894119873minus1 (119865119894)120591]minus1 (47)





+ 119862120591119905119870119905119862119905+ infinsum119894=1


(48)



119894=1

Ψ (Φ) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φ) (49)



119894=1


119894=1



119894=1


119894=1


(51)




119870+119895 = 119870+ isin 119878119899+119870+ = lim

119895rarrinfin119870+119895 = lim



= Ψ (Φ+119895 ) (53)


119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870+ (54)


lim119895rarrinfin


119870minus119895 = 119870minus isin 119878119899+ (55)




(119865119894)120591 (119905) Φ (119905) 119865119894 (119905) = 119873 (119905) infinsum119894=1

[119868119899 + 119870 (119905) 119865119894 (119905)119873minus1 (119865119894)120591 (119905)]minus1119870 (119905) le Φ (119905) (56)




119894=1


119894=1

(119864119894119905)120591Φ (119905) 119864119894119905+ 119877119905119870119879 = 119876

(57)








119861119905119864119894119905]120591119870119905 + 119870119905 [119860 119905 minus infinsum119894=1

119861119905119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591119873(119905) 119864119894119905+ 119877119905119870119879 = 119876

(58)





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






(i) int119877(1 and 1199092)](119889119909) lt infin

(ii) int(minus120576120576)119888





119871119894119905 = 119871 119905 119894 = 1sum0lt119904le119905

(Δ119871 119904)119894 119894 ge 2 (8)









119889119909119905 = 119887 (119905 119909119905 119910119905 119911119905 119903119905) 119889119905 + 119889sum119894=1

120590119894 (119905 119909119905 119910119905 119911119905 119903119905) 119889119861119894119905+ infinsum119894=1


119894=1

119911119894119905119889119861119894119905 minus infinsum119894=1

1199031198941199051198891198671198941199051199090 = 119886119910119879 = Φ (119909119879)

(10)


120582 = (119909119910119911)

119860 (119905 120582 119903) = (minus119866120591119891 (119905 120582 119903)119866119887 (119905 120582 119903)119866120590 (119905 120582 119903) ) (11)






119894=1

1003817100381710038171003817100381710038171198661205911199031198941003817100381710038171003817100381710038172) ⟨Φ (1199091) minus Φ (1199092) 119866 (1199091 minus 1199092)⟩ ge 1205831 |119866119909|2

(12)






119865 (119883119905) = 119865 (1198830) + int11990501198651015840 (119883119904minus) 119889119883119904

+ 12 int1199050 11986510158401015840 (119883119904) 119889 [119883]C119904+ sum0lt119904le119905



1198832119905 = 11988320 + int11990502119883119904minus119889119883119904 + int119905

0119889 [119883]119904

119883119905119884119905 = 11988301198840 + int1199050119883119904minus119889119884119904 + int119905

0119884119904minus119889119883119904

+ int1199050119889 [119883 119884]119904

(14)







119894=1

119865119894120591 (120596 119905) 119903119894119905) (15)


119869 (V119905) minus 119869 (119906119905) = 12sdot 119864 [int119879

0(⟨119877 (120596 119905) (119909V119905 minus 119909119905) 119909V119905 minus 119909119905⟩


(16)


0(⟨minus119877 (120596 119905) 119909119905 119909V119905 minus 119909119905⟩


⟨119865119894120591 (120596 119905) 119903119894119905 V119905 minus 119906119905⟩)119889119905(17)



119894=1

119865119894120591 (120596 119905) 119903119894119905)



+ 119863120591 (120596 119905) 119911119905 + infinsum119894=1


119894=1

119865119894120591 (120596 119905) 119903119894119905) V119905 minus 119906119905⟩ = 0




(1199091119905 + 11990921199052 1199101119905 + 11991021199052 1199111119905 + 11991121199052 1199031119905 + 11990321199052 ) (19)



+ 119864int1198790(⟨119877 (120596 119905) 1199091119905 minus 11990921199052 1199091119905 minus 11990921199052 ⟩


+ 1205752119864int1198790 100381610038161003816100381610038161199061119905 minus 1199062119905 100381610038161003816100381610038162 119889119905

(21)


100381610038161003816100381610038161199061119905 minus 1199062119905 100381610038161003816100381610038162 119889119905 le 0 (22)

hence 1199061119905 = 1199062119905 in1198722(0 119879 119877119896)


minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1






119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(23)




(119865119894119905)120591 119884119894119905]119909119905 (24)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (25)


119910119905 = 119870119905119909119905119911119905 = 119872119905119909119905119903119894119905 = 119884119894119905119909119905(26)



(119865119894119905)120591 119884119894119905]119909119905 (27)



0⟨119871 119905119910119905 119910119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩

minus ⟨119870119886 119886⟩= 119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905(28)


119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905 = minus119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 (29)

and then

119864int1198790⟨119877119905119909119905 119909119905⟩ 119889119905 + 119864int119879

0⟨119871 119905119910119905 119910119905⟩ 119889119905

+ 119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩ = ⟨119870119886 119886⟩ (30)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (31)


+ infinsum119894=1

(119864119894 (120596 119905) 119909119905minus

+ 119865119894 (120596 119905) 119906119905) 1198891198671198941199051199090 = 119886

(32)


119869 (119906) = 12119864 [int1198790 ⟨119877 (120596 119905) 119909119905 119909119905⟩ + ⟨119873 (120596 119905) 119906119905 119906119905⟩+ ⟨119876 (120596) 119909119879 119909119879⟩] (33)





119894=1

119865119894120591 (120596 119905) 119903119894119905] (34)


minus 119889119910119905 = [119860120591 (120596 119905) 119910119905 + 119862120591 (120596 119905) 119911119905 + infinsum119894=1


119894=1

119903119894119905119889119867119894119905119910119879 = 119876 (120596) 119909119879

(35)


+ infinsum119894=1


119894=1


119870119905119863119905119873minus1119905 (119865119894119905)120591 119884119894119905119884119894119905 = 119870119905119864119894119905 minus 119870119905119865i


119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(36)




(119865119894119905)120591 119884119894119905]119909119905 (37)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (38)




minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1



119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905 119872119905 = 119870119905119862119905119870119879 = 119876 119894 = 1 2 3

(39)



119894=1


119894=1


119894=1



119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(40)






119879 = 0119868119899 + infinsum119894=1

119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0 (119894 = 1 2 3 )

(42)

where

1198681 = infinsum119894=1



1198682 = infinsum119894=1


119894=1



119894=1





119894=1



119894=1


119894=1





119894=1





Φ119905 = Λ (119870) = [119868119899 + infinsum119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870 (44)



119861119905119873minus1119905 (119865119894119905)120591Φ119905119864119894119905]120591119870119905 + 119870119905 [119860 119905




(119864119894119905)120591Φ119905119864119894119905 + 119877119905119870119879 = 119876119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(45)


119894=1


(46)



119894=1


119894=1

119870119865119894119873minus1 (119865119894)120591)]minus1

= [119870minus1 + infinsum119894=1

119865119894119873minus1 (119865119894)120591]minus1 (47)





+ 119862120591119905119870119905119862119905+ infinsum119894=1


(48)



119894=1

Ψ (Φ) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φ) (49)



119894=1


119894=1



119894=1


119894=1


(51)




119870+119895 = 119870+ isin 119878119899+119870+ = lim

119895rarrinfin119870+119895 = lim



= Ψ (Φ+119895 ) (53)


119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870+ (54)


lim119895rarrinfin


119870minus119895 = 119870minus isin 119878119899+ (55)




(119865119894)120591 (119905) Φ (119905) 119865119894 (119905) = 119873 (119905) infinsum119894=1

[119868119899 + 119870 (119905) 119865119894 (119905)119873minus1 (119865119894)120591 (119905)]minus1119870 (119905) le Φ (119905) (56)




119894=1


119894=1

(119864119894119905)120591Φ (119905) 119864119894119905+ 119877119905119870119879 = 119876

(57)








119861119905119864119894119905]120591119870119905 + 119870119905 [119860 119905 minus infinsum119894=1

119861119905119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591119873(119905) 119864119894119905+ 119877119905119870119879 = 119876

(58)





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







119865 (119883119905) = 119865 (1198830) + int11990501198651015840 (119883119904minus) 119889119883119904

+ 12 int1199050 11986510158401015840 (119883119904) 119889 [119883]C119904+ sum0lt119904le119905



1198832119905 = 11988320 + int11990502119883119904minus119889119883119904 + int119905

0119889 [119883]119904

119883119905119884119905 = 11988301198840 + int1199050119883119904minus119889119884119904 + int119905

0119884119904minus119889119883119904

+ int1199050119889 [119883 119884]119904

(14)







119894=1

119865119894120591 (120596 119905) 119903119894119905) (15)


119869 (V119905) minus 119869 (119906119905) = 12sdot 119864 [int119879

0(⟨119877 (120596 119905) (119909V119905 minus 119909119905) 119909V119905 minus 119909119905⟩


(16)


0(⟨minus119877 (120596 119905) 119909119905 119909V119905 minus 119909119905⟩


⟨119865119894120591 (120596 119905) 119903119894119905 V119905 minus 119906119905⟩)119889119905(17)



119894=1

119865119894120591 (120596 119905) 119903119894119905)



+ 119863120591 (120596 119905) 119911119905 + infinsum119894=1


119894=1

119865119894120591 (120596 119905) 119903119894119905) V119905 minus 119906119905⟩ = 0




(1199091119905 + 11990921199052 1199101119905 + 11991021199052 1199111119905 + 11991121199052 1199031119905 + 11990321199052 ) (19)



+ 119864int1198790(⟨119877 (120596 119905) 1199091119905 minus 11990921199052 1199091119905 minus 11990921199052 ⟩


+ 1205752119864int1198790 100381610038161003816100381610038161199061119905 minus 1199062119905 100381610038161003816100381610038162 119889119905

(21)


100381610038161003816100381610038161199061119905 minus 1199062119905 100381610038161003816100381610038162 119889119905 le 0 (22)

hence 1199061119905 = 1199062119905 in1198722(0 119879 119877119896)


minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1






119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(23)




(119865119894119905)120591 119884119894119905]119909119905 (24)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (25)


119910119905 = 119870119905119909119905119911119905 = 119872119905119909119905119903119894119905 = 119884119894119905119909119905(26)



(119865119894119905)120591 119884119894119905]119909119905 (27)



0⟨119871 119905119910119905 119910119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩

minus ⟨119870119886 119886⟩= 119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905(28)


119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905 = minus119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 (29)

and then

119864int1198790⟨119877119905119909119905 119909119905⟩ 119889119905 + 119864int119879

0⟨119871 119905119910119905 119910119905⟩ 119889119905

+ 119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩ = ⟨119870119886 119886⟩ (30)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (31)


+ infinsum119894=1

(119864119894 (120596 119905) 119909119905minus

+ 119865119894 (120596 119905) 119906119905) 1198891198671198941199051199090 = 119886

(32)


119869 (119906) = 12119864 [int1198790 ⟨119877 (120596 119905) 119909119905 119909119905⟩ + ⟨119873 (120596 119905) 119906119905 119906119905⟩+ ⟨119876 (120596) 119909119879 119909119879⟩] (33)





119894=1

119865119894120591 (120596 119905) 119903119894119905] (34)


minus 119889119910119905 = [119860120591 (120596 119905) 119910119905 + 119862120591 (120596 119905) 119911119905 + infinsum119894=1


119894=1

119903119894119905119889119867119894119905119910119879 = 119876 (120596) 119909119879

(35)


+ infinsum119894=1


119894=1


119870119905119863119905119873minus1119905 (119865119894119905)120591 119884119894119905119884119894119905 = 119870119905119864119894119905 minus 119870119905119865i


119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(36)




(119865119894119905)120591 119884119894119905]119909119905 (37)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (38)




minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1



119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905 119872119905 = 119870119905119862119905119870119879 = 119876 119894 = 1 2 3

(39)



119894=1


119894=1


119894=1



119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(40)






119879 = 0119868119899 + infinsum119894=1

119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0 (119894 = 1 2 3 )

(42)

where

1198681 = infinsum119894=1



1198682 = infinsum119894=1


119894=1



119894=1





119894=1



119894=1


119894=1





119894=1





Φ119905 = Λ (119870) = [119868119899 + infinsum119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870 (44)



119861119905119873minus1119905 (119865119894119905)120591Φ119905119864119894119905]120591119870119905 + 119870119905 [119860 119905




(119864119894119905)120591Φ119905119864119894119905 + 119877119905119870119879 = 119876119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(45)


119894=1


(46)



119894=1


119894=1

119870119865119894119873minus1 (119865119894)120591)]minus1

= [119870minus1 + infinsum119894=1

119865119894119873minus1 (119865119894)120591]minus1 (47)





+ 119862120591119905119870119905119862119905+ infinsum119894=1


(48)



119894=1

Ψ (Φ) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φ) (49)



119894=1


119894=1



119894=1


119894=1


(51)




119870+119895 = 119870+ isin 119878119899+119870+ = lim

119895rarrinfin119870+119895 = lim



= Ψ (Φ+119895 ) (53)


119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870+ (54)


lim119895rarrinfin


119870minus119895 = 119870minus isin 119878119899+ (55)




(119865119894)120591 (119905) Φ (119905) 119865119894 (119905) = 119873 (119905) infinsum119894=1

[119868119899 + 119870 (119905) 119865119894 (119905)119873minus1 (119865119894)120591 (119905)]minus1119870 (119905) le Φ (119905) (56)




119894=1


119894=1

(119864119894119905)120591Φ (119905) 119864119894119905+ 119877119905119870119879 = 119876

(57)








119861119905119864119894119905]120591119870119905 + 119870119905 [119860 119905 minus infinsum119894=1

119861119905119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591119873(119905) 119864119894119905+ 119877119905119870119879 = 119876

(58)





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






+ 119863120591 (120596 119905) 119911119905 + infinsum119894=1


119894=1

119865119894120591 (120596 119905) 119903119894119905) V119905 minus 119906119905⟩ = 0




(1199091119905 + 11990921199052 1199101119905 + 11991021199052 1199111119905 + 11991121199052 1199031119905 + 11990321199052 ) (19)



+ 119864int1198790(⟨119877 (120596 119905) 1199091119905 minus 11990921199052 1199091119905 minus 11990921199052 ⟩


+ 1205752119864int1198790 100381610038161003816100381610038161199061119905 minus 1199062119905 100381610038161003816100381610038162 119889119905

(21)


100381610038161003816100381610038161199061119905 minus 1199062119905 100381610038161003816100381610038162 119889119905 le 0 (22)

hence 1199061119905 = 1199062119905 in1198722(0 119879 119877119896)


minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1






119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(23)




(119865119894119905)120591 119884119894119905]119909119905 (24)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (25)


119910119905 = 119870119905119909119905119911119905 = 119872119905119909119905119903119894119905 = 119884119894119905119909119905(26)



(119865119894119905)120591 119884119894119905]119909119905 (27)



0⟨119871 119905119910119905 119910119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩

minus ⟨119870119886 119886⟩= 119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905(28)


119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905 = minus119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 (29)

and then

119864int1198790⟨119877119905119909119905 119909119905⟩ 119889119905 + 119864int119879

0⟨119871 119905119910119905 119910119905⟩ 119889119905

+ 119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩ = ⟨119870119886 119886⟩ (30)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (31)


+ infinsum119894=1

(119864119894 (120596 119905) 119909119905minus

+ 119865119894 (120596 119905) 119906119905) 1198891198671198941199051199090 = 119886

(32)


119869 (119906) = 12119864 [int1198790 ⟨119877 (120596 119905) 119909119905 119909119905⟩ + ⟨119873 (120596 119905) 119906119905 119906119905⟩+ ⟨119876 (120596) 119909119879 119909119879⟩] (33)





119894=1

119865119894120591 (120596 119905) 119903119894119905] (34)


minus 119889119910119905 = [119860120591 (120596 119905) 119910119905 + 119862120591 (120596 119905) 119911119905 + infinsum119894=1


119894=1

119903119894119905119889119867119894119905119910119879 = 119876 (120596) 119909119879

(35)


+ infinsum119894=1


119894=1


119870119905119863119905119873minus1119905 (119865119894119905)120591 119884119894119905119884119894119905 = 119870119905119864119894119905 minus 119870119905119865i


119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(36)




(119865119894119905)120591 119884119894119905]119909119905 (37)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (38)




minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1



119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905 119872119905 = 119870119905119862119905119870119879 = 119876 119894 = 1 2 3

(39)



119894=1


119894=1


119894=1



119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(40)






119879 = 0119868119899 + infinsum119894=1

119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0 (119894 = 1 2 3 )

(42)

where

1198681 = infinsum119894=1



1198682 = infinsum119894=1


119894=1



119894=1





119894=1



119894=1


119894=1





119894=1





Φ119905 = Λ (119870) = [119868119899 + infinsum119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870 (44)



119861119905119873minus1119905 (119865119894119905)120591Φ119905119864119894119905]120591119870119905 + 119870119905 [119860 119905




(119864119894119905)120591Φ119905119864119894119905 + 119877119905119870119879 = 119876119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(45)


119894=1


(46)



119894=1


119894=1

119870119865119894119873minus1 (119865119894)120591)]minus1

= [119870minus1 + infinsum119894=1

119865119894119873minus1 (119865119894)120591]minus1 (47)





+ 119862120591119905119870119905119862119905+ infinsum119894=1


(48)



119894=1

Ψ (Φ) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φ) (49)



119894=1


119894=1



119894=1


119894=1


(51)




119870+119895 = 119870+ isin 119878119899+119870+ = lim

119895rarrinfin119870+119895 = lim



= Ψ (Φ+119895 ) (53)


119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870+ (54)


lim119895rarrinfin


119870minus119895 = 119870minus isin 119878119899+ (55)




(119865119894)120591 (119905) Φ (119905) 119865119894 (119905) = 119873 (119905) infinsum119894=1

[119868119899 + 119870 (119905) 119865119894 (119905)119873minus1 (119865119894)120591 (119905)]minus1119870 (119905) le Φ (119905) (56)




119894=1


119894=1

(119864119894119905)120591Φ (119905) 119864119894119905+ 119877119905119870119879 = 119876

(57)








119861119905119864119894119905]120591119870119905 + 119870119905 [119860 119905 minus infinsum119894=1

119861119905119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591119873(119905) 119864119894119905+ 119877119905119870119879 = 119876

(58)





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






0⟨119871 119905119910119905 119910119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩

minus ⟨119870119886 119886⟩= 119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905(28)


119864int1198790⟨119910119905 119861119905119906119905⟩ 119889119905 + 119864int119879

0⟨119911119905 119863119905119906119905⟩ 119889119905

+ 119864int1198790

infinsum119894=1

⟨119903119894119905 119865119894119905119906119905⟩ 119889119905 = minus119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 (29)

and then

119864int1198790⟨119877119905119909119905 119909119905⟩ 119889119905 + 119864int119879

0⟨119871 119905119910119905 119910119905⟩ 119889119905

+ 119864int1198790⟨119873119905119906119905 119906119905⟩ 119889119905 + ⟨119876119909119879 119909119879⟩ = ⟨119870119886 119886⟩ (30)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (31)


+ infinsum119894=1

(119864119894 (120596 119905) 119909119905minus

+ 119865119894 (120596 119905) 119906119905) 1198891198671198941199051199090 = 119886

(32)


119869 (119906) = 12119864 [int1198790 ⟨119877 (120596 119905) 119909119905 119909119905⟩ + ⟨119873 (120596 119905) 119906119905 119906119905⟩+ ⟨119876 (120596) 119909119879 119909119879⟩] (33)





119894=1

119865119894120591 (120596 119905) 119903119894119905] (34)


minus 119889119910119905 = [119860120591 (120596 119905) 119910119905 + 119862120591 (120596 119905) 119911119905 + infinsum119894=1


119894=1

119903119894119905119889119867119894119905119910119879 = 119876 (120596) 119909119879

(35)


+ infinsum119894=1


119894=1


119870119905119863119905119873minus1119905 (119865119894119905)120591 119884119894119905119884119894119905 = 119870119905119864119894119905 minus 119870119905119865i


119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905119870119879 = 119876

(36)




(119865119894119905)120591 119884119894119905]119909119905 (37)


119869 (119906119905) = 12 ⟨1198700119886 119886⟩ (38)




minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1



119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905 119872119905 = 119870119905119862119905119870119879 = 119876 119894 = 1 2 3

(39)



119894=1


119894=1


119894=1



119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(40)






119879 = 0119868119899 + infinsum119894=1

119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0 (119894 = 1 2 3 )

(42)

where

1198681 = infinsum119894=1



1198682 = infinsum119894=1


119894=1



119894=1





119894=1



119894=1


119894=1





119894=1





Φ119905 = Λ (119870) = [119868119899 + infinsum119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870 (44)



119861119905119873minus1119905 (119865119894119905)120591Φ119905119864119894119905]120591119870119905 + 119870119905 [119860 119905




(119864119894119905)120591Φ119905119864119894119905 + 119877119905119870119879 = 119876119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(45)


119894=1


(46)



119894=1


119894=1

119870119865119894119873minus1 (119865119894)120591)]minus1

= [119870minus1 + infinsum119894=1

119865119894119873minus1 (119865119894)120591]minus1 (47)





+ 119862120591119905119870119905119862119905+ infinsum119894=1


(48)



119894=1

Ψ (Φ) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φ) (49)



119894=1


119894=1



119894=1


119894=1


(51)




119870+119895 = 119870+ isin 119878119899+119870+ = lim

119895rarrinfin119870+119895 = lim



= Ψ (Φ+119895 ) (53)


119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870+ (54)


lim119895rarrinfin


119870minus119895 = 119870minus isin 119878119899+ (55)




(119865119894)120591 (119905) Φ (119905) 119865119894 (119905) = 119873 (119905) infinsum119894=1

[119868119899 + 119870 (119905) 119865119894 (119905)119873minus1 (119865119894)120591 (119905)]minus1119870 (119905) le Φ (119905) (56)




119894=1


119894=1

(119864119894119905)120591Φ (119905) 119864119894119905+ 119877119905119870119879 = 119876

(57)








119861119905119864119894119905]120591119870119905 + 119870119905 [119860 119905 minus infinsum119894=1

119861119905119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591119873(119905) 119864119894119905+ 119877119905119870119879 = 119876

(58)





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







minus119905 = 119860120591119905119870119905 + 119870119905119860 119905 + 119862120591119905119872119905 + infinsum119894=1



119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 119884119894119905 119872119905 = 119870119905119862119905119870119879 = 119876 119894 = 1 2 3

(39)



119894=1


119894=1


119894=1



119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(40)






119879 = 0119868119899 + infinsum119894=1

119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0 (119894 = 1 2 3 )

(42)

where

1198681 = infinsum119894=1



1198682 = infinsum119894=1


119894=1



119894=1





119894=1



119894=1


119894=1





119894=1





Φ119905 = Λ (119870) = [119868119899 + infinsum119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870 (44)



119861119905119873minus1119905 (119865119894119905)120591Φ119905119864119894119905]120591119870119905 + 119870119905 [119860 119905




(119864119894119905)120591Φ119905119864119894119905 + 119877119905119870119879 = 119876119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(45)


119894=1


(46)



119894=1


119894=1

119870119865119894119873minus1 (119865119894)120591)]minus1

= [119870minus1 + infinsum119894=1

119865119894119873minus1 (119865119894)120591]minus1 (47)





+ 119862120591119905119870119905119862119905+ infinsum119894=1


(48)



119894=1

Ψ (Φ) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φ) (49)



119894=1


119894=1



119894=1


119894=1


(51)




119870+119895 = 119870+ isin 119878119899+119870+ = lim

119895rarrinfin119870+119895 = lim



= Ψ (Φ+119895 ) (53)


119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870+ (54)


lim119895rarrinfin


119870minus119895 = 119870minus isin 119878119899+ (55)




(119865119894)120591 (119905) Φ (119905) 119865119894 (119905) = 119873 (119905) infinsum119894=1

[119868119899 + 119870 (119905) 119865119894 (119905)119873minus1 (119865119894)120591 (119905)]minus1119870 (119905) le Φ (119905) (56)




119894=1


119894=1

(119864119894119905)120591Φ (119905) 119864119894119905+ 119877119905119870119879 = 119876

(57)








119861119905119864119894119905]120591119870119905 + 119870119905 [119860 119905 minus infinsum119894=1

119861119905119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591119873(119905) 119864119894119905+ 119877119905119870119879 = 119876

(58)





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







119894=1



119894=1


119894=1





119894=1





Φ119905 = Λ (119870) = [119868119899 + infinsum119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870 (44)



119861119905119873minus1119905 (119865119894119905)120591Φ119905119864119894119905]120591119870119905 + 119870119905 [119860 119905




(119864119894119905)120591Φ119905119864119894119905 + 119877119905119870119879 = 119876119868119899 + infinsum119894=1

119870119905119865119894119905119873minus1119905 (119865119894119905)120591 gt 0(119894 = 1 2 3 )

(45)


119894=1


(46)



119894=1


119894=1

119870119865119894119873minus1 (119865119894)120591)]minus1

= [119870minus1 + infinsum119894=1

119865119894119873minus1 (119865119894)120591]minus1 (47)





+ 119862120591119905119870119905119862119905+ infinsum119894=1


(48)



119894=1

Ψ (Φ) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φ) (49)



119894=1


119894=1



119894=1


119894=1


(51)




119870+119895 = 119870+ isin 119878119899+119870+ = lim

119895rarrinfin119870+119895 = lim



= Ψ (Φ+119895 ) (53)


119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870+ (54)


lim119895rarrinfin


119870minus119895 = 119870minus isin 119878119899+ (55)




(119865119894)120591 (119905) Φ (119905) 119865119894 (119905) = 119873 (119905) infinsum119894=1

[119868119899 + 119870 (119905) 119865119894 (119905)119873minus1 (119865119894)120591 (119905)]minus1119870 (119905) le Φ (119905) (56)




119894=1


119894=1

(119864119894119905)120591Φ (119905) 119864119894119905+ 119877119905119870119879 = 119876

(57)








119861119905119864119894119905]120591119870119905 + 119870119905 [119860 119905 minus infinsum119894=1

119861119905119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591119873(119905) 119864119894119905+ 119877119905119870119879 = 119876

(58)





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





+ 119862120591119905119870119905119862119905+ infinsum119894=1


(48)



119894=1

Ψ (Φ) 119865119894119873minus1 (119865119894)120591]minus1Ψ (Φ) (49)



119894=1


119894=1



119894=1


119894=1


(51)




119870+119895 = 119870+ isin 119878119899+119870+ = lim

119895rarrinfin119870+119895 = lim



= Ψ (Φ+119895 ) (53)


119894=1

119870119865119894119873minus1 (119865119894)120591]minus1119870+ (54)


lim119895rarrinfin


119870minus119895 = 119870minus isin 119878119899+ (55)




(119865119894)120591 (119905) Φ (119905) 119865119894 (119905) = 119873 (119905) infinsum119894=1

[119868119899 + 119870 (119905) 119865119894 (119905)119873minus1 (119865119894)120591 (119905)]minus1119870 (119905) le Φ (119905) (56)




119894=1


119894=1

(119864119894119905)120591Φ (119905) 119864119894119905+ 119877119905119870119879 = 119876

(57)








119861119905119864119894119905]120591119870119905 + 119870119905 [119860 119905 minus infinsum119894=1

119861119905119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591119873(119905) 119864119894119905+ 119877119905119870119879 = 119876

(58)





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











119861119905119864119894119905]120591119870119905 + 119870119905 [119860 119905 minus infinsum119894=1

119861119905119864119894119905]minus 119870119905119871120591119905119870119905 + 119862120591119905119870119905119862119905 + infinsum

119894=1

(119864119894119905)120591119873(119905) 119864119894119905+ 119877119905119870119879 = 119876

(58)





Acknowledgments



References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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