ANecessaryConditionforOptimalControlofForward-Backward...

11
Research Article ANecessaryConditionforOptimalControlofForward-Backward StochasticControlSystemwithL´ evyProcessinNonconvex Control Domain Case HongHuang , 1 Xiangrong Wang , 2 andYingLi 3,4 1 School of Data and Computer Science, Shandong Women’s University, Jinan 250300, China 2 Institute of Financial Engineering, Shandong University of Science and Technology, Qingdao 266590, China 3 Office of Academic Research, Shandong Women’s University, Jinan 250300, China 4 College of Computer Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China Correspondence should be addressed to Ying Li; [email protected] Received 10 April 2020; Accepted 18 May 2020; Published 3 June 2020 Guest Editor: Yi Qi Copyright © 2020 Hong Huang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper analyzes one kind of optimal control problem which is described by forward-backward stochastic differential equations with L´ evy process (FBSDEL). We derive a necessary condition for the existence of the optimal control by means of spike variational technique, while the control domain is not necessarily convex. Simultaneously, we also get the maximum principle for this control system when there are some initial and terminal state constraints. Finally, a financial example is discussed to illustrate the application of our result. 1.Introduction Stochastic optimal control is an important matter that cannot be neglected in modern control theory in long days. As is known to all, Pontryagin’s [1] maximum principle is one of the main ways to settle the stochastic optimal control problem. By introducing the Hamiltonian function, a necessary condition for the optimal control of stochastic control systems was given by him, which was called the maximum condition. From that time, plenty of works on this issue have been done. Peng [2] was the first one to prove the general maximum principle of the forward-backward stochastic control system with diffusion coefficient containing the control variable by the technique of the second-order Taylor expansion and the second-order duality. He [3] was also the first one to demonstrate the maximum principle of forward-back- ward stochastic control systems from the view of backward stochastic differential equations (BSDE). In Peng’s paper [3], the control domain was convex (in local form); Xu [4] extended this conclusion to the case of the nonconvex control domain (in global form), but the control variables were not included in the diffusion coefficient. And these results were extended to the fully coupled case in the form of local and global by Shi and Wu [5, 6] in 1998 and in 2006, respectively. On the basis of these works, Situ [7] was the first to obtain the maximum principle of the forward stochastic control system with global form of random jumps in 1991. Shi and Wu [8] and Shi [9] acquired the maximum principle for a kind of forward-backward stochastic control system with Poisson jumps in the form of local and global, respectively. e fully coupled forward-backward sto- chastic control system was extended by Liu et al. [10] at the base of Shi and Wu [8], and in the meanwhile, they also obtained the maximum principle with the control system be constrained about initial-terminal state constraints. Considering that in real life, the decision makers could only get partial information but not complete information in most cases; many scholars have paid attention to the partial observable stochastic optimal control problem and have achieved many results (see, for ex- ample, [11–13]). Traditionally, when using a stochastic partial differential equation called the Zakai equation to transform a full-information optimal control problem to the partially ob- servable case, scholars will encounter a difficult problem: an infinite-dimensional optimal control problem. Wang and Wu [14] proposed a backward separation approach and replaced the original state and observation equation with the Zakai equation, Hindawi Mathematical Problems in Engineering Volume 2020, Article ID 1768507, 11 pages https://doi.org/10.1155/2020/1768507

Transcript of ANecessaryConditionforOptimalControlofForward-Backward...

Page 1: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

Research ArticleANecessary Condition for Optimal Control of Forward-BackwardStochastic Control System with Levy Process in NonconvexControl Domain Case

Hong Huang 1 Xiangrong Wang 2 and Ying Li 34

1School of Data and Computer Science Shandong Womenrsquos University Jinan 250300 China2Institute of Financial Engineering Shandong University of Science and Technology Qingdao 266590 China3Office of Academic Research Shandong Womenrsquos University Jinan 250300 China4College of Computer Science and Engineering Shandong University of Science and Technology Qingdao 266590 China

Correspondence should be addressed to Ying Li cherry_jn126com

Received 10 April 2020 Accepted 18 May 2020 Published 3 June 2020

Guest Editor Yi Qi

Copyright copy 2020 Hong Huang et al is 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

is paper analyzes one kind of optimal control problem which is described by forward-backward stochastic differential equations withLevy process (FBSDEL)Wederive a necessary condition for the existence of the optimal control bymeans of spike variational techniquewhile the control domain is not necessarily convex Simultaneously we also get the maximum principle for this control system whenthere are some initial and terminal state constraints Finally a financial example is discussed to illustrate the application of our result

1 Introduction

Stochastic optimal control is an importantmatter that cannot beneglected inmodern control theory in long days As is known toall Pontryaginrsquos [1]maximum principle is one of themain waysto settle the stochastic optimal control problem By introducingtheHamiltonian function a necessary condition for the optimalcontrol of stochastic control systems was given by him whichwas called the maximum condition From that time plenty ofworks on this issue have been done Peng [2] was the first one toprove the general maximum principle of the forward-backwardstochastic control system with diffusion coefficient containingthe control variable by the technique of the second-order Taylorexpansion and the second-order duality He [3] was also the firstone to demonstrate the maximum principle of forward-back-ward stochastic control systems from the view of backwardstochastic differential equations (BSDE) In Pengrsquos paper [3] thecontrol domainwas convex (in local form) Xu [4] extended thisconclusion to the case of the nonconvex control domain (inglobal form) but the control variables were not included in thediffusion coefficient And these results were extended to thefully coupled case in the form of local and global by Shi andWu

[5 6] in 1998 and in 2006 respectively On the basis of theseworks Situ [7] was the first to obtain the maximum principle ofthe forward stochastic control system with global form ofrandom jumps in 1991 Shi andWu [8] and Shi [9] acquired themaximum principle for a kind of forward-backward stochasticcontrol system with Poisson jumps in the form of local andglobal respectively e fully coupled forward-backward sto-chastic control systemwas extended by Liu et al [10] at the baseof Shi andWu [8] and in the meanwhile they also obtained themaximum principle with the control system be constrainedabout initial-terminal state constraints Considering that in reallife the decision makers could only get partial information butnot complete information in most cases many scholars havepaid attention to the partial observable stochastic optimalcontrol problem and have achieved many results (see for ex-ample [11ndash13]) Traditionally when using a stochastic partialdifferential equation called the Zakai equation to transform afull-information optimal control problem to the partially ob-servable case scholars will encounter a difficult problem aninfinite-dimensional optimal control problem Wang and Wu[14] proposed a backward separation approach and replaced theoriginal state and observation equation with the Zakai equation

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 1768507 11 pageshttpsdoiorg10115520201768507

and lots of complicated stochastic calculi in infinite-dimensionalspaces were avoided in this way Based on this approach Xiao[15] studied a partially observed optimal control of forward-backward stochastic systems with random jumps and obtainedthe maximum principle and sufficient conditions of an optimalcontrol under some certain convexity assumptions Wang et al[16] proved the maximum principles for forward-backwardstochastic control systems with correlated state and observationnoises More recent conclusions of the partially observed sto-chastic control problem can be seen from the studies conductedby Wang et al [17] Zhang et al [18] and Xiong et al [19]

In these years through the study of mathematicaleconomics and mathematical finance many scholars turntheir attention to the stochastic control system driven byLevy process In 2000 Nualart and Schoutens [20] built apair of pairwise strongly orthonormal martingales whichwas called the Teugels martingale Meanwhile under someexponential moment conditions they also obtained amartingale representation in that paper Under these twoimportant conclusions for BSDE driven by the Teugelsmartingale they [21] proved the existence and uniquenesstheorem of its solution in the next year From then on anumber of important results were proved Meng and Tang[22] obtained the maximum principle of the forwardstochastic control system driven by Levy process Anecessary and sufficient condition for the existence of theoptimal control of backward stochastic control systemsdriven by Levy process was deduced by Tang and Zhang[23] through convex variation methods and dualitytechniques For the forward-backward stochastic controlsystem driven by Levy process there are also a lot ofachievements based on the existence and uniquenesstheorem of FBSDEL [24] Zhang et al [25] obtained anecessary condition of the optimal control and verifica-tion theorem but in their control system the backwardstate variables yt and zt did not enter the forward partWang and Huang [26] extended this result to the fullycoupled control system and obtained the continuity resultdepending on parameters about FBSDEL and the localform maximum principle Subsequently Huang et al [27]studied this control system with terminal state constraintsand obtained the corresponding necessary maximumprinciple using Ekelandrsquos variational For more recentconclusions about the stochastic control problem drivenby Levy process please refer to [28ndash30]

In this paper we will study the optimal control problemfor forward-backward stochastic control systems driven byLevy process which could be considered as a nonconvexcontrol domain case that is extended from the result of [25]

With the technique of spike variation and Ekelandrsquos vari-ational principle the maximum principle of this type ofcontrol system and the control system with initial and finalstate constraints are obtained

e structure of this paper is as follows Section 2 de-scribes some of the preparations used in this paper emaximum principle and the one with initial and terminalstate constraints as the major results of this paper will beshown in Sections 3 and 4 As an application of the max-imum principle Section 5 gives an optimal consumptionproblem in the financial market Section 6 is the summary ofthis article

2 Preliminary Statement

Let (ΩFt P) be a complete probability space which sat-isfied the usual conditions and the information structure isgiven byFt which is generated by two processes a standardBrownian motion Bt1113864 11138650le tleT valued in Rd and an inde-pendent 1-dimensional Levy process Lt1113864 11138650le tleT of the formLt bt + lt here lt is a pure jump process And assume thatLevy measure ] satisfies the following two conditionsthereby Levy process Lt1113864 11138650le tleT has moments in all orders

(i) 1113938R(1and x2)](dx)ltinfin

(ii) 1113938(minus εε)c eλ |x|](dx)ltinfin forallεgt 0 and some λgt 0

Denote L1t Lt Lt Lt minus Ltminus and Li

t 11139360ltslet(Ls)i

for ige 2 And let Yit Li

t minus E[Lit] (ige 1) be the compensated

power jump process of order i then Teugels martingale isdefined by Hi

t 1113936ij1 cijY

jt the coefficients cij correspond to

orthonormalization of the polynomials 1 x x2 withrespect to the measure μ(dx) υ(dx) + σ2δ0(dx)

en Hit1113864 1113865infini1 are pairwise strongly orthogonal martin-

gales and their predictable quadratic variation processes arelangHi

t Hjt rang δijt δij is an indicator function here And

[Hi Hj]t minus langHit H

jt rang is an Ftminus martingale for more details

of the Teugels martingale see Nualart and Schoutens [20]In the following of this section we shall assume some

notations for a Hilbert space H

l2(H) ϕ |H minus valued 1113936infini1 ϕi2 ltinfin1113966 1113967

L2(ΩH) ξ |

H valuedFT minus measurable E|ξ|2 ltinfinl2(0 TH) ϕi

t | l2(H)minus1113864

valuedFt minus measurable 1113936infini1 E 1113938

T

0 ϕit2dtltinfin

M2(0 TH) ϕ(middot) |1113864

H minus valuedFt minus measurable E 1113938T

0 |ϕt|2dtltinfin

For the following FBSDEL

dxt b t xt yt zt rt( 1113857dt + σ t xt yt zt rt( 1113857dBt + 1113936infin

i1gi t xtminus ytminus zt rt( 1113857dHi

t

minus dyt f t xt yt zt rt( 1113857dt minus ztdBt minus 1113936infin

i1ri

tdHit

x0 a yT Φ xT( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(1)

2 Mathematical Problems in Engineering

where (xt yt zt rt) take the value inΩ times [0 T] times Rn times Rm times

Rmtimesd times l2(Rm) and mappings b σ g and f take the value inRn Rntimesd l2(Rn) and Rm respectively Convenient forwriting set column vector α (x y z)T andA(t α r) (minus Mτf(t α r) Mb(t α r) Mσ(t α r))Twhere M is a m times n full rank matrix

Assumption 1

(i) All mappings in equation (1) are uniformly Lip-schitz continuous in their own argumentsrespectively

(ii) For all (ω t) isin Ω times [0 T] l(ω t 0 0 0 0)

isinM2(0 T Rn+m+mtimesd) times l2(0 T Rm) for l b f σrespectively and g(ω t 0 0 0 0) isin l2(Rn)

(iii) langΦ(x) minus Φ(x) M(x minus x)rangge β|M1113954x|2(iv) langA(tα r) minus A(tα r)α minus αrang + 1113936

infini1langM 1113954gi 1113954rirangle minus μ

1|M1113954x|2 minus μ2(|Mτ1113954y|2 + |Mτ1113954z|2 + 1113936infini1 Mτ 1113954ri2)

where α (x y z) 1113954x x minus x 1113954y y minus y 1113954z z minus z1113954ri ri minus ri 1113954gi gi(tα r) minus gi(tα r) μ1 μ2 and βare nonnegative constants which satisfiedμ1 + μ2 gt 0 μ2 + βgt 0 and μ1 gt 0βgt 0(resp μ2 gt 0)when mgtn (resp ngtm)

en the following existence and uniqueness of thesolution conclusion holds

Lemma 1 9ere exists a unique solution in M2(0 TH)

satisfying FBSDEL (1) under Assumption 19e detailed certification process of this conclusion can be

seen in [24]

3 Stochastic Maximum Principle

In this section for any given admissible control u(middot) weconsider the following stochastic control system

dxt b t xt ut( 1113857dt + σ t xt( 1113857dBt + 1113936infin

i1gi t xtminus( 1113857dHi

t

minus dyt f t xt yt zt rt ut1113872 1113873dt minus ztdBt minus 1113936infin

i1ri

tdHit

x0 a yT Φ xT( 1113857 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where a isin Rn is given An admissible controlu(middot) isinM2(0 T Rp) is an Ft-predictable process whichtakes values in a nonempty subset U of Rp Uad is the set ofall admissible controls

And the performance criterion is

J(u) Ec y0( 1113857 (3)

where c Rm⟶ R is a given Frechet differential functionOur optimal control problem amounts to determining

an admissible control ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (4)

In order to get the necessary conditions for the optimalcontrol we assume ulowastt is the optimal control and thecorresponding solution of (2) is recorded as (xlowastt ylowastt zlowastt rlowastt )

and introduce the ldquospike variational controlrdquo as follows

uεt

vt τ le tle τ + ε

ulowastt otherwise1113896 (5)

and (xεt yε

t zεt rεt) are the state trajectories of uε

t here vt is anarbitrary admissible control and ε is a sufficiently smallconstant

We also need the following assumption and variationalequation (6)

Assumption 2

(i) b f g σ Φ and c are continuously differentiablewith respect to (x y z r u) and the derivatives areall bounded

(ii) ere exists a constant Cgt 0 and it holds that|cy|leC(1 + |y|)

dXt bx(t)Xt + b t uεt( 1113857 minus b t ulowastt( 11138571113858 1113859dt + σx(t)XtdBt + 1113936

infin

i1gi

x(t)XtdHit

minus dYt fx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t uεt( 1113857 minus f t ulowastt( 11138571113960 1113961dt

minus ZtdBt minus 1113936infin

i1Ri

tdHit

X0 0

YT Φx(t)XT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

Mathematical Problems in Engineering 3

Here bx(t) bx(txlowastt ulowastt ) σx(t) σx(txlowastt ) gix(t)

gix(txlowastt ) b(tuε

t) b(txεt u

εt) b(tulowastt ) b (txlowastt ulowastt )

fw(t) fw(txlowastt ylowastt zlowastt rlowastt ulowastt ) (w xyz middot r) f(tuεt)

f(xεt y

εt z

εt r

εt u

εt) and f(tulowastt ) f(xlowastt ylowastt zlowastt rlowastt ulowastt )

Lemma 2 Suppose Assumptions 1 and 2 hold for the first-order variation X Y Z R we have the following estimations

sup0letleT

E Xt

111386811138681113868111386811138681113868111386811138682 leCε2 (7)

sup0letleT

E Xt

111386811138681113868111386811138681113868111386811138684 leCε4 (8)

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138682 leCε2 (9)

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138684 leCε4 (10)

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 (11)

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682ds1113888 1113889

2

leCε4 (12)

sup0letleT

E 1113946T

0Rt

2dsleCε2 (13)

sup0letleT

E 1113946T

0Rt

2ds1113888 1113889

2

leCε4 (14)

Proof We first prove inequations (7) and (8) For theforward part of the first-order variation equation we have

E Xt

111386811138681113868111386811138681113868111386811138682

E 1113946t

0bx(s)Xs + b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds + σx(s)XsdBs + 1113944

infin

i1g

ix(s)XsdH

is

⎧⎨

⎩

⎫⎬

⎭

2

le 4 E 1113946t

0bx(s)Xsds1113888 1113889

2

+ E 1113946t

0b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds1113888 1113889

2

+ E 1113946t

0σx(s)Xs1113858 1113859

2ds + E 1113946t

01113944

infin

i1g

ix(s)Xs

⎡⎣ ⎤⎦2

ds⎧⎨

⎩

⎫⎬

⎭

le 12C2TE 1113946

t

0X

2sds + 4E 1113946

t

0b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds1113888 1113889

2

(15)

Applying Gronwallrsquos inequation we have

E Xt

111386811138681113868111386811138681113868111386811138682 leCε2 for all t isin [0 T] (16)

Similarly (8) holdsWe next estimate Yt Zt and Rt the backward part of the

first-order variation equation can be rewritten as

Yt + 1113946T

tZsdBs + 1113946

T

t1113944

infin

i1R

isdH

is Φx(t)XT + 1113946

T

tfx(t)Xt1113858

+ fy(t)Yt + fz(t)Zt

+ fr(t)Rt

+ f t uεt( 1113857 minus f t u

lowastt( 1113857]dt

(17)

Squaring both sides of (17) and using the fact of

EYt 1113946T

tZsdBs 0

EYt 1113946T

t1113944

infin

i1R

isdH

is 0

E 1113946T

tZsdBs 1113946

T

t1113944

infin

i1R

isdH

is 0

(18)

we get

E Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tZ2sds + E 1113946

T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

E Φx(t)XT + 1113946T

tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t u

εt( 1113857 minus f t u

lowastt( 11138571113960 1113961dt1113896 1113897

2

le 6C2EX

2T + 6C

2TE 1113946

T

tX

2sds + 6C

2TE 1113946

T

tY2sds + 6C

2(T minus t)E 1113946

T

tZ2sds

+ 6C2(T minus t)E 1113946

T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

ds + 6E 1113946T

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(19)

4 Mathematical Problems in Engineering

When t isin [T minus δ T] with δ 112C2 we have

E Yt

111386811138681113868111386811138681113868111386811138682

+12

E 1113946T

tZ2sds +

12

E 1113946T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

le 6C2EX

2T + 6C

2TE 1113946

T

tX

2sds + 6C

2TE

middot 1113946T

tY2sds + 6E 1113946

T

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(20)

Applying Gronwallrsquos inequation we have

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus δ T]

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus δ T]

sup0letleT

E 1113946T

0Rt

2dsleCε2 t isin [T minus δ T]

(21)

Consider the BSDE of the first-order variation equationin the interval [t T minus δ]

Yt + 1113946Tminus δ

tZsdBs + 1113946

Tminus δ

t1113944

infin

i1R

isdH

is YTminus δ

+ 1113946Tminus δ

tfx(t)Xt + fy(t)Yt + fz(t)Zt1113960

+ fr(t)Rt + f t uεt( 1113857 minus f t u

lowastt( 11138571113859dt

(22)

us

E Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946Tminus δ

tZ2sds + E 1113946

Tminus δ

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

E YTminus δ + 1113946Tminus δ

tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt11139601113896

+ f t uεt( 1113857 minus f t u

lowastt( 11138571113859dt1113865

2

le 6C2EX

2T + 6C

2TE 1113946

Tminus δ

tX

2sds + 6C

2TE 1113946

Tminus δ

tY2sds

+ 6C2(T minus t)E 1113946

Tminus δ

tZ2sds + 6C

2(T minus t)E

middot 1113946Tminus δ

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

ds + 6E 1113946Tminus δ

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(23)

So when t isin [T minus 2δ T] with δ 112C2 we have

sup0letleTE Yt

111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus 2δ T]

sup0letleTE 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus 2δ T]

sup0letleTE 1113946T

0Rt

2dsleCε2 t isin [T minus 2δ T]

(24)

After a finite number of iterations (9) (11) and (13) areobtained And (10) (12) and (14) can be proved by using a

similar method and the following inequalitiesE(1113938

T

tZsdBs)

4 ge β1E(1113938T

tZ2

sds)2 β1 gt 0E(1113938

T

t1113936infini1 Ri

sdHis)4 ge β2E( 1113938

T

t(1113936infini1 Ri

s)2ds)2 β2 gt 0

Lemma 3 Under hypothesis Assumptions 1 and 2 it holdsthe following four estimations

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(25)

sup0letleT

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(26)

E 1113946T

0zεt minus zlowastt minus Zt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

(27)

E 1113946T

trεt minus rlowastt minus Rt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(28)

Proof To prove (25) we observe that

1113946t

0b s x

lowasts + Xs u

εs( 1113857ds + 1113946

t

0σ s x

lowasts + Xs( 1113857dBs

+ 1113946t

01113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

1113946t

0b s x

lowasts u

εs( 1113857 + 1113946

1

0bx s x

lowasts + λXs u

εs( 1113857dλXs1113890 1113891ds

+ 1113946t

0σ s x

lowasts( 1113857 + 1113946

1

0σx s x

lowasts + λXs( 1113857dλXs1113890 1113891dBs

+ 1113946t

01113944

infin

i1g

is xlowasts( 1113857 + 1113946

1

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857dλXs

⎡⎣ ⎤⎦dHis

1113946t

0b s x

lowasts ulowasts( 1113857ds + 1113946

t

0σ s x

lowasts( 1113857dBs + 1113946

t

01113944

infin

i1g

is xlowasts( 1113857dH

is

+ 1113946t

0bx s x

lowasts ulowasts( 1113857Xsds + 1113946

t

0σx s x

lowasts( 1113857XsdBs

+ 1113946t

01113944

infin

i1g

ix s x

lowastsminus( 1113857dH

is

+ 1113946t

0b s x

lowasts u

εs( 1113857 minus b s x

lowasts ulowasts( 11138571113858 1113859ds + 1113946

t

0Aεds

+ 1113946t

0BεdBs + 1113946

t

0CεdH

is

xlowastt minus x0 + Xt + 1113946

t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

(29)

Mathematical Problems in Engineering 5

where

Aε

11139461

0bx s x

lowasts + λXs u

εs( 1113857 minus bx s x

lowasts ulowasts( 11138571113858 1113859dλXs

Bε

11139461

0σx s x

lowasts + λXs( 1113857 minus σx s x

lowasts( 11138571113858 1113859dλXs

Cε

11139461

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857 minus g

ix s x

lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs

(30)

It follows easily from Lemma 2 that

sup0letleT

E 1113946t

0A

εds1113888 1113889

2

+ 1113946t

0BεdBs1113888 1113889

2

+ 1113946t

0CεdH

is1113888 1113889

2⎡⎣ ⎤⎦ o ε21113872 1113873

(31)

Since

xεt minus x0 1113946

t

0b s x

εs u

εs( 1113857ds + 1113946

t

0σ s x

εs( 1113857dBs + 1113946

t

01113944

infin

i1g

is x

εsminus( 1113857dH

is

(32)

then

xεt minus xlowastt minus Xt 1113946

t

0b s x

εs u

εs( 1113857 minus b s x

lowasts + Xs u

εs( 11138571113858 1113859ds

+ 1113946t

0σ s x

εs( 1113857 minus σ s x

lowasts + Xs( 11138571113858 1113859dBs

+ 1113946t

01113944

infin

i1g

is x

εsminus( 1113857 minus g

is xlowastsminus + Xsminus( 11138571113872 1113873dH

is

+ 1113946t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

1113946t

0D

εxεs minus xlowasts minus Xs( 1113857ds + 1113946

t

0Eε

xεs minus xlowasts minus Xs( 1113857dBs

+ 1113946t

0Fε

xεs minus xlowasts minus Xs( 1113857dH

is

(33)

with Dε 111393810 bx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs) uεs)dλ

Eε 111393810 σx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs))dλ and Fε 111393810

1113936infini1 gi

x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ

By Gronwallrsquos inequation we have

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(34)

Next we prove (26) (27) (28) it can be easily checkedthat

minus 1113946T

tf s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 1113857ds

+ 1113946T

ts zlowasts + Zs( 1113857dBs + 1113946

T

t1113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

Φ xlowastT( 1113857 minus y

lowastt +Φx x

lowastT( 1113857XT minus Yt minus 1113946

T

tGεds

(35)

Here

Gε

11139461

0fx s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fx(s)1113857dλXs

+ 11139461

0fy s x

lowasts + λXs y

lowasts + λYs z

lowasts(1113872

+ λZs rlowasts + λRs u

εs1113857 minus fy(s)1113873dλYs

+ 11139461

0fz s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fz(s)1113857dλZs

+ 11139461

0fr s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fr(s)1113857dλRs

(36)

Since

yεt Φ x

εT( 1113857 + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857ds minus 1113946

T

tzsdBs

minus 1113946T

t1113944

infin

i1r

iεs dH

is

(37)

then

yεt minus ylowastt minus Yt Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs(1113858 1113859ds

minus 1113946T

tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946

T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873dH

is + 1113946

T

tGεds

(38)

6 Mathematical Problems in Engineering

Squaring both sides of the equation above we get

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tzεs minus zlowasts minus Zs( 1113857

2ds minus E 1113946T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873

2ds

E f s xεs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 11138571113858 1113859ds +Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tGεds1113896 1113897

2

(39)

From Lemma 2 and equation (25) we have

sup0letleTE 1113946T

tGεds1113888 1113889

2

o ε21113872 1113873

E Φ xεT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT1113858 1113859

2 o ε21113872 1113873

(40)

en we can get (26) (27) and (28) by applying theiterative method to the above relations

Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality

Ecy ylowast0( 1113857Y0 ge o(ε) (41)

Proof From the four estimations in Lemma 3 we have thefollowing estimation

E c yε0( 1113857 minus c y

lowast0 + Y0( 11138571113858 1113859 o(ε) (42)

erefore

0leE c ylowast0 + Y0( 1113857 minus c y

lowast0( 11138571113858 1113859 + o(ε) Ecy y

lowast0( 1113857Y0 + o(ε)

(43)

We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times

Rntimesdtimes l2(Rn) as

H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang

+langk g(t x)rang minus langq f(t x y z r u)rang

(44)

and the following adjoint equation

dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin

i1Hi

r ulowastt( 1113857dHit

minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin

i1ki

tminus dHit

q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

where

Hx ulowastt( 1113857 Hx t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hy ulowastt( 1113857 Hy t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hz ulowastt( 1113857 Hz t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hr ulowastt( 1113857 Hr t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

(46)

It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times

l2(0 T Rm)en we get the main result of this section

Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have

H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857

geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)

Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that

E 1113946T

0H t xt yt zt rt u

εt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y

lowast0( 1113857Y0 ge o(ε)

(48)

By the definition of uεt we know that for any

u isin Uad[0 T] the following inequation holds

E H t xt yt zt rt uεt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0

(49)

en (47) can be easily checked

4 Stochastic Control Problem withState Constraints

In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows

EG1 xT( 1113857 0

EG0 y0( 1113857 0(50)

Mathematical Problems in Engineering 7

where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (51)

subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions

Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded

For forallu1(middot) u2(middot) isin Uad let

d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868

11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967

11138681113868111386811138681113868

11138681113868111386811138681113868 (52)

Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0

Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ E G0 y0( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868

11138681113868111386811138682

1113966 111396712

(53)

It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad

Jρ(u)ge 0 Jρ ulowast

( 1113857 ρ

Jρ ulowast

( 1113857le infu(middot)isinUad

Jρ(u) + ρ(54)

It can be obtained by Ekelandrsquos variational principle thatthere exists u

ρt isin Uad such that

(i) Jρ uρ( )le Jρ ulowast( ) ρ

(ii) d uρ ulowast( )le ρradic

(iii) Jρ(v)ge Jρ uρ( ) minusρradic

d uρ v( ) for forallv(middot) isin Uad

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

For fixed ρ and admissible control uρt we define the spike

variation as follows

uρεt

vt τ le tle τ + ε

uρt otherwise

1113896 (56)

for any εgt 0 and it is easy to check from (iii) of (55) that

Jρ uρε( ) minus Jρ uρ( ) +ρradic

d uρε uρ( )ge 0 (57)

Let (xρt y

ρt z

ρt r

ρt ) be the trajectories corresponding to u

ρt

and (xρεt y

ρεt z

ρεt r

ρεt ) be the trajectories corresponding to

uρεt e variational equation we used in this section is the

same as the one in Section 3 with ulowastt uρt and

(xlowastt ylowastt zlowastt rlowastt ) (xρt y

ρt z

ρt r

ρt ) And we also assume that

the solution of this variational equation is (Xρt Y

ρt Z

ρt R

ρt )

Similar to the approach in Lemmas 2 and 3 it can be shownthat

sup0letleTE xρεt minus x

ρt minus X

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

sup0letleTE yρεt minus y

ρt minus Y

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0zρεt minus z

ρt minus Z

ρt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0rρεt minus r

ρt minus R

ρt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(58)

en by (57) the following variational inequality holds

Jρ uρε

( 1113857 minus Jρ uρ

( 1113857 +ρ

radicd u

ρε u

ρ( 1113857

J2ρ uρε( ) minus J2ρ uρ( )

Jρ uρε( ) + Jρ uρ( )+ ε

ρ

radic

langhρε1 E G1x x

ρT( 11138571113858 1113859X

ρTrang +langhρε

0 E G0y yρ0( 11138571113960 1113961Y

ρ0rang

+ hρε

E cy yρ0( 1113857Y

ρ01113960 1113961 + ε

ρ

radic+ o(ε)

(59)

where

hρε1

2E G1 xρT( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε0

2E G0 yρ0( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε

2E c y

ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859

Jρ uρε( ) + Jρ uρ( )

(60)

Now let (pρεt q

ρεt w

ρεt k

ρεt ) be the solution of

minus dpρεt b

ρx(t)( 1113857

τpρεt minus f

ρx(t)( 1113857

τqρεt + σρx(t)( 1113857

τw

ρεt + 1113936infin

i1gρx(t)( 1113857

τkρεt )i

1113890 1113891dt

minus wρεt dBt minus 1113936

infin

i1kρεt( 1113857

idHit

dqρεt f

ρy(t)( 1113857

τqρεt dt + f

ρz(t)( 1113857

τqρεt dBt + 1113936

infin

i1fρir (t)1113872 1113873

τ)q

ρεt dHi

t

q0 minus G0y yρ0( 1113857h

ρε0 + cy y

ρ0( 1113857hρε pT G1x x

ρT( 1113857h

ρε1 minus Φx x

ρT( 1113857q

ρεT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

8 Mathematical Problems in Engineering

Here bρx(t) bx(t x

ρt u

ρt ) σρx(t) σx(t x

ρt )

fρy(t) fy(t x

ρt y

ρt z

ρt r

ρt u

ρt ) etc Applying It 1113954os formula

to langXρt q

ρεt rang + langY

ρt p

ρεt rang variational inequality (59) can be

rewritten as

H t xρt y

ρt z

ρt r

ρt u

ρεt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

+ ερ

radic+ o(ε)ge 0 forallvt isin U ae as

(62)

where the Hamiltonian function H is defined as (44) Since

limε⟶0 hρε0

111386811138681113868111386811138681113868111386811138682

+ hρε1

111386811138681113868111386811138681113868111386811138682

+ hρε1113868111386811138681113868

11138681113868111386811138682

1113872 1113873 1 (63)

there exists a convergent subsequence still denoted by(h

ρε0 h

ρε1 hρε) such that (h

ρε0 h

ρε1 hρε)⟶ (h

ρ0 h

ρ1 hρ) when

ε⟶ 0 with |hρ0|2 + |h

ρ1|2 + |hρ|2 1

Let (pρt q

ρt w

ρt k

ρt ) be the solution of

minus dpρt b

ρx(t)( 1113857

τpρt minus f

ρx(t)( 1113857

τqρt + σρx(t)( 1113857

τw

ρt + 1113936infin

i1gρx(t)( 1113857

τkρt )i

1113890 1113891dt

minus wρtdBt minus 1113936

infin

i1kρt( 1113857

idHit

dqρt f

ρy(t)( 1113857

τqρtdt + f

ρz(t)( 1113857

τqρtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889q

ρtdHi

t

q0 minus G0y yρ0( 1113857h

ρ0 + cy y

ρ0( 1113857hρ pT G1x x

ρT( 1113857h

ρ1 minus Φx x

ρT( 1113857q

ρT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(64)

From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p

ρεt q

ρεt w

ρεt k

ρεt )⟶ (p

ρt q

ρt w

ρt k

ρt ) then in equation

(62) it implies that

H t xρt y

ρt z

ρt r

ρt vt p

ρt q

ρt w

ρt k

ρt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρt q

ρt w

ρt k

ρt( 1113857

+ρ

radic ge 0 forallvt isin U ae as

(65)

Similarly there exists a convergent subsequence(h

ρ0 h

ρ1 h

ρ) such that (h

ρ0 h

ρ1 h

ρ)⟶ (h0 h1 h) when

ρ⟶ 0 with |h0|2 + |h1|

2 + |h|2 1 Since uρt⟶ ulowastt as

ρ⟶ 0 we have (xρt y

ρt z

ρt r

ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and

(Xρt Y

ρt Z

ρt R

ρt )⟶ (Xt Yt Zt Rt) which is the solution of

the variational equation as same as (6)e following adjoint equation is introduced

minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857

τqt + σx(t)( 1113857

τwt + 1113936infin

i1gx(t)( 1113857

τkt)

i1113890 1113891dt

minus wtdBt minus 1113936infin

i1kt( 1113857

idHit

dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857

τqtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889qtdHi

t

q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(66)

Similarly it can be proved that(p

ρt q

ρt w

ρt k

ρt )⟶ (pt qt wt kt) then inequation (65)

impliesH t x

lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857

minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as

(67)

en we get the following theorem

Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint

equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|

2 + |h1|2 + |h|2 1

such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds

9is conclusion can be drawn from the above analysisdirectly

5 A Financial Example

In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3

Mathematical Problems in Engineering 9

Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

x0 Wgt 0 t isin [0 T]

⎧⎪⎪⎨

⎪⎪⎩(68)

where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility

J(C(middot)) E y01113858 1113859 (69)

where yt is the following backward stochastic process

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(70)

with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)

If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

x0 Wgt 0 yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(71)

which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)

gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R

t (1 minus R)) minus ryt

We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to

H t xlowastt ylowastt Ct p

lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx

lowastt minus Ctrang

+langwlowastt σtxlowastt rang +langklowastt g

itxlowastt rang minus langqlowastt Le

minus rt C1minus Rt

1 minus Rminus rylowastt rang

(72)

and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation

dqt rqtdt

minus dpt μtpt + σtwt + 1113936infin

i1ki

t( 1113857τgi

t1113890 1113891dt minus wtdBt minus 1113936infin

i1ki

tdHit

q0 1 pT qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(73)

According to the maximum principle (47) we have

plowastt minus Le

minus rtClowastt( 1113857

minus Rqlowastt (74)

and the optimal consumption rate

Clowastt minus

ert

L

plowasttqlowastt

1113888 1113889

minus 1R

(75)

By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is

Clowastt minus

erT

Leμ(Tminus t)

1113888 1113889

minus 1R

t isin [0 T] (76)

6 Conclusions

In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)

References

[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962

[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990

10 Mathematical Problems in Engineering

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11

Page 2: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

and lots of complicated stochastic calculi in infinite-dimensionalspaces were avoided in this way Based on this approach Xiao[15] studied a partially observed optimal control of forward-backward stochastic systems with random jumps and obtainedthe maximum principle and sufficient conditions of an optimalcontrol under some certain convexity assumptions Wang et al[16] proved the maximum principles for forward-backwardstochastic control systems with correlated state and observationnoises More recent conclusions of the partially observed sto-chastic control problem can be seen from the studies conductedby Wang et al [17] Zhang et al [18] and Xiong et al [19]

In these years through the study of mathematicaleconomics and mathematical finance many scholars turntheir attention to the stochastic control system driven byLevy process In 2000 Nualart and Schoutens [20] built apair of pairwise strongly orthonormal martingales whichwas called the Teugels martingale Meanwhile under someexponential moment conditions they also obtained amartingale representation in that paper Under these twoimportant conclusions for BSDE driven by the Teugelsmartingale they [21] proved the existence and uniquenesstheorem of its solution in the next year From then on anumber of important results were proved Meng and Tang[22] obtained the maximum principle of the forwardstochastic control system driven by Levy process Anecessary and sufficient condition for the existence of theoptimal control of backward stochastic control systemsdriven by Levy process was deduced by Tang and Zhang[23] through convex variation methods and dualitytechniques For the forward-backward stochastic controlsystem driven by Levy process there are also a lot ofachievements based on the existence and uniquenesstheorem of FBSDEL [24] Zhang et al [25] obtained anecessary condition of the optimal control and verifica-tion theorem but in their control system the backwardstate variables yt and zt did not enter the forward partWang and Huang [26] extended this result to the fullycoupled control system and obtained the continuity resultdepending on parameters about FBSDEL and the localform maximum principle Subsequently Huang et al [27]studied this control system with terminal state constraintsand obtained the corresponding necessary maximumprinciple using Ekelandrsquos variational For more recentconclusions about the stochastic control problem drivenby Levy process please refer to [28ndash30]

In this paper we will study the optimal control problemfor forward-backward stochastic control systems driven byLevy process which could be considered as a nonconvexcontrol domain case that is extended from the result of [25]

With the technique of spike variation and Ekelandrsquos vari-ational principle the maximum principle of this type ofcontrol system and the control system with initial and finalstate constraints are obtained

e structure of this paper is as follows Section 2 de-scribes some of the preparations used in this paper emaximum principle and the one with initial and terminalstate constraints as the major results of this paper will beshown in Sections 3 and 4 As an application of the max-imum principle Section 5 gives an optimal consumptionproblem in the financial market Section 6 is the summary ofthis article

2 Preliminary Statement

Let (ΩFt P) be a complete probability space which sat-isfied the usual conditions and the information structure isgiven byFt which is generated by two processes a standardBrownian motion Bt1113864 11138650le tleT valued in Rd and an inde-pendent 1-dimensional Levy process Lt1113864 11138650le tleT of the formLt bt + lt here lt is a pure jump process And assume thatLevy measure ] satisfies the following two conditionsthereby Levy process Lt1113864 11138650le tleT has moments in all orders

(i) 1113938R(1and x2)](dx)ltinfin

(ii) 1113938(minus εε)c eλ |x|](dx)ltinfin forallεgt 0 and some λgt 0

Denote L1t Lt Lt Lt minus Ltminus and Li

t 11139360ltslet(Ls)i

for ige 2 And let Yit Li

t minus E[Lit] (ige 1) be the compensated

power jump process of order i then Teugels martingale isdefined by Hi

t 1113936ij1 cijY

jt the coefficients cij correspond to

orthonormalization of the polynomials 1 x x2 withrespect to the measure μ(dx) υ(dx) + σ2δ0(dx)

en Hit1113864 1113865infini1 are pairwise strongly orthogonal martin-

gales and their predictable quadratic variation processes arelangHi

t Hjt rang δijt δij is an indicator function here And

[Hi Hj]t minus langHit H

jt rang is an Ftminus martingale for more details

of the Teugels martingale see Nualart and Schoutens [20]In the following of this section we shall assume some

notations for a Hilbert space H

l2(H) ϕ |H minus valued 1113936infini1 ϕi2 ltinfin1113966 1113967

L2(ΩH) ξ |

H valuedFT minus measurable E|ξ|2 ltinfinl2(0 TH) ϕi

t | l2(H)minus1113864

valuedFt minus measurable 1113936infini1 E 1113938

T

0 ϕit2dtltinfin

M2(0 TH) ϕ(middot) |1113864

H minus valuedFt minus measurable E 1113938T

0 |ϕt|2dtltinfin

For the following FBSDEL

dxt b t xt yt zt rt( 1113857dt + σ t xt yt zt rt( 1113857dBt + 1113936infin

i1gi t xtminus ytminus zt rt( 1113857dHi

t

minus dyt f t xt yt zt rt( 1113857dt minus ztdBt minus 1113936infin

i1ri

tdHit

x0 a yT Φ xT( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(1)

2 Mathematical Problems in Engineering

where (xt yt zt rt) take the value inΩ times [0 T] times Rn times Rm times

Rmtimesd times l2(Rm) and mappings b σ g and f take the value inRn Rntimesd l2(Rn) and Rm respectively Convenient forwriting set column vector α (x y z)T andA(t α r) (minus Mτf(t α r) Mb(t α r) Mσ(t α r))Twhere M is a m times n full rank matrix

Assumption 1

(i) All mappings in equation (1) are uniformly Lip-schitz continuous in their own argumentsrespectively

(ii) For all (ω t) isin Ω times [0 T] l(ω t 0 0 0 0)

isinM2(0 T Rn+m+mtimesd) times l2(0 T Rm) for l b f σrespectively and g(ω t 0 0 0 0) isin l2(Rn)

(iii) langΦ(x) minus Φ(x) M(x minus x)rangge β|M1113954x|2(iv) langA(tα r) minus A(tα r)α minus αrang + 1113936

infini1langM 1113954gi 1113954rirangle minus μ

1|M1113954x|2 minus μ2(|Mτ1113954y|2 + |Mτ1113954z|2 + 1113936infini1 Mτ 1113954ri2)

where α (x y z) 1113954x x minus x 1113954y y minus y 1113954z z minus z1113954ri ri minus ri 1113954gi gi(tα r) minus gi(tα r) μ1 μ2 and βare nonnegative constants which satisfiedμ1 + μ2 gt 0 μ2 + βgt 0 and μ1 gt 0βgt 0(resp μ2 gt 0)when mgtn (resp ngtm)

en the following existence and uniqueness of thesolution conclusion holds

Lemma 1 9ere exists a unique solution in M2(0 TH)

satisfying FBSDEL (1) under Assumption 19e detailed certification process of this conclusion can be

seen in [24]

3 Stochastic Maximum Principle

In this section for any given admissible control u(middot) weconsider the following stochastic control system

dxt b t xt ut( 1113857dt + σ t xt( 1113857dBt + 1113936infin

i1gi t xtminus( 1113857dHi

t

minus dyt f t xt yt zt rt ut1113872 1113873dt minus ztdBt minus 1113936infin

i1ri

tdHit

x0 a yT Φ xT( 1113857 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where a isin Rn is given An admissible controlu(middot) isinM2(0 T Rp) is an Ft-predictable process whichtakes values in a nonempty subset U of Rp Uad is the set ofall admissible controls

And the performance criterion is

J(u) Ec y0( 1113857 (3)

where c Rm⟶ R is a given Frechet differential functionOur optimal control problem amounts to determining

an admissible control ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (4)

In order to get the necessary conditions for the optimalcontrol we assume ulowastt is the optimal control and thecorresponding solution of (2) is recorded as (xlowastt ylowastt zlowastt rlowastt )

and introduce the ldquospike variational controlrdquo as follows

uεt

vt τ le tle τ + ε

ulowastt otherwise1113896 (5)

and (xεt yε

t zεt rεt) are the state trajectories of uε

t here vt is anarbitrary admissible control and ε is a sufficiently smallconstant

We also need the following assumption and variationalequation (6)

Assumption 2

(i) b f g σ Φ and c are continuously differentiablewith respect to (x y z r u) and the derivatives areall bounded

(ii) ere exists a constant Cgt 0 and it holds that|cy|leC(1 + |y|)

dXt bx(t)Xt + b t uεt( 1113857 minus b t ulowastt( 11138571113858 1113859dt + σx(t)XtdBt + 1113936

infin

i1gi

x(t)XtdHit

minus dYt fx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t uεt( 1113857 minus f t ulowastt( 11138571113960 1113961dt

minus ZtdBt minus 1113936infin

i1Ri

tdHit

X0 0

YT Φx(t)XT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

Mathematical Problems in Engineering 3

Here bx(t) bx(txlowastt ulowastt ) σx(t) σx(txlowastt ) gix(t)

gix(txlowastt ) b(tuε

t) b(txεt u

εt) b(tulowastt ) b (txlowastt ulowastt )

fw(t) fw(txlowastt ylowastt zlowastt rlowastt ulowastt ) (w xyz middot r) f(tuεt)

f(xεt y

εt z

εt r

εt u

εt) and f(tulowastt ) f(xlowastt ylowastt zlowastt rlowastt ulowastt )

Lemma 2 Suppose Assumptions 1 and 2 hold for the first-order variation X Y Z R we have the following estimations

sup0letleT

E Xt

111386811138681113868111386811138681113868111386811138682 leCε2 (7)

sup0letleT

E Xt

111386811138681113868111386811138681113868111386811138684 leCε4 (8)

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138682 leCε2 (9)

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138684 leCε4 (10)

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 (11)

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682ds1113888 1113889

2

leCε4 (12)

sup0letleT

E 1113946T

0Rt

2dsleCε2 (13)

sup0letleT

E 1113946T

0Rt

2ds1113888 1113889

2

leCε4 (14)

Proof We first prove inequations (7) and (8) For theforward part of the first-order variation equation we have

E Xt

111386811138681113868111386811138681113868111386811138682

E 1113946t

0bx(s)Xs + b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds + σx(s)XsdBs + 1113944

infin

i1g

ix(s)XsdH

is

⎧⎨

⎩

⎫⎬

⎭

2

le 4 E 1113946t

0bx(s)Xsds1113888 1113889

2

+ E 1113946t

0b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds1113888 1113889

2

+ E 1113946t

0σx(s)Xs1113858 1113859

2ds + E 1113946t

01113944

infin

i1g

ix(s)Xs

⎡⎣ ⎤⎦2

ds⎧⎨

⎩

⎫⎬

⎭

le 12C2TE 1113946

t

0X

2sds + 4E 1113946

t

0b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds1113888 1113889

2

(15)

Applying Gronwallrsquos inequation we have

E Xt

111386811138681113868111386811138681113868111386811138682 leCε2 for all t isin [0 T] (16)

Similarly (8) holdsWe next estimate Yt Zt and Rt the backward part of the

first-order variation equation can be rewritten as

Yt + 1113946T

tZsdBs + 1113946

T

t1113944

infin

i1R

isdH

is Φx(t)XT + 1113946

T

tfx(t)Xt1113858

+ fy(t)Yt + fz(t)Zt

+ fr(t)Rt

+ f t uεt( 1113857 minus f t u

lowastt( 1113857]dt

(17)

Squaring both sides of (17) and using the fact of

EYt 1113946T

tZsdBs 0

EYt 1113946T

t1113944

infin

i1R

isdH

is 0

E 1113946T

tZsdBs 1113946

T

t1113944

infin

i1R

isdH

is 0

(18)

we get

E Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tZ2sds + E 1113946

T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

E Φx(t)XT + 1113946T

tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t u

εt( 1113857 minus f t u

lowastt( 11138571113960 1113961dt1113896 1113897

2

le 6C2EX

2T + 6C

2TE 1113946

T

tX

2sds + 6C

2TE 1113946

T

tY2sds + 6C

2(T minus t)E 1113946

T

tZ2sds

+ 6C2(T minus t)E 1113946

T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

ds + 6E 1113946T

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(19)

4 Mathematical Problems in Engineering

When t isin [T minus δ T] with δ 112C2 we have

E Yt

111386811138681113868111386811138681113868111386811138682

+12

E 1113946T

tZ2sds +

12

E 1113946T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

le 6C2EX

2T + 6C

2TE 1113946

T

tX

2sds + 6C

2TE

middot 1113946T

tY2sds + 6E 1113946

T

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(20)

Applying Gronwallrsquos inequation we have

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus δ T]

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus δ T]

sup0letleT

E 1113946T

0Rt

2dsleCε2 t isin [T minus δ T]

(21)

Consider the BSDE of the first-order variation equationin the interval [t T minus δ]

Yt + 1113946Tminus δ

tZsdBs + 1113946

Tminus δ

t1113944

infin

i1R

isdH

is YTminus δ

+ 1113946Tminus δ

tfx(t)Xt + fy(t)Yt + fz(t)Zt1113960

+ fr(t)Rt + f t uεt( 1113857 minus f t u

lowastt( 11138571113859dt

(22)

us

E Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946Tminus δ

tZ2sds + E 1113946

Tminus δ

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

E YTminus δ + 1113946Tminus δ

tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt11139601113896

+ f t uεt( 1113857 minus f t u

lowastt( 11138571113859dt1113865

2

le 6C2EX

2T + 6C

2TE 1113946

Tminus δ

tX

2sds + 6C

2TE 1113946

Tminus δ

tY2sds

+ 6C2(T minus t)E 1113946

Tminus δ

tZ2sds + 6C

2(T minus t)E

middot 1113946Tminus δ

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

ds + 6E 1113946Tminus δ

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(23)

So when t isin [T minus 2δ T] with δ 112C2 we have

sup0letleTE Yt

111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus 2δ T]

sup0letleTE 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus 2δ T]

sup0letleTE 1113946T

0Rt

2dsleCε2 t isin [T minus 2δ T]

(24)

After a finite number of iterations (9) (11) and (13) areobtained And (10) (12) and (14) can be proved by using a

similar method and the following inequalitiesE(1113938

T

tZsdBs)

4 ge β1E(1113938T

tZ2

sds)2 β1 gt 0E(1113938

T

t1113936infini1 Ri

sdHis)4 ge β2E( 1113938

T

t(1113936infini1 Ri

s)2ds)2 β2 gt 0

Lemma 3 Under hypothesis Assumptions 1 and 2 it holdsthe following four estimations

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(25)

sup0letleT

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(26)

E 1113946T

0zεt minus zlowastt minus Zt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

(27)

E 1113946T

trεt minus rlowastt minus Rt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(28)

Proof To prove (25) we observe that

1113946t

0b s x

lowasts + Xs u

εs( 1113857ds + 1113946

t

0σ s x

lowasts + Xs( 1113857dBs

+ 1113946t

01113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

1113946t

0b s x

lowasts u

εs( 1113857 + 1113946

1

0bx s x

lowasts + λXs u

εs( 1113857dλXs1113890 1113891ds

+ 1113946t

0σ s x

lowasts( 1113857 + 1113946

1

0σx s x

lowasts + λXs( 1113857dλXs1113890 1113891dBs

+ 1113946t

01113944

infin

i1g

is xlowasts( 1113857 + 1113946

1

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857dλXs

⎡⎣ ⎤⎦dHis

1113946t

0b s x

lowasts ulowasts( 1113857ds + 1113946

t

0σ s x

lowasts( 1113857dBs + 1113946

t

01113944

infin

i1g

is xlowasts( 1113857dH

is

+ 1113946t

0bx s x

lowasts ulowasts( 1113857Xsds + 1113946

t

0σx s x

lowasts( 1113857XsdBs

+ 1113946t

01113944

infin

i1g

ix s x

lowastsminus( 1113857dH

is

+ 1113946t

0b s x

lowasts u

εs( 1113857 minus b s x

lowasts ulowasts( 11138571113858 1113859ds + 1113946

t

0Aεds

+ 1113946t

0BεdBs + 1113946

t

0CεdH

is

xlowastt minus x0 + Xt + 1113946

t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

(29)

Mathematical Problems in Engineering 5

where

Aε

11139461

0bx s x

lowasts + λXs u

εs( 1113857 minus bx s x

lowasts ulowasts( 11138571113858 1113859dλXs

Bε

11139461

0σx s x

lowasts + λXs( 1113857 minus σx s x

lowasts( 11138571113858 1113859dλXs

Cε

11139461

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857 minus g

ix s x

lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs

(30)

It follows easily from Lemma 2 that

sup0letleT

E 1113946t

0A

εds1113888 1113889

2

+ 1113946t

0BεdBs1113888 1113889

2

+ 1113946t

0CεdH

is1113888 1113889

2⎡⎣ ⎤⎦ o ε21113872 1113873

(31)

Since

xεt minus x0 1113946

t

0b s x

εs u

εs( 1113857ds + 1113946

t

0σ s x

εs( 1113857dBs + 1113946

t

01113944

infin

i1g

is x

εsminus( 1113857dH

is

(32)

then

xεt minus xlowastt minus Xt 1113946

t

0b s x

εs u

εs( 1113857 minus b s x

lowasts + Xs u

εs( 11138571113858 1113859ds

+ 1113946t

0σ s x

εs( 1113857 minus σ s x

lowasts + Xs( 11138571113858 1113859dBs

+ 1113946t

01113944

infin

i1g

is x

εsminus( 1113857 minus g

is xlowastsminus + Xsminus( 11138571113872 1113873dH

is

+ 1113946t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

1113946t

0D

εxεs minus xlowasts minus Xs( 1113857ds + 1113946

t

0Eε

xεs minus xlowasts minus Xs( 1113857dBs

+ 1113946t

0Fε

xεs minus xlowasts minus Xs( 1113857dH

is

(33)

with Dε 111393810 bx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs) uεs)dλ

Eε 111393810 σx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs))dλ and Fε 111393810

1113936infini1 gi

x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ

By Gronwallrsquos inequation we have

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(34)

Next we prove (26) (27) (28) it can be easily checkedthat

minus 1113946T

tf s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 1113857ds

+ 1113946T

ts zlowasts + Zs( 1113857dBs + 1113946

T

t1113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

Φ xlowastT( 1113857 minus y

lowastt +Φx x

lowastT( 1113857XT minus Yt minus 1113946

T

tGεds

(35)

Here

Gε

11139461

0fx s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fx(s)1113857dλXs

+ 11139461

0fy s x

lowasts + λXs y

lowasts + λYs z

lowasts(1113872

+ λZs rlowasts + λRs u

εs1113857 minus fy(s)1113873dλYs

+ 11139461

0fz s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fz(s)1113857dλZs

+ 11139461

0fr s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fr(s)1113857dλRs

(36)

Since

yεt Φ x

εT( 1113857 + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857ds minus 1113946

T

tzsdBs

minus 1113946T

t1113944

infin

i1r

iεs dH

is

(37)

then

yεt minus ylowastt minus Yt Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs(1113858 1113859ds

minus 1113946T

tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946

T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873dH

is + 1113946

T

tGεds

(38)

6 Mathematical Problems in Engineering

Squaring both sides of the equation above we get

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tzεs minus zlowasts minus Zs( 1113857

2ds minus E 1113946T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873

2ds

E f s xεs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 11138571113858 1113859ds +Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tGεds1113896 1113897

2

(39)

From Lemma 2 and equation (25) we have

sup0letleTE 1113946T

tGεds1113888 1113889

2

o ε21113872 1113873

E Φ xεT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT1113858 1113859

2 o ε21113872 1113873

(40)

en we can get (26) (27) and (28) by applying theiterative method to the above relations

Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality

Ecy ylowast0( 1113857Y0 ge o(ε) (41)

Proof From the four estimations in Lemma 3 we have thefollowing estimation

E c yε0( 1113857 minus c y

lowast0 + Y0( 11138571113858 1113859 o(ε) (42)

erefore

0leE c ylowast0 + Y0( 1113857 minus c y

lowast0( 11138571113858 1113859 + o(ε) Ecy y

lowast0( 1113857Y0 + o(ε)

(43)

We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times

Rntimesdtimes l2(Rn) as

H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang

+langk g(t x)rang minus langq f(t x y z r u)rang

(44)

and the following adjoint equation

dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin

i1Hi

r ulowastt( 1113857dHit

minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin

i1ki

tminus dHit

q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

where

Hx ulowastt( 1113857 Hx t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hy ulowastt( 1113857 Hy t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hz ulowastt( 1113857 Hz t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hr ulowastt( 1113857 Hr t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

(46)

It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times

l2(0 T Rm)en we get the main result of this section

Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have

H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857

geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)

Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that

E 1113946T

0H t xt yt zt rt u

εt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y

lowast0( 1113857Y0 ge o(ε)

(48)

By the definition of uεt we know that for any

u isin Uad[0 T] the following inequation holds

E H t xt yt zt rt uεt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0

(49)

en (47) can be easily checked

4 Stochastic Control Problem withState Constraints

In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows

EG1 xT( 1113857 0

EG0 y0( 1113857 0(50)

Mathematical Problems in Engineering 7

where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (51)

subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions

Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded

For forallu1(middot) u2(middot) isin Uad let

d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868

11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967

11138681113868111386811138681113868

11138681113868111386811138681113868 (52)

Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0

Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ E G0 y0( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868

11138681113868111386811138682

1113966 111396712

(53)

It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad

Jρ(u)ge 0 Jρ ulowast

( 1113857 ρ

Jρ ulowast

( 1113857le infu(middot)isinUad

Jρ(u) + ρ(54)

It can be obtained by Ekelandrsquos variational principle thatthere exists u

ρt isin Uad such that

(i) Jρ uρ( )le Jρ ulowast( ) ρ

(ii) d uρ ulowast( )le ρradic

(iii) Jρ(v)ge Jρ uρ( ) minusρradic

d uρ v( ) for forallv(middot) isin Uad

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

For fixed ρ and admissible control uρt we define the spike

variation as follows

uρεt

vt τ le tle τ + ε

uρt otherwise

1113896 (56)

for any εgt 0 and it is easy to check from (iii) of (55) that

Jρ uρε( ) minus Jρ uρ( ) +ρradic

d uρε uρ( )ge 0 (57)

Let (xρt y

ρt z

ρt r

ρt ) be the trajectories corresponding to u

ρt

and (xρεt y

ρεt z

ρεt r

ρεt ) be the trajectories corresponding to

uρεt e variational equation we used in this section is the

same as the one in Section 3 with ulowastt uρt and

(xlowastt ylowastt zlowastt rlowastt ) (xρt y

ρt z

ρt r

ρt ) And we also assume that

the solution of this variational equation is (Xρt Y

ρt Z

ρt R

ρt )

Similar to the approach in Lemmas 2 and 3 it can be shownthat

sup0letleTE xρεt minus x

ρt minus X

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

sup0letleTE yρεt minus y

ρt minus Y

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0zρεt minus z

ρt minus Z

ρt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0rρεt minus r

ρt minus R

ρt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(58)

en by (57) the following variational inequality holds

Jρ uρε

( 1113857 minus Jρ uρ

( 1113857 +ρ

radicd u

ρε u

ρ( 1113857

J2ρ uρε( ) minus J2ρ uρ( )

Jρ uρε( ) + Jρ uρ( )+ ε

ρ

radic

langhρε1 E G1x x

ρT( 11138571113858 1113859X

ρTrang +langhρε

0 E G0y yρ0( 11138571113960 1113961Y

ρ0rang

+ hρε

E cy yρ0( 1113857Y

ρ01113960 1113961 + ε

ρ

radic+ o(ε)

(59)

where

hρε1

2E G1 xρT( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε0

2E G0 yρ0( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε

2E c y

ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859

Jρ uρε( ) + Jρ uρ( )

(60)

Now let (pρεt q

ρεt w

ρεt k

ρεt ) be the solution of

minus dpρεt b

ρx(t)( 1113857

τpρεt minus f

ρx(t)( 1113857

τqρεt + σρx(t)( 1113857

τw

ρεt + 1113936infin

i1gρx(t)( 1113857

τkρεt )i

1113890 1113891dt

minus wρεt dBt minus 1113936

infin

i1kρεt( 1113857

idHit

dqρεt f

ρy(t)( 1113857

τqρεt dt + f

ρz(t)( 1113857

τqρεt dBt + 1113936

infin

i1fρir (t)1113872 1113873

τ)q

ρεt dHi

t

q0 minus G0y yρ0( 1113857h

ρε0 + cy y

ρ0( 1113857hρε pT G1x x

ρT( 1113857h

ρε1 minus Φx x

ρT( 1113857q

ρεT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

8 Mathematical Problems in Engineering

Here bρx(t) bx(t x

ρt u

ρt ) σρx(t) σx(t x

ρt )

fρy(t) fy(t x

ρt y

ρt z

ρt r

ρt u

ρt ) etc Applying It 1113954os formula

to langXρt q

ρεt rang + langY

ρt p

ρεt rang variational inequality (59) can be

rewritten as

H t xρt y

ρt z

ρt r

ρt u

ρεt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

+ ερ

radic+ o(ε)ge 0 forallvt isin U ae as

(62)

where the Hamiltonian function H is defined as (44) Since

limε⟶0 hρε0

111386811138681113868111386811138681113868111386811138682

+ hρε1

111386811138681113868111386811138681113868111386811138682

+ hρε1113868111386811138681113868

11138681113868111386811138682

1113872 1113873 1 (63)

there exists a convergent subsequence still denoted by(h

ρε0 h

ρε1 hρε) such that (h

ρε0 h

ρε1 hρε)⟶ (h

ρ0 h

ρ1 hρ) when

ε⟶ 0 with |hρ0|2 + |h

ρ1|2 + |hρ|2 1

Let (pρt q

ρt w

ρt k

ρt ) be the solution of

minus dpρt b

ρx(t)( 1113857

τpρt minus f

ρx(t)( 1113857

τqρt + σρx(t)( 1113857

τw

ρt + 1113936infin

i1gρx(t)( 1113857

τkρt )i

1113890 1113891dt

minus wρtdBt minus 1113936

infin

i1kρt( 1113857

idHit

dqρt f

ρy(t)( 1113857

τqρtdt + f

ρz(t)( 1113857

τqρtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889q

ρtdHi

t

q0 minus G0y yρ0( 1113857h

ρ0 + cy y

ρ0( 1113857hρ pT G1x x

ρT( 1113857h

ρ1 minus Φx x

ρT( 1113857q

ρT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(64)

From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p

ρεt q

ρεt w

ρεt k

ρεt )⟶ (p

ρt q

ρt w

ρt k

ρt ) then in equation

(62) it implies that

H t xρt y

ρt z

ρt r

ρt vt p

ρt q

ρt w

ρt k

ρt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρt q

ρt w

ρt k

ρt( 1113857

+ρ

radic ge 0 forallvt isin U ae as

(65)

Similarly there exists a convergent subsequence(h

ρ0 h

ρ1 h

ρ) such that (h

ρ0 h

ρ1 h

ρ)⟶ (h0 h1 h) when

ρ⟶ 0 with |h0|2 + |h1|

2 + |h|2 1 Since uρt⟶ ulowastt as

ρ⟶ 0 we have (xρt y

ρt z

ρt r

ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and

(Xρt Y

ρt Z

ρt R

ρt )⟶ (Xt Yt Zt Rt) which is the solution of

the variational equation as same as (6)e following adjoint equation is introduced

minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857

τqt + σx(t)( 1113857

τwt + 1113936infin

i1gx(t)( 1113857

τkt)

i1113890 1113891dt

minus wtdBt minus 1113936infin

i1kt( 1113857

idHit

dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857

τqtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889qtdHi

t

q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(66)

Similarly it can be proved that(p

ρt q

ρt w

ρt k

ρt )⟶ (pt qt wt kt) then inequation (65)

impliesH t x

lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857

minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as

(67)

en we get the following theorem

Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint

equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|

2 + |h1|2 + |h|2 1

such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds

9is conclusion can be drawn from the above analysisdirectly

5 A Financial Example

In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3

Mathematical Problems in Engineering 9

Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

x0 Wgt 0 t isin [0 T]

⎧⎪⎪⎨

⎪⎪⎩(68)

where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility

J(C(middot)) E y01113858 1113859 (69)

where yt is the following backward stochastic process

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(70)

with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)

If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

x0 Wgt 0 yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(71)

which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)

gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R

t (1 minus R)) minus ryt

We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to

H t xlowastt ylowastt Ct p

lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx

lowastt minus Ctrang

+langwlowastt σtxlowastt rang +langklowastt g

itxlowastt rang minus langqlowastt Le

minus rt C1minus Rt

1 minus Rminus rylowastt rang

(72)

and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation

dqt rqtdt

minus dpt μtpt + σtwt + 1113936infin

i1ki

t( 1113857τgi

t1113890 1113891dt minus wtdBt minus 1113936infin

i1ki

tdHit

q0 1 pT qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(73)

According to the maximum principle (47) we have

plowastt minus Le

minus rtClowastt( 1113857

minus Rqlowastt (74)

and the optimal consumption rate

Clowastt minus

ert

L

plowasttqlowastt

1113888 1113889

minus 1R

(75)

By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is

Clowastt minus

erT

Leμ(Tminus t)

1113888 1113889

minus 1R

t isin [0 T] (76)

6 Conclusions

In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)

References

[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962

[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990

10 Mathematical Problems in Engineering

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11

Page 3: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

where (xt yt zt rt) take the value inΩ times [0 T] times Rn times Rm times

Rmtimesd times l2(Rm) and mappings b σ g and f take the value inRn Rntimesd l2(Rn) and Rm respectively Convenient forwriting set column vector α (x y z)T andA(t α r) (minus Mτf(t α r) Mb(t α r) Mσ(t α r))Twhere M is a m times n full rank matrix

Assumption 1

(i) All mappings in equation (1) are uniformly Lip-schitz continuous in their own argumentsrespectively

(ii) For all (ω t) isin Ω times [0 T] l(ω t 0 0 0 0)

isinM2(0 T Rn+m+mtimesd) times l2(0 T Rm) for l b f σrespectively and g(ω t 0 0 0 0) isin l2(Rn)

(iii) langΦ(x) minus Φ(x) M(x minus x)rangge β|M1113954x|2(iv) langA(tα r) minus A(tα r)α minus αrang + 1113936

infini1langM 1113954gi 1113954rirangle minus μ

1|M1113954x|2 minus μ2(|Mτ1113954y|2 + |Mτ1113954z|2 + 1113936infini1 Mτ 1113954ri2)

where α (x y z) 1113954x x minus x 1113954y y minus y 1113954z z minus z1113954ri ri minus ri 1113954gi gi(tα r) minus gi(tα r) μ1 μ2 and βare nonnegative constants which satisfiedμ1 + μ2 gt 0 μ2 + βgt 0 and μ1 gt 0βgt 0(resp μ2 gt 0)when mgtn (resp ngtm)

en the following existence and uniqueness of thesolution conclusion holds

Lemma 1 9ere exists a unique solution in M2(0 TH)

satisfying FBSDEL (1) under Assumption 19e detailed certification process of this conclusion can be

seen in [24]

3 Stochastic Maximum Principle

In this section for any given admissible control u(middot) weconsider the following stochastic control system

dxt b t xt ut( 1113857dt + σ t xt( 1113857dBt + 1113936infin

i1gi t xtminus( 1113857dHi

t

minus dyt f t xt yt zt rt ut1113872 1113873dt minus ztdBt minus 1113936infin

i1ri

tdHit

x0 a yT Φ xT( 1113857 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where a isin Rn is given An admissible controlu(middot) isinM2(0 T Rp) is an Ft-predictable process whichtakes values in a nonempty subset U of Rp Uad is the set ofall admissible controls

And the performance criterion is

J(u) Ec y0( 1113857 (3)

where c Rm⟶ R is a given Frechet differential functionOur optimal control problem amounts to determining

an admissible control ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (4)

In order to get the necessary conditions for the optimalcontrol we assume ulowastt is the optimal control and thecorresponding solution of (2) is recorded as (xlowastt ylowastt zlowastt rlowastt )

and introduce the ldquospike variational controlrdquo as follows

uεt

vt τ le tle τ + ε

ulowastt otherwise1113896 (5)

and (xεt yε

t zεt rεt) are the state trajectories of uε

t here vt is anarbitrary admissible control and ε is a sufficiently smallconstant

We also need the following assumption and variationalequation (6)

Assumption 2

(i) b f g σ Φ and c are continuously differentiablewith respect to (x y z r u) and the derivatives areall bounded

(ii) ere exists a constant Cgt 0 and it holds that|cy|leC(1 + |y|)

dXt bx(t)Xt + b t uεt( 1113857 minus b t ulowastt( 11138571113858 1113859dt + σx(t)XtdBt + 1113936

infin

i1gi

x(t)XtdHit

minus dYt fx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t uεt( 1113857 minus f t ulowastt( 11138571113960 1113961dt

minus ZtdBt minus 1113936infin

i1Ri

tdHit

X0 0

YT Φx(t)XT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

Mathematical Problems in Engineering 3

Here bx(t) bx(txlowastt ulowastt ) σx(t) σx(txlowastt ) gix(t)

gix(txlowastt ) b(tuε

t) b(txεt u

εt) b(tulowastt ) b (txlowastt ulowastt )

fw(t) fw(txlowastt ylowastt zlowastt rlowastt ulowastt ) (w xyz middot r) f(tuεt)

f(xεt y

εt z

εt r

εt u

εt) and f(tulowastt ) f(xlowastt ylowastt zlowastt rlowastt ulowastt )

Lemma 2 Suppose Assumptions 1 and 2 hold for the first-order variation X Y Z R we have the following estimations

sup0letleT

E Xt

111386811138681113868111386811138681113868111386811138682 leCε2 (7)

sup0letleT

E Xt

111386811138681113868111386811138681113868111386811138684 leCε4 (8)

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138682 leCε2 (9)

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138684 leCε4 (10)

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 (11)

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682ds1113888 1113889

2

leCε4 (12)

sup0letleT

E 1113946T

0Rt

2dsleCε2 (13)

sup0letleT

E 1113946T

0Rt

2ds1113888 1113889

2

leCε4 (14)

Proof We first prove inequations (7) and (8) For theforward part of the first-order variation equation we have

E Xt

111386811138681113868111386811138681113868111386811138682

E 1113946t

0bx(s)Xs + b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds + σx(s)XsdBs + 1113944

infin

i1g

ix(s)XsdH

is

⎧⎨

⎩

⎫⎬

⎭

2

le 4 E 1113946t

0bx(s)Xsds1113888 1113889

2

+ E 1113946t

0b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds1113888 1113889

2

+ E 1113946t

0σx(s)Xs1113858 1113859

2ds + E 1113946t

01113944

infin

i1g

ix(s)Xs

⎡⎣ ⎤⎦2

ds⎧⎨

⎩

⎫⎬

⎭

le 12C2TE 1113946

t

0X

2sds + 4E 1113946

t

0b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds1113888 1113889

2

(15)

Applying Gronwallrsquos inequation we have

E Xt

111386811138681113868111386811138681113868111386811138682 leCε2 for all t isin [0 T] (16)

Similarly (8) holdsWe next estimate Yt Zt and Rt the backward part of the

first-order variation equation can be rewritten as

Yt + 1113946T

tZsdBs + 1113946

T

t1113944

infin

i1R

isdH

is Φx(t)XT + 1113946

T

tfx(t)Xt1113858

+ fy(t)Yt + fz(t)Zt

+ fr(t)Rt

+ f t uεt( 1113857 minus f t u

lowastt( 1113857]dt

(17)

Squaring both sides of (17) and using the fact of

EYt 1113946T

tZsdBs 0

EYt 1113946T

t1113944

infin

i1R

isdH

is 0

E 1113946T

tZsdBs 1113946

T

t1113944

infin

i1R

isdH

is 0

(18)

we get

E Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tZ2sds + E 1113946

T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

E Φx(t)XT + 1113946T

tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t u

εt( 1113857 minus f t u

lowastt( 11138571113960 1113961dt1113896 1113897

2

le 6C2EX

2T + 6C

2TE 1113946

T

tX

2sds + 6C

2TE 1113946

T

tY2sds + 6C

2(T minus t)E 1113946

T

tZ2sds

+ 6C2(T minus t)E 1113946

T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

ds + 6E 1113946T

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(19)

4 Mathematical Problems in Engineering

When t isin [T minus δ T] with δ 112C2 we have

E Yt

111386811138681113868111386811138681113868111386811138682

+12

E 1113946T

tZ2sds +

12

E 1113946T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

le 6C2EX

2T + 6C

2TE 1113946

T

tX

2sds + 6C

2TE

middot 1113946T

tY2sds + 6E 1113946

T

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(20)

Applying Gronwallrsquos inequation we have

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus δ T]

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus δ T]

sup0letleT

E 1113946T

0Rt

2dsleCε2 t isin [T minus δ T]

(21)

Consider the BSDE of the first-order variation equationin the interval [t T minus δ]

Yt + 1113946Tminus δ

tZsdBs + 1113946

Tminus δ

t1113944

infin

i1R

isdH

is YTminus δ

+ 1113946Tminus δ

tfx(t)Xt + fy(t)Yt + fz(t)Zt1113960

+ fr(t)Rt + f t uεt( 1113857 minus f t u

lowastt( 11138571113859dt

(22)

us

E Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946Tminus δ

tZ2sds + E 1113946

Tminus δ

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

E YTminus δ + 1113946Tminus δ

tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt11139601113896

+ f t uεt( 1113857 minus f t u

lowastt( 11138571113859dt1113865

2

le 6C2EX

2T + 6C

2TE 1113946

Tminus δ

tX

2sds + 6C

2TE 1113946

Tminus δ

tY2sds

+ 6C2(T minus t)E 1113946

Tminus δ

tZ2sds + 6C

2(T minus t)E

middot 1113946Tminus δ

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

ds + 6E 1113946Tminus δ

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(23)

So when t isin [T minus 2δ T] with δ 112C2 we have

sup0letleTE Yt

111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus 2δ T]

sup0letleTE 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus 2δ T]

sup0letleTE 1113946T

0Rt

2dsleCε2 t isin [T minus 2δ T]

(24)

After a finite number of iterations (9) (11) and (13) areobtained And (10) (12) and (14) can be proved by using a

similar method and the following inequalitiesE(1113938

T

tZsdBs)

4 ge β1E(1113938T

tZ2

sds)2 β1 gt 0E(1113938

T

t1113936infini1 Ri

sdHis)4 ge β2E( 1113938

T

t(1113936infini1 Ri

s)2ds)2 β2 gt 0

Lemma 3 Under hypothesis Assumptions 1 and 2 it holdsthe following four estimations

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(25)

sup0letleT

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(26)

E 1113946T

0zεt minus zlowastt minus Zt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

(27)

E 1113946T

trεt minus rlowastt minus Rt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(28)

Proof To prove (25) we observe that

1113946t

0b s x

lowasts + Xs u

εs( 1113857ds + 1113946

t

0σ s x

lowasts + Xs( 1113857dBs

+ 1113946t

01113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

1113946t

0b s x

lowasts u

εs( 1113857 + 1113946

1

0bx s x

lowasts + λXs u

εs( 1113857dλXs1113890 1113891ds

+ 1113946t

0σ s x

lowasts( 1113857 + 1113946

1

0σx s x

lowasts + λXs( 1113857dλXs1113890 1113891dBs

+ 1113946t

01113944

infin

i1g

is xlowasts( 1113857 + 1113946

1

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857dλXs

⎡⎣ ⎤⎦dHis

1113946t

0b s x

lowasts ulowasts( 1113857ds + 1113946

t

0σ s x

lowasts( 1113857dBs + 1113946

t

01113944

infin

i1g

is xlowasts( 1113857dH

is

+ 1113946t

0bx s x

lowasts ulowasts( 1113857Xsds + 1113946

t

0σx s x

lowasts( 1113857XsdBs

+ 1113946t

01113944

infin

i1g

ix s x

lowastsminus( 1113857dH

is

+ 1113946t

0b s x

lowasts u

εs( 1113857 minus b s x

lowasts ulowasts( 11138571113858 1113859ds + 1113946

t

0Aεds

+ 1113946t

0BεdBs + 1113946

t

0CεdH

is

xlowastt minus x0 + Xt + 1113946

t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

(29)

Mathematical Problems in Engineering 5

where

Aε

11139461

0bx s x

lowasts + λXs u

εs( 1113857 minus bx s x

lowasts ulowasts( 11138571113858 1113859dλXs

Bε

11139461

0σx s x

lowasts + λXs( 1113857 minus σx s x

lowasts( 11138571113858 1113859dλXs

Cε

11139461

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857 minus g

ix s x

lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs

(30)

It follows easily from Lemma 2 that

sup0letleT

E 1113946t

0A

εds1113888 1113889

2

+ 1113946t

0BεdBs1113888 1113889

2

+ 1113946t

0CεdH

is1113888 1113889

2⎡⎣ ⎤⎦ o ε21113872 1113873

(31)

Since

xεt minus x0 1113946

t

0b s x

εs u

εs( 1113857ds + 1113946

t

0σ s x

εs( 1113857dBs + 1113946

t

01113944

infin

i1g

is x

εsminus( 1113857dH

is

(32)

then

xεt minus xlowastt minus Xt 1113946

t

0b s x

εs u

εs( 1113857 minus b s x

lowasts + Xs u

εs( 11138571113858 1113859ds

+ 1113946t

0σ s x

εs( 1113857 minus σ s x

lowasts + Xs( 11138571113858 1113859dBs

+ 1113946t

01113944

infin

i1g

is x

εsminus( 1113857 minus g

is xlowastsminus + Xsminus( 11138571113872 1113873dH

is

+ 1113946t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

1113946t

0D

εxεs minus xlowasts minus Xs( 1113857ds + 1113946

t

0Eε

xεs minus xlowasts minus Xs( 1113857dBs

+ 1113946t

0Fε

xεs minus xlowasts minus Xs( 1113857dH

is

(33)

with Dε 111393810 bx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs) uεs)dλ

Eε 111393810 σx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs))dλ and Fε 111393810

1113936infini1 gi

x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ

By Gronwallrsquos inequation we have

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(34)

Next we prove (26) (27) (28) it can be easily checkedthat

minus 1113946T

tf s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 1113857ds

+ 1113946T

ts zlowasts + Zs( 1113857dBs + 1113946

T

t1113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

Φ xlowastT( 1113857 minus y

lowastt +Φx x

lowastT( 1113857XT minus Yt minus 1113946

T

tGεds

(35)

Here

Gε

11139461

0fx s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fx(s)1113857dλXs

+ 11139461

0fy s x

lowasts + λXs y

lowasts + λYs z

lowasts(1113872

+ λZs rlowasts + λRs u

εs1113857 minus fy(s)1113873dλYs

+ 11139461

0fz s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fz(s)1113857dλZs

+ 11139461

0fr s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fr(s)1113857dλRs

(36)

Since

yεt Φ x

εT( 1113857 + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857ds minus 1113946

T

tzsdBs

minus 1113946T

t1113944

infin

i1r

iεs dH

is

(37)

then

yεt minus ylowastt minus Yt Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs(1113858 1113859ds

minus 1113946T

tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946

T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873dH

is + 1113946

T

tGεds

(38)

6 Mathematical Problems in Engineering

Squaring both sides of the equation above we get

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tzεs minus zlowasts minus Zs( 1113857

2ds minus E 1113946T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873

2ds

E f s xεs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 11138571113858 1113859ds +Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tGεds1113896 1113897

2

(39)

From Lemma 2 and equation (25) we have

sup0letleTE 1113946T

tGεds1113888 1113889

2

o ε21113872 1113873

E Φ xεT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT1113858 1113859

2 o ε21113872 1113873

(40)

en we can get (26) (27) and (28) by applying theiterative method to the above relations

Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality

Ecy ylowast0( 1113857Y0 ge o(ε) (41)

Proof From the four estimations in Lemma 3 we have thefollowing estimation

E c yε0( 1113857 minus c y

lowast0 + Y0( 11138571113858 1113859 o(ε) (42)

erefore

0leE c ylowast0 + Y0( 1113857 minus c y

lowast0( 11138571113858 1113859 + o(ε) Ecy y

lowast0( 1113857Y0 + o(ε)

(43)

We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times

Rntimesdtimes l2(Rn) as

H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang

+langk g(t x)rang minus langq f(t x y z r u)rang

(44)

and the following adjoint equation

dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin

i1Hi

r ulowastt( 1113857dHit

minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin

i1ki

tminus dHit

q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

where

Hx ulowastt( 1113857 Hx t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hy ulowastt( 1113857 Hy t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hz ulowastt( 1113857 Hz t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hr ulowastt( 1113857 Hr t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

(46)

It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times

l2(0 T Rm)en we get the main result of this section

Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have

H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857

geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)

Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that

E 1113946T

0H t xt yt zt rt u

εt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y

lowast0( 1113857Y0 ge o(ε)

(48)

By the definition of uεt we know that for any

u isin Uad[0 T] the following inequation holds

E H t xt yt zt rt uεt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0

(49)

en (47) can be easily checked

4 Stochastic Control Problem withState Constraints

In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows

EG1 xT( 1113857 0

EG0 y0( 1113857 0(50)

Mathematical Problems in Engineering 7

where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (51)

subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions

Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded

For forallu1(middot) u2(middot) isin Uad let

d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868

11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967

11138681113868111386811138681113868

11138681113868111386811138681113868 (52)

Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0

Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ E G0 y0( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868

11138681113868111386811138682

1113966 111396712

(53)

It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad

Jρ(u)ge 0 Jρ ulowast

( 1113857 ρ

Jρ ulowast

( 1113857le infu(middot)isinUad

Jρ(u) + ρ(54)

It can be obtained by Ekelandrsquos variational principle thatthere exists u

ρt isin Uad such that

(i) Jρ uρ( )le Jρ ulowast( ) ρ

(ii) d uρ ulowast( )le ρradic

(iii) Jρ(v)ge Jρ uρ( ) minusρradic

d uρ v( ) for forallv(middot) isin Uad

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

For fixed ρ and admissible control uρt we define the spike

variation as follows

uρεt

vt τ le tle τ + ε

uρt otherwise

1113896 (56)

for any εgt 0 and it is easy to check from (iii) of (55) that

Jρ uρε( ) minus Jρ uρ( ) +ρradic

d uρε uρ( )ge 0 (57)

Let (xρt y

ρt z

ρt r

ρt ) be the trajectories corresponding to u

ρt

and (xρεt y

ρεt z

ρεt r

ρεt ) be the trajectories corresponding to

uρεt e variational equation we used in this section is the

same as the one in Section 3 with ulowastt uρt and

(xlowastt ylowastt zlowastt rlowastt ) (xρt y

ρt z

ρt r

ρt ) And we also assume that

the solution of this variational equation is (Xρt Y

ρt Z

ρt R

ρt )

Similar to the approach in Lemmas 2 and 3 it can be shownthat

sup0letleTE xρεt minus x

ρt minus X

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

sup0letleTE yρεt minus y

ρt minus Y

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0zρεt minus z

ρt minus Z

ρt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0rρεt minus r

ρt minus R

ρt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(58)

en by (57) the following variational inequality holds

Jρ uρε

( 1113857 minus Jρ uρ

( 1113857 +ρ

radicd u

ρε u

ρ( 1113857

J2ρ uρε( ) minus J2ρ uρ( )

Jρ uρε( ) + Jρ uρ( )+ ε

ρ

radic

langhρε1 E G1x x

ρT( 11138571113858 1113859X

ρTrang +langhρε

0 E G0y yρ0( 11138571113960 1113961Y

ρ0rang

+ hρε

E cy yρ0( 1113857Y

ρ01113960 1113961 + ε

ρ

radic+ o(ε)

(59)

where

hρε1

2E G1 xρT( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε0

2E G0 yρ0( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε

2E c y

ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859

Jρ uρε( ) + Jρ uρ( )

(60)

Now let (pρεt q

ρεt w

ρεt k

ρεt ) be the solution of

minus dpρεt b

ρx(t)( 1113857

τpρεt minus f

ρx(t)( 1113857

τqρεt + σρx(t)( 1113857

τw

ρεt + 1113936infin

i1gρx(t)( 1113857

τkρεt )i

1113890 1113891dt

minus wρεt dBt minus 1113936

infin

i1kρεt( 1113857

idHit

dqρεt f

ρy(t)( 1113857

τqρεt dt + f

ρz(t)( 1113857

τqρεt dBt + 1113936

infin

i1fρir (t)1113872 1113873

τ)q

ρεt dHi

t

q0 minus G0y yρ0( 1113857h

ρε0 + cy y

ρ0( 1113857hρε pT G1x x

ρT( 1113857h

ρε1 minus Φx x

ρT( 1113857q

ρεT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

8 Mathematical Problems in Engineering

Here bρx(t) bx(t x

ρt u

ρt ) σρx(t) σx(t x

ρt )

fρy(t) fy(t x

ρt y

ρt z

ρt r

ρt u

ρt ) etc Applying It 1113954os formula

to langXρt q

ρεt rang + langY

ρt p

ρεt rang variational inequality (59) can be

rewritten as

H t xρt y

ρt z

ρt r

ρt u

ρεt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

+ ερ

radic+ o(ε)ge 0 forallvt isin U ae as

(62)

where the Hamiltonian function H is defined as (44) Since

limε⟶0 hρε0

111386811138681113868111386811138681113868111386811138682

+ hρε1

111386811138681113868111386811138681113868111386811138682

+ hρε1113868111386811138681113868

11138681113868111386811138682

1113872 1113873 1 (63)

there exists a convergent subsequence still denoted by(h

ρε0 h

ρε1 hρε) such that (h

ρε0 h

ρε1 hρε)⟶ (h

ρ0 h

ρ1 hρ) when

ε⟶ 0 with |hρ0|2 + |h

ρ1|2 + |hρ|2 1

Let (pρt q

ρt w

ρt k

ρt ) be the solution of

minus dpρt b

ρx(t)( 1113857

τpρt minus f

ρx(t)( 1113857

τqρt + σρx(t)( 1113857

τw

ρt + 1113936infin

i1gρx(t)( 1113857

τkρt )i

1113890 1113891dt

minus wρtdBt minus 1113936

infin

i1kρt( 1113857

idHit

dqρt f

ρy(t)( 1113857

τqρtdt + f

ρz(t)( 1113857

τqρtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889q

ρtdHi

t

q0 minus G0y yρ0( 1113857h

ρ0 + cy y

ρ0( 1113857hρ pT G1x x

ρT( 1113857h

ρ1 minus Φx x

ρT( 1113857q

ρT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(64)

From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p

ρεt q

ρεt w

ρεt k

ρεt )⟶ (p

ρt q

ρt w

ρt k

ρt ) then in equation

(62) it implies that

H t xρt y

ρt z

ρt r

ρt vt p

ρt q

ρt w

ρt k

ρt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρt q

ρt w

ρt k

ρt( 1113857

+ρ

radic ge 0 forallvt isin U ae as

(65)

Similarly there exists a convergent subsequence(h

ρ0 h

ρ1 h

ρ) such that (h

ρ0 h

ρ1 h

ρ)⟶ (h0 h1 h) when

ρ⟶ 0 with |h0|2 + |h1|

2 + |h|2 1 Since uρt⟶ ulowastt as

ρ⟶ 0 we have (xρt y

ρt z

ρt r

ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and

(Xρt Y

ρt Z

ρt R

ρt )⟶ (Xt Yt Zt Rt) which is the solution of

the variational equation as same as (6)e following adjoint equation is introduced

minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857

τqt + σx(t)( 1113857

τwt + 1113936infin

i1gx(t)( 1113857

τkt)

i1113890 1113891dt

minus wtdBt minus 1113936infin

i1kt( 1113857

idHit

dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857

τqtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889qtdHi

t

q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(66)

Similarly it can be proved that(p

ρt q

ρt w

ρt k

ρt )⟶ (pt qt wt kt) then inequation (65)

impliesH t x

lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857

minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as

(67)

en we get the following theorem

Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint

equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|

2 + |h1|2 + |h|2 1

such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds

9is conclusion can be drawn from the above analysisdirectly

5 A Financial Example

In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3

Mathematical Problems in Engineering 9

Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

x0 Wgt 0 t isin [0 T]

⎧⎪⎪⎨

⎪⎪⎩(68)

where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility

J(C(middot)) E y01113858 1113859 (69)

where yt is the following backward stochastic process

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(70)

with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)

If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

x0 Wgt 0 yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(71)

which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)

gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R

t (1 minus R)) minus ryt

We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to

H t xlowastt ylowastt Ct p

lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx

lowastt minus Ctrang

+langwlowastt σtxlowastt rang +langklowastt g

itxlowastt rang minus langqlowastt Le

minus rt C1minus Rt

1 minus Rminus rylowastt rang

(72)

and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation

dqt rqtdt

minus dpt μtpt + σtwt + 1113936infin

i1ki

t( 1113857τgi

t1113890 1113891dt minus wtdBt minus 1113936infin

i1ki

tdHit

q0 1 pT qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(73)

According to the maximum principle (47) we have

plowastt minus Le

minus rtClowastt( 1113857

minus Rqlowastt (74)

and the optimal consumption rate

Clowastt minus

ert

L

plowasttqlowastt

1113888 1113889

minus 1R

(75)

By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is

Clowastt minus

erT

Leμ(Tminus t)

1113888 1113889

minus 1R

t isin [0 T] (76)

6 Conclusions

In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)

References

[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962

[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990

10 Mathematical Problems in Engineering

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11

Page 4: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

Here bx(t) bx(txlowastt ulowastt ) σx(t) σx(txlowastt ) gix(t)

gix(txlowastt ) b(tuε

t) b(txεt u

εt) b(tulowastt ) b (txlowastt ulowastt )

fw(t) fw(txlowastt ylowastt zlowastt rlowastt ulowastt ) (w xyz middot r) f(tuεt)

f(xεt y

εt z

εt r

εt u

εt) and f(tulowastt ) f(xlowastt ylowastt zlowastt rlowastt ulowastt )

Lemma 2 Suppose Assumptions 1 and 2 hold for the first-order variation X Y Z R we have the following estimations

sup0letleT

E Xt

111386811138681113868111386811138681113868111386811138682 leCε2 (7)

sup0letleT

E Xt

111386811138681113868111386811138681113868111386811138684 leCε4 (8)

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138682 leCε2 (9)

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138684 leCε4 (10)

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 (11)

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682ds1113888 1113889

2

leCε4 (12)

sup0letleT

E 1113946T

0Rt

2dsleCε2 (13)

sup0letleT

E 1113946T

0Rt

2ds1113888 1113889

2

leCε4 (14)

Proof We first prove inequations (7) and (8) For theforward part of the first-order variation equation we have

E Xt

111386811138681113868111386811138681113868111386811138682

E 1113946t

0bx(s)Xs + b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds + σx(s)XsdBs + 1113944

infin

i1g

ix(s)XsdH

is

⎧⎨

⎩

⎫⎬

⎭

2

le 4 E 1113946t

0bx(s)Xsds1113888 1113889

2

+ E 1113946t

0b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds1113888 1113889

2

+ E 1113946t

0σx(s)Xs1113858 1113859

2ds + E 1113946t

01113944

infin

i1g

ix(s)Xs

⎡⎣ ⎤⎦2

ds⎧⎨

⎩

⎫⎬

⎭

le 12C2TE 1113946

t

0X

2sds + 4E 1113946

t

0b s u

εs( 1113857 minus b s u

lowasts( 11138571113858 1113859ds1113888 1113889

2

(15)

Applying Gronwallrsquos inequation we have

E Xt

111386811138681113868111386811138681113868111386811138682 leCε2 for all t isin [0 T] (16)

Similarly (8) holdsWe next estimate Yt Zt and Rt the backward part of the

first-order variation equation can be rewritten as

Yt + 1113946T

tZsdBs + 1113946

T

t1113944

infin

i1R

isdH

is Φx(t)XT + 1113946

T

tfx(t)Xt1113858

+ fy(t)Yt + fz(t)Zt

+ fr(t)Rt

+ f t uεt( 1113857 minus f t u

lowastt( 1113857]dt

(17)

Squaring both sides of (17) and using the fact of

EYt 1113946T

tZsdBs 0

EYt 1113946T

t1113944

infin

i1R

isdH

is 0

E 1113946T

tZsdBs 1113946

T

t1113944

infin

i1R

isdH

is 0

(18)

we get

E Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tZ2sds + E 1113946

T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

E Φx(t)XT + 1113946T

tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt + f t u

εt( 1113857 minus f t u

lowastt( 11138571113960 1113961dt1113896 1113897

2

le 6C2EX

2T + 6C

2TE 1113946

T

tX

2sds + 6C

2TE 1113946

T

tY2sds + 6C

2(T minus t)E 1113946

T

tZ2sds

+ 6C2(T minus t)E 1113946

T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

ds + 6E 1113946T

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(19)

4 Mathematical Problems in Engineering

When t isin [T minus δ T] with δ 112C2 we have

E Yt

111386811138681113868111386811138681113868111386811138682

+12

E 1113946T

tZ2sds +

12

E 1113946T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

le 6C2EX

2T + 6C

2TE 1113946

T

tX

2sds + 6C

2TE

middot 1113946T

tY2sds + 6E 1113946

T

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(20)

Applying Gronwallrsquos inequation we have

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus δ T]

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus δ T]

sup0letleT

E 1113946T

0Rt

2dsleCε2 t isin [T minus δ T]

(21)

Consider the BSDE of the first-order variation equationin the interval [t T minus δ]

Yt + 1113946Tminus δ

tZsdBs + 1113946

Tminus δ

t1113944

infin

i1R

isdH

is YTminus δ

+ 1113946Tminus δ

tfx(t)Xt + fy(t)Yt + fz(t)Zt1113960

+ fr(t)Rt + f t uεt( 1113857 minus f t u

lowastt( 11138571113859dt

(22)

us

E Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946Tminus δ

tZ2sds + E 1113946

Tminus δ

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

E YTminus δ + 1113946Tminus δ

tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt11139601113896

+ f t uεt( 1113857 minus f t u

lowastt( 11138571113859dt1113865

2

le 6C2EX

2T + 6C

2TE 1113946

Tminus δ

tX

2sds + 6C

2TE 1113946

Tminus δ

tY2sds

+ 6C2(T minus t)E 1113946

Tminus δ

tZ2sds + 6C

2(T minus t)E

middot 1113946Tminus δ

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

ds + 6E 1113946Tminus δ

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(23)

So when t isin [T minus 2δ T] with δ 112C2 we have

sup0letleTE Yt

111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus 2δ T]

sup0letleTE 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus 2δ T]

sup0letleTE 1113946T

0Rt

2dsleCε2 t isin [T minus 2δ T]

(24)

After a finite number of iterations (9) (11) and (13) areobtained And (10) (12) and (14) can be proved by using a

similar method and the following inequalitiesE(1113938

T

tZsdBs)

4 ge β1E(1113938T

tZ2

sds)2 β1 gt 0E(1113938

T

t1113936infini1 Ri

sdHis)4 ge β2E( 1113938

T

t(1113936infini1 Ri

s)2ds)2 β2 gt 0

Lemma 3 Under hypothesis Assumptions 1 and 2 it holdsthe following four estimations

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(25)

sup0letleT

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(26)

E 1113946T

0zεt minus zlowastt minus Zt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

(27)

E 1113946T

trεt minus rlowastt minus Rt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(28)

Proof To prove (25) we observe that

1113946t

0b s x

lowasts + Xs u

εs( 1113857ds + 1113946

t

0σ s x

lowasts + Xs( 1113857dBs

+ 1113946t

01113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

1113946t

0b s x

lowasts u

εs( 1113857 + 1113946

1

0bx s x

lowasts + λXs u

εs( 1113857dλXs1113890 1113891ds

+ 1113946t

0σ s x

lowasts( 1113857 + 1113946

1

0σx s x

lowasts + λXs( 1113857dλXs1113890 1113891dBs

+ 1113946t

01113944

infin

i1g

is xlowasts( 1113857 + 1113946

1

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857dλXs

⎡⎣ ⎤⎦dHis

1113946t

0b s x

lowasts ulowasts( 1113857ds + 1113946

t

0σ s x

lowasts( 1113857dBs + 1113946

t

01113944

infin

i1g

is xlowasts( 1113857dH

is

+ 1113946t

0bx s x

lowasts ulowasts( 1113857Xsds + 1113946

t

0σx s x

lowasts( 1113857XsdBs

+ 1113946t

01113944

infin

i1g

ix s x

lowastsminus( 1113857dH

is

+ 1113946t

0b s x

lowasts u

εs( 1113857 minus b s x

lowasts ulowasts( 11138571113858 1113859ds + 1113946

t

0Aεds

+ 1113946t

0BεdBs + 1113946

t

0CεdH

is

xlowastt minus x0 + Xt + 1113946

t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

(29)

Mathematical Problems in Engineering 5

where

Aε

11139461

0bx s x

lowasts + λXs u

εs( 1113857 minus bx s x

lowasts ulowasts( 11138571113858 1113859dλXs

Bε

11139461

0σx s x

lowasts + λXs( 1113857 minus σx s x

lowasts( 11138571113858 1113859dλXs

Cε

11139461

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857 minus g

ix s x

lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs

(30)

It follows easily from Lemma 2 that

sup0letleT

E 1113946t

0A

εds1113888 1113889

2

+ 1113946t

0BεdBs1113888 1113889

2

+ 1113946t

0CεdH

is1113888 1113889

2⎡⎣ ⎤⎦ o ε21113872 1113873

(31)

Since

xεt minus x0 1113946

t

0b s x

εs u

εs( 1113857ds + 1113946

t

0σ s x

εs( 1113857dBs + 1113946

t

01113944

infin

i1g

is x

εsminus( 1113857dH

is

(32)

then

xεt minus xlowastt minus Xt 1113946

t

0b s x

εs u

εs( 1113857 minus b s x

lowasts + Xs u

εs( 11138571113858 1113859ds

+ 1113946t

0σ s x

εs( 1113857 minus σ s x

lowasts + Xs( 11138571113858 1113859dBs

+ 1113946t

01113944

infin

i1g

is x

εsminus( 1113857 minus g

is xlowastsminus + Xsminus( 11138571113872 1113873dH

is

+ 1113946t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

1113946t

0D

εxεs minus xlowasts minus Xs( 1113857ds + 1113946

t

0Eε

xεs minus xlowasts minus Xs( 1113857dBs

+ 1113946t

0Fε

xεs minus xlowasts minus Xs( 1113857dH

is

(33)

with Dε 111393810 bx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs) uεs)dλ

Eε 111393810 σx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs))dλ and Fε 111393810

1113936infini1 gi

x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ

By Gronwallrsquos inequation we have

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(34)

Next we prove (26) (27) (28) it can be easily checkedthat

minus 1113946T

tf s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 1113857ds

+ 1113946T

ts zlowasts + Zs( 1113857dBs + 1113946

T

t1113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

Φ xlowastT( 1113857 minus y

lowastt +Φx x

lowastT( 1113857XT minus Yt minus 1113946

T

tGεds

(35)

Here

Gε

11139461

0fx s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fx(s)1113857dλXs

+ 11139461

0fy s x

lowasts + λXs y

lowasts + λYs z

lowasts(1113872

+ λZs rlowasts + λRs u

εs1113857 minus fy(s)1113873dλYs

+ 11139461

0fz s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fz(s)1113857dλZs

+ 11139461

0fr s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fr(s)1113857dλRs

(36)

Since

yεt Φ x

εT( 1113857 + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857ds minus 1113946

T

tzsdBs

minus 1113946T

t1113944

infin

i1r

iεs dH

is

(37)

then

yεt minus ylowastt minus Yt Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs(1113858 1113859ds

minus 1113946T

tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946

T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873dH

is + 1113946

T

tGεds

(38)

6 Mathematical Problems in Engineering

Squaring both sides of the equation above we get

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tzεs minus zlowasts minus Zs( 1113857

2ds minus E 1113946T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873

2ds

E f s xεs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 11138571113858 1113859ds +Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tGεds1113896 1113897

2

(39)

From Lemma 2 and equation (25) we have

sup0letleTE 1113946T

tGεds1113888 1113889

2

o ε21113872 1113873

E Φ xεT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT1113858 1113859

2 o ε21113872 1113873

(40)

en we can get (26) (27) and (28) by applying theiterative method to the above relations

Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality

Ecy ylowast0( 1113857Y0 ge o(ε) (41)

Proof From the four estimations in Lemma 3 we have thefollowing estimation

E c yε0( 1113857 minus c y

lowast0 + Y0( 11138571113858 1113859 o(ε) (42)

erefore

0leE c ylowast0 + Y0( 1113857 minus c y

lowast0( 11138571113858 1113859 + o(ε) Ecy y

lowast0( 1113857Y0 + o(ε)

(43)

We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times

Rntimesdtimes l2(Rn) as

H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang

+langk g(t x)rang minus langq f(t x y z r u)rang

(44)

and the following adjoint equation

dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin

i1Hi

r ulowastt( 1113857dHit

minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin

i1ki

tminus dHit

q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

where

Hx ulowastt( 1113857 Hx t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hy ulowastt( 1113857 Hy t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hz ulowastt( 1113857 Hz t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hr ulowastt( 1113857 Hr t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

(46)

It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times

l2(0 T Rm)en we get the main result of this section

Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have

H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857

geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)

Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that

E 1113946T

0H t xt yt zt rt u

εt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y

lowast0( 1113857Y0 ge o(ε)

(48)

By the definition of uεt we know that for any

u isin Uad[0 T] the following inequation holds

E H t xt yt zt rt uεt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0

(49)

en (47) can be easily checked

4 Stochastic Control Problem withState Constraints

In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows

EG1 xT( 1113857 0

EG0 y0( 1113857 0(50)

Mathematical Problems in Engineering 7

where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (51)

subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions

Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded

For forallu1(middot) u2(middot) isin Uad let

d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868

11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967

11138681113868111386811138681113868

11138681113868111386811138681113868 (52)

Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0

Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ E G0 y0( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868

11138681113868111386811138682

1113966 111396712

(53)

It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad

Jρ(u)ge 0 Jρ ulowast

( 1113857 ρ

Jρ ulowast

( 1113857le infu(middot)isinUad

Jρ(u) + ρ(54)

It can be obtained by Ekelandrsquos variational principle thatthere exists u

ρt isin Uad such that

(i) Jρ uρ( )le Jρ ulowast( ) ρ

(ii) d uρ ulowast( )le ρradic

(iii) Jρ(v)ge Jρ uρ( ) minusρradic

d uρ v( ) for forallv(middot) isin Uad

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

For fixed ρ and admissible control uρt we define the spike

variation as follows

uρεt

vt τ le tle τ + ε

uρt otherwise

1113896 (56)

for any εgt 0 and it is easy to check from (iii) of (55) that

Jρ uρε( ) minus Jρ uρ( ) +ρradic

d uρε uρ( )ge 0 (57)

Let (xρt y

ρt z

ρt r

ρt ) be the trajectories corresponding to u

ρt

and (xρεt y

ρεt z

ρεt r

ρεt ) be the trajectories corresponding to

uρεt e variational equation we used in this section is the

same as the one in Section 3 with ulowastt uρt and

(xlowastt ylowastt zlowastt rlowastt ) (xρt y

ρt z

ρt r

ρt ) And we also assume that

the solution of this variational equation is (Xρt Y

ρt Z

ρt R

ρt )

Similar to the approach in Lemmas 2 and 3 it can be shownthat

sup0letleTE xρεt minus x

ρt minus X

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

sup0letleTE yρεt minus y

ρt minus Y

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0zρεt minus z

ρt minus Z

ρt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0rρεt minus r

ρt minus R

ρt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(58)

en by (57) the following variational inequality holds

Jρ uρε

( 1113857 minus Jρ uρ

( 1113857 +ρ

radicd u

ρε u

ρ( 1113857

J2ρ uρε( ) minus J2ρ uρ( )

Jρ uρε( ) + Jρ uρ( )+ ε

ρ

radic

langhρε1 E G1x x

ρT( 11138571113858 1113859X

ρTrang +langhρε

0 E G0y yρ0( 11138571113960 1113961Y

ρ0rang

+ hρε

E cy yρ0( 1113857Y

ρ01113960 1113961 + ε

ρ

radic+ o(ε)

(59)

where

hρε1

2E G1 xρT( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε0

2E G0 yρ0( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε

2E c y

ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859

Jρ uρε( ) + Jρ uρ( )

(60)

Now let (pρεt q

ρεt w

ρεt k

ρεt ) be the solution of

minus dpρεt b

ρx(t)( 1113857

τpρεt minus f

ρx(t)( 1113857

τqρεt + σρx(t)( 1113857

τw

ρεt + 1113936infin

i1gρx(t)( 1113857

τkρεt )i

1113890 1113891dt

minus wρεt dBt minus 1113936

infin

i1kρεt( 1113857

idHit

dqρεt f

ρy(t)( 1113857

τqρεt dt + f

ρz(t)( 1113857

τqρεt dBt + 1113936

infin

i1fρir (t)1113872 1113873

τ)q

ρεt dHi

t

q0 minus G0y yρ0( 1113857h

ρε0 + cy y

ρ0( 1113857hρε pT G1x x

ρT( 1113857h

ρε1 minus Φx x

ρT( 1113857q

ρεT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

8 Mathematical Problems in Engineering

Here bρx(t) bx(t x

ρt u

ρt ) σρx(t) σx(t x

ρt )

fρy(t) fy(t x

ρt y

ρt z

ρt r

ρt u

ρt ) etc Applying It 1113954os formula

to langXρt q

ρεt rang + langY

ρt p

ρεt rang variational inequality (59) can be

rewritten as

H t xρt y

ρt z

ρt r

ρt u

ρεt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

+ ερ

radic+ o(ε)ge 0 forallvt isin U ae as

(62)

where the Hamiltonian function H is defined as (44) Since

limε⟶0 hρε0

111386811138681113868111386811138681113868111386811138682

+ hρε1

111386811138681113868111386811138681113868111386811138682

+ hρε1113868111386811138681113868

11138681113868111386811138682

1113872 1113873 1 (63)

there exists a convergent subsequence still denoted by(h

ρε0 h

ρε1 hρε) such that (h

ρε0 h

ρε1 hρε)⟶ (h

ρ0 h

ρ1 hρ) when

ε⟶ 0 with |hρ0|2 + |h

ρ1|2 + |hρ|2 1

Let (pρt q

ρt w

ρt k

ρt ) be the solution of

minus dpρt b

ρx(t)( 1113857

τpρt minus f

ρx(t)( 1113857

τqρt + σρx(t)( 1113857

τw

ρt + 1113936infin

i1gρx(t)( 1113857

τkρt )i

1113890 1113891dt

minus wρtdBt minus 1113936

infin

i1kρt( 1113857

idHit

dqρt f

ρy(t)( 1113857

τqρtdt + f

ρz(t)( 1113857

τqρtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889q

ρtdHi

t

q0 minus G0y yρ0( 1113857h

ρ0 + cy y

ρ0( 1113857hρ pT G1x x

ρT( 1113857h

ρ1 minus Φx x

ρT( 1113857q

ρT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(64)

From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p

ρεt q

ρεt w

ρεt k

ρεt )⟶ (p

ρt q

ρt w

ρt k

ρt ) then in equation

(62) it implies that

H t xρt y

ρt z

ρt r

ρt vt p

ρt q

ρt w

ρt k

ρt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρt q

ρt w

ρt k

ρt( 1113857

+ρ

radic ge 0 forallvt isin U ae as

(65)

Similarly there exists a convergent subsequence(h

ρ0 h

ρ1 h

ρ) such that (h

ρ0 h

ρ1 h

ρ)⟶ (h0 h1 h) when

ρ⟶ 0 with |h0|2 + |h1|

2 + |h|2 1 Since uρt⟶ ulowastt as

ρ⟶ 0 we have (xρt y

ρt z

ρt r

ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and

(Xρt Y

ρt Z

ρt R

ρt )⟶ (Xt Yt Zt Rt) which is the solution of

the variational equation as same as (6)e following adjoint equation is introduced

minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857

τqt + σx(t)( 1113857

τwt + 1113936infin

i1gx(t)( 1113857

τkt)

i1113890 1113891dt

minus wtdBt minus 1113936infin

i1kt( 1113857

idHit

dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857

τqtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889qtdHi

t

q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(66)

Similarly it can be proved that(p

ρt q

ρt w

ρt k

ρt )⟶ (pt qt wt kt) then inequation (65)

impliesH t x

lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857

minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as

(67)

en we get the following theorem

Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint

equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|

2 + |h1|2 + |h|2 1

such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds

9is conclusion can be drawn from the above analysisdirectly

5 A Financial Example

In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3

Mathematical Problems in Engineering 9

Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

x0 Wgt 0 t isin [0 T]

⎧⎪⎪⎨

⎪⎪⎩(68)

where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility

J(C(middot)) E y01113858 1113859 (69)

where yt is the following backward stochastic process

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(70)

with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)

If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

x0 Wgt 0 yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(71)

which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)

gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R

t (1 minus R)) minus ryt

We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to

H t xlowastt ylowastt Ct p

lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx

lowastt minus Ctrang

+langwlowastt σtxlowastt rang +langklowastt g

itxlowastt rang minus langqlowastt Le

minus rt C1minus Rt

1 minus Rminus rylowastt rang

(72)

and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation

dqt rqtdt

minus dpt μtpt + σtwt + 1113936infin

i1ki

t( 1113857τgi

t1113890 1113891dt minus wtdBt minus 1113936infin

i1ki

tdHit

q0 1 pT qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(73)

According to the maximum principle (47) we have

plowastt minus Le

minus rtClowastt( 1113857

minus Rqlowastt (74)

and the optimal consumption rate

Clowastt minus

ert

L

plowasttqlowastt

1113888 1113889

minus 1R

(75)

By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is

Clowastt minus

erT

Leμ(Tminus t)

1113888 1113889

minus 1R

t isin [0 T] (76)

6 Conclusions

In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)

References

[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962

[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990

10 Mathematical Problems in Engineering

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11

Page 5: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

When t isin [T minus δ T] with δ 112C2 we have

E Yt

111386811138681113868111386811138681113868111386811138682

+12

E 1113946T

tZ2sds +

12

E 1113946T

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

le 6C2EX

2T + 6C

2TE 1113946

T

tX

2sds + 6C

2TE

middot 1113946T

tY2sds + 6E 1113946

T

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(20)

Applying Gronwallrsquos inequation we have

sup0letleT

E Yt

111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus δ T]

sup0letleT

E 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus δ T]

sup0letleT

E 1113946T

0Rt

2dsleCε2 t isin [T minus δ T]

(21)

Consider the BSDE of the first-order variation equationin the interval [t T minus δ]

Yt + 1113946Tminus δ

tZsdBs + 1113946

Tminus δ

t1113944

infin

i1R

isdH

is YTminus δ

+ 1113946Tminus δ

tfx(t)Xt + fy(t)Yt + fz(t)Zt1113960

+ fr(t)Rt + f t uεt( 1113857 minus f t u

lowastt( 11138571113859dt

(22)

us

E Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946Tminus δ

tZ2sds + E 1113946

Tminus δ

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

dHis

E YTminus δ + 1113946Tminus δ

tfx(t)Xt + fy(t)Yt + fz(t)Zt + fr(t)Rt11139601113896

+ f t uεt( 1113857 minus f t u

lowastt( 11138571113859dt1113865

2

le 6C2EX

2T + 6C

2TE 1113946

Tminus δ

tX

2sds + 6C

2TE 1113946

Tminus δ

tY2sds

+ 6C2(T minus t)E 1113946

Tminus δ

tZ2sds + 6C

2(T minus t)E

middot 1113946Tminus δ

t1113944

infin

i1R

is

⎛⎝ ⎞⎠

2

ds + 6E 1113946Tminus δ

tf s u

εs( 1113857 minus f s u

lowasts( 1113857( 1113857ds1113888 1113889

2

(23)

So when t isin [T minus 2δ T] with δ 112C2 we have

sup0letleTE Yt

111386811138681113868111386811138681113868111386811138682 leCε2 t isin [T minus 2δ T]

sup0letleTE 1113946T

0Zt

111386811138681113868111386811138681113868111386811138682dsleCε2 t isin [T minus 2δ T]

sup0letleTE 1113946T

0Rt

2dsleCε2 t isin [T minus 2δ T]

(24)

After a finite number of iterations (9) (11) and (13) areobtained And (10) (12) and (14) can be proved by using a

similar method and the following inequalitiesE(1113938

T

tZsdBs)

4 ge β1E(1113938T

tZ2

sds)2 β1 gt 0E(1113938

T

t1113936infini1 Ri

sdHis)4 ge β2E( 1113938

T

t(1113936infini1 Ri

s)2ds)2 β2 gt 0

Lemma 3 Under hypothesis Assumptions 1 and 2 it holdsthe following four estimations

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(25)

sup0letleT

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(26)

E 1113946T

0zεt minus zlowastt minus Zt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

(27)

E 1113946T

trεt minus rlowastt minus Rt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(28)

Proof To prove (25) we observe that

1113946t

0b s x

lowasts + Xs u

εs( 1113857ds + 1113946

t

0σ s x

lowasts + Xs( 1113857dBs

+ 1113946t

01113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

1113946t

0b s x

lowasts u

εs( 1113857 + 1113946

1

0bx s x

lowasts + λXs u

εs( 1113857dλXs1113890 1113891ds

+ 1113946t

0σ s x

lowasts( 1113857 + 1113946

1

0σx s x

lowasts + λXs( 1113857dλXs1113890 1113891dBs

+ 1113946t

01113944

infin

i1g

is xlowasts( 1113857 + 1113946

1

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857dλXs

⎡⎣ ⎤⎦dHis

1113946t

0b s x

lowasts ulowasts( 1113857ds + 1113946

t

0σ s x

lowasts( 1113857dBs + 1113946

t

01113944

infin

i1g

is xlowasts( 1113857dH

is

+ 1113946t

0bx s x

lowasts ulowasts( 1113857Xsds + 1113946

t

0σx s x

lowasts( 1113857XsdBs

+ 1113946t

01113944

infin

i1g

ix s x

lowastsminus( 1113857dH

is

+ 1113946t

0b s x

lowasts u

εs( 1113857 minus b s x

lowasts ulowasts( 11138571113858 1113859ds + 1113946

t

0Aεds

+ 1113946t

0BεdBs + 1113946

t

0CεdH

is

xlowastt minus x0 + Xt + 1113946

t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

(29)

Mathematical Problems in Engineering 5

where

Aε

11139461

0bx s x

lowasts + λXs u

εs( 1113857 minus bx s x

lowasts ulowasts( 11138571113858 1113859dλXs

Bε

11139461

0σx s x

lowasts + λXs( 1113857 minus σx s x

lowasts( 11138571113858 1113859dλXs

Cε

11139461

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857 minus g

ix s x

lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs

(30)

It follows easily from Lemma 2 that

sup0letleT

E 1113946t

0A

εds1113888 1113889

2

+ 1113946t

0BεdBs1113888 1113889

2

+ 1113946t

0CεdH

is1113888 1113889

2⎡⎣ ⎤⎦ o ε21113872 1113873

(31)

Since

xεt minus x0 1113946

t

0b s x

εs u

εs( 1113857ds + 1113946

t

0σ s x

εs( 1113857dBs + 1113946

t

01113944

infin

i1g

is x

εsminus( 1113857dH

is

(32)

then

xεt minus xlowastt minus Xt 1113946

t

0b s x

εs u

εs( 1113857 minus b s x

lowasts + Xs u

εs( 11138571113858 1113859ds

+ 1113946t

0σ s x

εs( 1113857 minus σ s x

lowasts + Xs( 11138571113858 1113859dBs

+ 1113946t

01113944

infin

i1g

is x

εsminus( 1113857 minus g

is xlowastsminus + Xsminus( 11138571113872 1113873dH

is

+ 1113946t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

1113946t

0D

εxεs minus xlowasts minus Xs( 1113857ds + 1113946

t

0Eε

xεs minus xlowasts minus Xs( 1113857dBs

+ 1113946t

0Fε

xεs minus xlowasts minus Xs( 1113857dH

is

(33)

with Dε 111393810 bx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs) uεs)dλ

Eε 111393810 σx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs))dλ and Fε 111393810

1113936infini1 gi

x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ

By Gronwallrsquos inequation we have

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(34)

Next we prove (26) (27) (28) it can be easily checkedthat

minus 1113946T

tf s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 1113857ds

+ 1113946T

ts zlowasts + Zs( 1113857dBs + 1113946

T

t1113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

Φ xlowastT( 1113857 minus y

lowastt +Φx x

lowastT( 1113857XT minus Yt minus 1113946

T

tGεds

(35)

Here

Gε

11139461

0fx s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fx(s)1113857dλXs

+ 11139461

0fy s x

lowasts + λXs y

lowasts + λYs z

lowasts(1113872

+ λZs rlowasts + λRs u

εs1113857 minus fy(s)1113873dλYs

+ 11139461

0fz s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fz(s)1113857dλZs

+ 11139461

0fr s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fr(s)1113857dλRs

(36)

Since

yεt Φ x

εT( 1113857 + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857ds minus 1113946

T

tzsdBs

minus 1113946T

t1113944

infin

i1r

iεs dH

is

(37)

then

yεt minus ylowastt minus Yt Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs(1113858 1113859ds

minus 1113946T

tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946

T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873dH

is + 1113946

T

tGεds

(38)

6 Mathematical Problems in Engineering

Squaring both sides of the equation above we get

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tzεs minus zlowasts minus Zs( 1113857

2ds minus E 1113946T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873

2ds

E f s xεs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 11138571113858 1113859ds +Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tGεds1113896 1113897

2

(39)

From Lemma 2 and equation (25) we have

sup0letleTE 1113946T

tGεds1113888 1113889

2

o ε21113872 1113873

E Φ xεT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT1113858 1113859

2 o ε21113872 1113873

(40)

en we can get (26) (27) and (28) by applying theiterative method to the above relations

Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality

Ecy ylowast0( 1113857Y0 ge o(ε) (41)

Proof From the four estimations in Lemma 3 we have thefollowing estimation

E c yε0( 1113857 minus c y

lowast0 + Y0( 11138571113858 1113859 o(ε) (42)

erefore

0leE c ylowast0 + Y0( 1113857 minus c y

lowast0( 11138571113858 1113859 + o(ε) Ecy y

lowast0( 1113857Y0 + o(ε)

(43)

We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times

Rntimesdtimes l2(Rn) as

H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang

+langk g(t x)rang minus langq f(t x y z r u)rang

(44)

and the following adjoint equation

dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin

i1Hi

r ulowastt( 1113857dHit

minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin

i1ki

tminus dHit

q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

where

Hx ulowastt( 1113857 Hx t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hy ulowastt( 1113857 Hy t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hz ulowastt( 1113857 Hz t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hr ulowastt( 1113857 Hr t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

(46)

It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times

l2(0 T Rm)en we get the main result of this section

Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have

H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857

geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)

Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that

E 1113946T

0H t xt yt zt rt u

εt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y

lowast0( 1113857Y0 ge o(ε)

(48)

By the definition of uεt we know that for any

u isin Uad[0 T] the following inequation holds

E H t xt yt zt rt uεt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0

(49)

en (47) can be easily checked

4 Stochastic Control Problem withState Constraints

In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows

EG1 xT( 1113857 0

EG0 y0( 1113857 0(50)

Mathematical Problems in Engineering 7

where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (51)

subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions

Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded

For forallu1(middot) u2(middot) isin Uad let

d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868

11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967

11138681113868111386811138681113868

11138681113868111386811138681113868 (52)

Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0

Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ E G0 y0( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868

11138681113868111386811138682

1113966 111396712

(53)

It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad

Jρ(u)ge 0 Jρ ulowast

( 1113857 ρ

Jρ ulowast

( 1113857le infu(middot)isinUad

Jρ(u) + ρ(54)

It can be obtained by Ekelandrsquos variational principle thatthere exists u

ρt isin Uad such that

(i) Jρ uρ( )le Jρ ulowast( ) ρ

(ii) d uρ ulowast( )le ρradic

(iii) Jρ(v)ge Jρ uρ( ) minusρradic

d uρ v( ) for forallv(middot) isin Uad

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

For fixed ρ and admissible control uρt we define the spike

variation as follows

uρεt

vt τ le tle τ + ε

uρt otherwise

1113896 (56)

for any εgt 0 and it is easy to check from (iii) of (55) that

Jρ uρε( ) minus Jρ uρ( ) +ρradic

d uρε uρ( )ge 0 (57)

Let (xρt y

ρt z

ρt r

ρt ) be the trajectories corresponding to u

ρt

and (xρεt y

ρεt z

ρεt r

ρεt ) be the trajectories corresponding to

uρεt e variational equation we used in this section is the

same as the one in Section 3 with ulowastt uρt and

(xlowastt ylowastt zlowastt rlowastt ) (xρt y

ρt z

ρt r

ρt ) And we also assume that

the solution of this variational equation is (Xρt Y

ρt Z

ρt R

ρt )

Similar to the approach in Lemmas 2 and 3 it can be shownthat

sup0letleTE xρεt minus x

ρt minus X

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

sup0letleTE yρεt minus y

ρt minus Y

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0zρεt minus z

ρt minus Z

ρt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0rρεt minus r

ρt minus R

ρt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(58)

en by (57) the following variational inequality holds

Jρ uρε

( 1113857 minus Jρ uρ

( 1113857 +ρ

radicd u

ρε u

ρ( 1113857

J2ρ uρε( ) minus J2ρ uρ( )

Jρ uρε( ) + Jρ uρ( )+ ε

ρ

radic

langhρε1 E G1x x

ρT( 11138571113858 1113859X

ρTrang +langhρε

0 E G0y yρ0( 11138571113960 1113961Y

ρ0rang

+ hρε

E cy yρ0( 1113857Y

ρ01113960 1113961 + ε

ρ

radic+ o(ε)

(59)

where

hρε1

2E G1 xρT( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε0

2E G0 yρ0( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε

2E c y

ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859

Jρ uρε( ) + Jρ uρ( )

(60)

Now let (pρεt q

ρεt w

ρεt k

ρεt ) be the solution of

minus dpρεt b

ρx(t)( 1113857

τpρεt minus f

ρx(t)( 1113857

τqρεt + σρx(t)( 1113857

τw

ρεt + 1113936infin

i1gρx(t)( 1113857

τkρεt )i

1113890 1113891dt

minus wρεt dBt minus 1113936

infin

i1kρεt( 1113857

idHit

dqρεt f

ρy(t)( 1113857

τqρεt dt + f

ρz(t)( 1113857

τqρεt dBt + 1113936

infin

i1fρir (t)1113872 1113873

τ)q

ρεt dHi

t

q0 minus G0y yρ0( 1113857h

ρε0 + cy y

ρ0( 1113857hρε pT G1x x

ρT( 1113857h

ρε1 minus Φx x

ρT( 1113857q

ρεT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

8 Mathematical Problems in Engineering

Here bρx(t) bx(t x

ρt u

ρt ) σρx(t) σx(t x

ρt )

fρy(t) fy(t x

ρt y

ρt z

ρt r

ρt u

ρt ) etc Applying It 1113954os formula

to langXρt q

ρεt rang + langY

ρt p

ρεt rang variational inequality (59) can be

rewritten as

H t xρt y

ρt z

ρt r

ρt u

ρεt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

+ ερ

radic+ o(ε)ge 0 forallvt isin U ae as

(62)

where the Hamiltonian function H is defined as (44) Since

limε⟶0 hρε0

111386811138681113868111386811138681113868111386811138682

+ hρε1

111386811138681113868111386811138681113868111386811138682

+ hρε1113868111386811138681113868

11138681113868111386811138682

1113872 1113873 1 (63)

there exists a convergent subsequence still denoted by(h

ρε0 h

ρε1 hρε) such that (h

ρε0 h

ρε1 hρε)⟶ (h

ρ0 h

ρ1 hρ) when

ε⟶ 0 with |hρ0|2 + |h

ρ1|2 + |hρ|2 1

Let (pρt q

ρt w

ρt k

ρt ) be the solution of

minus dpρt b

ρx(t)( 1113857

τpρt minus f

ρx(t)( 1113857

τqρt + σρx(t)( 1113857

τw

ρt + 1113936infin

i1gρx(t)( 1113857

τkρt )i

1113890 1113891dt

minus wρtdBt minus 1113936

infin

i1kρt( 1113857

idHit

dqρt f

ρy(t)( 1113857

τqρtdt + f

ρz(t)( 1113857

τqρtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889q

ρtdHi

t

q0 minus G0y yρ0( 1113857h

ρ0 + cy y

ρ0( 1113857hρ pT G1x x

ρT( 1113857h

ρ1 minus Φx x

ρT( 1113857q

ρT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(64)

From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p

ρεt q

ρεt w

ρεt k

ρεt )⟶ (p

ρt q

ρt w

ρt k

ρt ) then in equation

(62) it implies that

H t xρt y

ρt z

ρt r

ρt vt p

ρt q

ρt w

ρt k

ρt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρt q

ρt w

ρt k

ρt( 1113857

+ρ

radic ge 0 forallvt isin U ae as

(65)

Similarly there exists a convergent subsequence(h

ρ0 h

ρ1 h

ρ) such that (h

ρ0 h

ρ1 h

ρ)⟶ (h0 h1 h) when

ρ⟶ 0 with |h0|2 + |h1|

2 + |h|2 1 Since uρt⟶ ulowastt as

ρ⟶ 0 we have (xρt y

ρt z

ρt r

ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and

(Xρt Y

ρt Z

ρt R

ρt )⟶ (Xt Yt Zt Rt) which is the solution of

the variational equation as same as (6)e following adjoint equation is introduced

minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857

τqt + σx(t)( 1113857

τwt + 1113936infin

i1gx(t)( 1113857

τkt)

i1113890 1113891dt

minus wtdBt minus 1113936infin

i1kt( 1113857

idHit

dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857

τqtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889qtdHi

t

q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(66)

Similarly it can be proved that(p

ρt q

ρt w

ρt k

ρt )⟶ (pt qt wt kt) then inequation (65)

impliesH t x

lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857

minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as

(67)

en we get the following theorem

Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint

equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|

2 + |h1|2 + |h|2 1

such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds

9is conclusion can be drawn from the above analysisdirectly

5 A Financial Example

In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3

Mathematical Problems in Engineering 9

Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

x0 Wgt 0 t isin [0 T]

⎧⎪⎪⎨

⎪⎪⎩(68)

where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility

J(C(middot)) E y01113858 1113859 (69)

where yt is the following backward stochastic process

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(70)

with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)

If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

x0 Wgt 0 yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(71)

which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)

gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R

t (1 minus R)) minus ryt

We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to

H t xlowastt ylowastt Ct p

lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx

lowastt minus Ctrang

+langwlowastt σtxlowastt rang +langklowastt g

itxlowastt rang minus langqlowastt Le

minus rt C1minus Rt

1 minus Rminus rylowastt rang

(72)

and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation

dqt rqtdt

minus dpt μtpt + σtwt + 1113936infin

i1ki

t( 1113857τgi

t1113890 1113891dt minus wtdBt minus 1113936infin

i1ki

tdHit

q0 1 pT qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(73)

According to the maximum principle (47) we have

plowastt minus Le

minus rtClowastt( 1113857

minus Rqlowastt (74)

and the optimal consumption rate

Clowastt minus

ert

L

plowasttqlowastt

1113888 1113889

minus 1R

(75)

By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is

Clowastt minus

erT

Leμ(Tminus t)

1113888 1113889

minus 1R

t isin [0 T] (76)

6 Conclusions

In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)

References

[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962

[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990

10 Mathematical Problems in Engineering

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11

Page 6: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

where

Aε

11139461

0bx s x

lowasts + λXs u

εs( 1113857 minus bx s x

lowasts ulowasts( 11138571113858 1113859dλXs

Bε

11139461

0σx s x

lowasts + λXs( 1113857 minus σx s x

lowasts( 11138571113858 1113859dλXs

Cε

11139461

01113944

infin

i1g

ix s x

lowasts + λXs( 1113857 minus g

ix s x

lowasts( 11138571113872 1113873⎡⎣ ⎤⎦dλXs

(30)

It follows easily from Lemma 2 that

sup0letleT

E 1113946t

0A

εds1113888 1113889

2

+ 1113946t

0BεdBs1113888 1113889

2

+ 1113946t

0CεdH

is1113888 1113889

2⎡⎣ ⎤⎦ o ε21113872 1113873

(31)

Since

xεt minus x0 1113946

t

0b s x

εs u

εs( 1113857ds + 1113946

t

0σ s x

εs( 1113857dBs + 1113946

t

01113944

infin

i1g

is x

εsminus( 1113857dH

is

(32)

then

xεt minus xlowastt minus Xt 1113946

t

0b s x

εs u

εs( 1113857 minus b s x

lowasts + Xs u

εs( 11138571113858 1113859ds

+ 1113946t

0σ s x

εs( 1113857 minus σ s x

lowasts + Xs( 11138571113858 1113859dBs

+ 1113946t

01113944

infin

i1g

is x

εsminus( 1113857 minus g

is xlowastsminus + Xsminus( 11138571113872 1113873dH

is

+ 1113946t

0Aεds + 1113946

t

0BεdBs + 1113946

t

0CεdH

is

1113946t

0D

εxεs minus xlowasts minus Xs( 1113857ds + 1113946

t

0Eε

xεs minus xlowasts minus Xs( 1113857dBs

+ 1113946t

0Fε

xεs minus xlowasts minus Xs( 1113857dH

is

(33)

with Dε 111393810 bx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs) uεs)dλ

Eε 111393810 σx(s xlowasts + Xs + λ(xε

s minus xlowasts minus Xs))dλ and Fε 111393810

1113936infini1 gi

x(s xlowasts + Xs + λ(xεs minus xlowasts minus Xs))dλ

By Gronwallrsquos inequation we have

sup0letleT

E xεt minus xlowastt minus Xt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

(34)

Next we prove (26) (27) (28) it can be easily checkedthat

minus 1113946T

tf s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 1113857ds

+ 1113946T

ts zlowasts + Zs( 1113857dBs + 1113946

T

t1113944

infin

i1g

is xlowastsminus + Xsminus( 1113857dH

is

Φ xlowastT( 1113857 minus y

lowastt +Φx x

lowastT( 1113857XT minus Yt minus 1113946

T

tGεds

(35)

Here

Gε

11139461

0fx s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fx(s)1113857dλXs

+ 11139461

0fy s x

lowasts + λXs y

lowasts + λYs z

lowasts(1113872

+ λZs rlowasts + λRs u

εs1113857 minus fy(s)1113873dλYs

+ 11139461

0fz s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fz(s)1113857dλZs

+ 11139461

0fr s x

lowasts + λXs y

lowasts + λYs z

lowasts((

+ λZs rlowasts + λRs u

εs1113857 minus fr(s)1113857dλRs

(36)

Since

yεt Φ x

εT( 1113857 + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857ds minus 1113946

T

tzsdBs

minus 1113946T

t1113944

infin

i1r

iεs dH

is

(37)

then

yεt minus ylowastt minus Yt Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tf s x

εs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs(1113858 1113859ds

minus 1113946T

tzεs minus zlowasts minus Zs( 1113857dBs minus 1113946

T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873dH

is + 1113946

T

tGεds

(38)

6 Mathematical Problems in Engineering

Squaring both sides of the equation above we get

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tzεs minus zlowasts minus Zs( 1113857

2ds minus E 1113946T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873

2ds

E f s xεs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 11138571113858 1113859ds +Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tGεds1113896 1113897

2

(39)

From Lemma 2 and equation (25) we have

sup0letleTE 1113946T

tGεds1113888 1113889

2

o ε21113872 1113873

E Φ xεT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT1113858 1113859

2 o ε21113872 1113873

(40)

en we can get (26) (27) and (28) by applying theiterative method to the above relations

Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality

Ecy ylowast0( 1113857Y0 ge o(ε) (41)

Proof From the four estimations in Lemma 3 we have thefollowing estimation

E c yε0( 1113857 minus c y

lowast0 + Y0( 11138571113858 1113859 o(ε) (42)

erefore

0leE c ylowast0 + Y0( 1113857 minus c y

lowast0( 11138571113858 1113859 + o(ε) Ecy y

lowast0( 1113857Y0 + o(ε)

(43)

We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times

Rntimesdtimes l2(Rn) as

H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang

+langk g(t x)rang minus langq f(t x y z r u)rang

(44)

and the following adjoint equation

dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin

i1Hi

r ulowastt( 1113857dHit

minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin

i1ki

tminus dHit

q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

where

Hx ulowastt( 1113857 Hx t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hy ulowastt( 1113857 Hy t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hz ulowastt( 1113857 Hz t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hr ulowastt( 1113857 Hr t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

(46)

It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times

l2(0 T Rm)en we get the main result of this section

Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have

H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857

geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)

Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that

E 1113946T

0H t xt yt zt rt u

εt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y

lowast0( 1113857Y0 ge o(ε)

(48)

By the definition of uεt we know that for any

u isin Uad[0 T] the following inequation holds

E H t xt yt zt rt uεt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0

(49)

en (47) can be easily checked

4 Stochastic Control Problem withState Constraints

In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows

EG1 xT( 1113857 0

EG0 y0( 1113857 0(50)

Mathematical Problems in Engineering 7

where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (51)

subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions

Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded

For forallu1(middot) u2(middot) isin Uad let

d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868

11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967

11138681113868111386811138681113868

11138681113868111386811138681113868 (52)

Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0

Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ E G0 y0( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868

11138681113868111386811138682

1113966 111396712

(53)

It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad

Jρ(u)ge 0 Jρ ulowast

( 1113857 ρ

Jρ ulowast

( 1113857le infu(middot)isinUad

Jρ(u) + ρ(54)

It can be obtained by Ekelandrsquos variational principle thatthere exists u

ρt isin Uad such that

(i) Jρ uρ( )le Jρ ulowast( ) ρ

(ii) d uρ ulowast( )le ρradic

(iii) Jρ(v)ge Jρ uρ( ) minusρradic

d uρ v( ) for forallv(middot) isin Uad

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

For fixed ρ and admissible control uρt we define the spike

variation as follows

uρεt

vt τ le tle τ + ε

uρt otherwise

1113896 (56)

for any εgt 0 and it is easy to check from (iii) of (55) that

Jρ uρε( ) minus Jρ uρ( ) +ρradic

d uρε uρ( )ge 0 (57)

Let (xρt y

ρt z

ρt r

ρt ) be the trajectories corresponding to u

ρt

and (xρεt y

ρεt z

ρεt r

ρεt ) be the trajectories corresponding to

uρεt e variational equation we used in this section is the

same as the one in Section 3 with ulowastt uρt and

(xlowastt ylowastt zlowastt rlowastt ) (xρt y

ρt z

ρt r

ρt ) And we also assume that

the solution of this variational equation is (Xρt Y

ρt Z

ρt R

ρt )

Similar to the approach in Lemmas 2 and 3 it can be shownthat

sup0letleTE xρεt minus x

ρt minus X

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

sup0letleTE yρεt minus y

ρt minus Y

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0zρεt minus z

ρt minus Z

ρt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0rρεt minus r

ρt minus R

ρt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(58)

en by (57) the following variational inequality holds

Jρ uρε

( 1113857 minus Jρ uρ

( 1113857 +ρ

radicd u

ρε u

ρ( 1113857

J2ρ uρε( ) minus J2ρ uρ( )

Jρ uρε( ) + Jρ uρ( )+ ε

ρ

radic

langhρε1 E G1x x

ρT( 11138571113858 1113859X

ρTrang +langhρε

0 E G0y yρ0( 11138571113960 1113961Y

ρ0rang

+ hρε

E cy yρ0( 1113857Y

ρ01113960 1113961 + ε

ρ

radic+ o(ε)

(59)

where

hρε1

2E G1 xρT( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε0

2E G0 yρ0( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε

2E c y

ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859

Jρ uρε( ) + Jρ uρ( )

(60)

Now let (pρεt q

ρεt w

ρεt k

ρεt ) be the solution of

minus dpρεt b

ρx(t)( 1113857

τpρεt minus f

ρx(t)( 1113857

τqρεt + σρx(t)( 1113857

τw

ρεt + 1113936infin

i1gρx(t)( 1113857

τkρεt )i

1113890 1113891dt

minus wρεt dBt minus 1113936

infin

i1kρεt( 1113857

idHit

dqρεt f

ρy(t)( 1113857

τqρεt dt + f

ρz(t)( 1113857

τqρεt dBt + 1113936

infin

i1fρir (t)1113872 1113873

τ)q

ρεt dHi

t

q0 minus G0y yρ0( 1113857h

ρε0 + cy y

ρ0( 1113857hρε pT G1x x

ρT( 1113857h

ρε1 minus Φx x

ρT( 1113857q

ρεT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

8 Mathematical Problems in Engineering

Here bρx(t) bx(t x

ρt u

ρt ) σρx(t) σx(t x

ρt )

fρy(t) fy(t x

ρt y

ρt z

ρt r

ρt u

ρt ) etc Applying It 1113954os formula

to langXρt q

ρεt rang + langY

ρt p

ρεt rang variational inequality (59) can be

rewritten as

H t xρt y

ρt z

ρt r

ρt u

ρεt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

+ ερ

radic+ o(ε)ge 0 forallvt isin U ae as

(62)

where the Hamiltonian function H is defined as (44) Since

limε⟶0 hρε0

111386811138681113868111386811138681113868111386811138682

+ hρε1

111386811138681113868111386811138681113868111386811138682

+ hρε1113868111386811138681113868

11138681113868111386811138682

1113872 1113873 1 (63)

there exists a convergent subsequence still denoted by(h

ρε0 h

ρε1 hρε) such that (h

ρε0 h

ρε1 hρε)⟶ (h

ρ0 h

ρ1 hρ) when

ε⟶ 0 with |hρ0|2 + |h

ρ1|2 + |hρ|2 1

Let (pρt q

ρt w

ρt k

ρt ) be the solution of

minus dpρt b

ρx(t)( 1113857

τpρt minus f

ρx(t)( 1113857

τqρt + σρx(t)( 1113857

τw

ρt + 1113936infin

i1gρx(t)( 1113857

τkρt )i

1113890 1113891dt

minus wρtdBt minus 1113936

infin

i1kρt( 1113857

idHit

dqρt f

ρy(t)( 1113857

τqρtdt + f

ρz(t)( 1113857

τqρtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889q

ρtdHi

t

q0 minus G0y yρ0( 1113857h

ρ0 + cy y

ρ0( 1113857hρ pT G1x x

ρT( 1113857h

ρ1 minus Φx x

ρT( 1113857q

ρT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(64)

From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p

ρεt q

ρεt w

ρεt k

ρεt )⟶ (p

ρt q

ρt w

ρt k

ρt ) then in equation

(62) it implies that

H t xρt y

ρt z

ρt r

ρt vt p

ρt q

ρt w

ρt k

ρt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρt q

ρt w

ρt k

ρt( 1113857

+ρ

radic ge 0 forallvt isin U ae as

(65)

Similarly there exists a convergent subsequence(h

ρ0 h

ρ1 h

ρ) such that (h

ρ0 h

ρ1 h

ρ)⟶ (h0 h1 h) when

ρ⟶ 0 with |h0|2 + |h1|

2 + |h|2 1 Since uρt⟶ ulowastt as

ρ⟶ 0 we have (xρt y

ρt z

ρt r

ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and

(Xρt Y

ρt Z

ρt R

ρt )⟶ (Xt Yt Zt Rt) which is the solution of

the variational equation as same as (6)e following adjoint equation is introduced

minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857

τqt + σx(t)( 1113857

τwt + 1113936infin

i1gx(t)( 1113857

τkt)

i1113890 1113891dt

minus wtdBt minus 1113936infin

i1kt( 1113857

idHit

dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857

τqtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889qtdHi

t

q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(66)

Similarly it can be proved that(p

ρt q

ρt w

ρt k

ρt )⟶ (pt qt wt kt) then inequation (65)

impliesH t x

lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857

minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as

(67)

en we get the following theorem

Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint

equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|

2 + |h1|2 + |h|2 1

such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds

9is conclusion can be drawn from the above analysisdirectly

5 A Financial Example

In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3

Mathematical Problems in Engineering 9

Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

x0 Wgt 0 t isin [0 T]

⎧⎪⎪⎨

⎪⎪⎩(68)

where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility

J(C(middot)) E y01113858 1113859 (69)

where yt is the following backward stochastic process

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(70)

with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)

If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

x0 Wgt 0 yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(71)

which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)

gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R

t (1 minus R)) minus ryt

We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to

H t xlowastt ylowastt Ct p

lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx

lowastt minus Ctrang

+langwlowastt σtxlowastt rang +langklowastt g

itxlowastt rang minus langqlowastt Le

minus rt C1minus Rt

1 minus Rminus rylowastt rang

(72)

and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation

dqt rqtdt

minus dpt μtpt + σtwt + 1113936infin

i1ki

t( 1113857τgi

t1113890 1113891dt minus wtdBt minus 1113936infin

i1ki

tdHit

q0 1 pT qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(73)

According to the maximum principle (47) we have

plowastt minus Le

minus rtClowastt( 1113857

minus Rqlowastt (74)

and the optimal consumption rate

Clowastt minus

ert

L

plowasttqlowastt

1113888 1113889

minus 1R

(75)

By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is

Clowastt minus

erT

Leμ(Tminus t)

1113888 1113889

minus 1R

t isin [0 T] (76)

6 Conclusions

In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)

References

[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962

[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990

10 Mathematical Problems in Engineering

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11

Page 7: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

Squaring both sides of the equation above we get

E yεt minus ylowastt minus Yt

111386811138681113868111386811138681113868111386811138682

+ E 1113946T

tzεs minus zlowasts minus Zs( 1113857

2ds minus E 1113946T

t1113944

infin

i1r

iεs minus r

ilowasts minus Rs1113872 1113873

2ds

E f s xεs y

εs z

εs r

εs u

εs( 1113857 minus f s x

lowasts + Xs y

lowasts + Ys z

lowasts + Zs r

lowasts + Rs u

εs( 11138571113858 1113859ds +Φ x

εT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT + 1113946

T

tGεds1113896 1113897

2

(39)

From Lemma 2 and equation (25) we have

sup0letleTE 1113946T

tGεds1113888 1113889

2

o ε21113872 1113873

E Φ xεT( 1113857 minus Φ x

lowastT( 1113857 minus Φx x

lowastT( 1113857XT1113858 1113859

2 o ε21113872 1113873

(40)

en we can get (26) (27) and (28) by applying theiterative method to the above relations

Lemma 4 (variational inequality) Under the conditionsthat Assumptions 1 and 2 are established we can get thefollowing variational inequality

Ecy ylowast0( 1113857Y0 ge o(ε) (41)

Proof From the four estimations in Lemma 3 we have thefollowing estimation

E c yε0( 1113857 minus c y

lowast0 + Y0( 11138571113858 1113859 o(ε) (42)

erefore

0leE c ylowast0 + Y0( 1113857 minus c y

lowast0( 11138571113858 1113859 + o(ε) Ecy y

lowast0( 1113857Y0 + o(ε)

(43)

We introduce the following Hamiltonian functionH [0 T] times Rn times Rm times Rmtimesd times l2(Rm) times U times Rn times Rm times

Rntimesdtimes l2(Rn) as

H(t x y z r u p q w k) langp b(t x u)rang +langw σ(t x)rang

+langk g(t x)rang minus langq f(t x y z r u)rang

(44)

and the following adjoint equation

dqt Hy ulowastt( 1113857dt + Hz ulowastt( 1113857dBt + 1113936infin

i1Hi

r ulowastt( 1113857dHit

minus dpt Hx ulowastt( 1113857dt minus wtdBt minus 1113936infin

i1ki

tminus dHit

q0 cy ylowast0( 1113857 pT minus Φx xlowastT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

where

Hx ulowastt( 1113857 Hx t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hy ulowastt( 1113857 Hy t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hz ulowastt( 1113857 Hz t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

Hr ulowastt( 1113857 Hr t x

lowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857

(46)

It is easily check that adjoint equation (45) has a uniquesolution quartet (pt qt wt kt) isinM2(0 T Rn+m+mtimesd) times

l2(0 T Rm)en we get the main result of this section

Theorem 1 Let hypothesis Assumptions 1 and2 hold ulowastt isan optimal control and the corresponding optimal statetrajectories are (xlowastt ylowastt zlowastt rlowastt ) let (pt qt wt kt) be thesolution of adjoint equation (45) and then for each ad-missible control ut isin Uad[0 T] we have

H txlowastt ylowastt zlowastt rlowastt utpt qtwtkt( 1113857

geH txlowastt ylowastt zlowastt rlowastt ulowastt pt qtwtkt( 1113857asae(47)

Proof Applying It 1113954os formula to langp Xrang and langq Yrang itfollows from (6) (45) and the variational inequality that

E 1113946T

0H t xt yt zt rt u

εt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859dt Ecy y

lowast0( 1113857Y0 ge o(ε)

(48)

By the definition of uεt we know that for any

u isin Uad[0 T] the following inequation holds

E H t xt yt zt rt uεt pt qt wt kt( 11138571113858

minus H t xt yt zt rt ulowastt pt qt wt kt( 11138571113859ge 0

(49)

en (47) can be easily checked

4 Stochastic Control Problem withState Constraints

In this part we are going to discuss stochastic controlproblems with state constraints in control system (2)Specifically the initial state constraints and final stateconstraints are as follows

EG1 xT( 1113857 0

EG0 y0( 1113857 0(50)

Mathematical Problems in Engineering 7

where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (51)

subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions

Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded

For forallu1(middot) u2(middot) isin Uad let

d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868

11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967

11138681113868111386811138681113868

11138681113868111386811138681113868 (52)

Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0

Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ E G0 y0( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868

11138681113868111386811138682

1113966 111396712

(53)

It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad

Jρ(u)ge 0 Jρ ulowast

( 1113857 ρ

Jρ ulowast

( 1113857le infu(middot)isinUad

Jρ(u) + ρ(54)

It can be obtained by Ekelandrsquos variational principle thatthere exists u

ρt isin Uad such that

(i) Jρ uρ( )le Jρ ulowast( ) ρ

(ii) d uρ ulowast( )le ρradic

(iii) Jρ(v)ge Jρ uρ( ) minusρradic

d uρ v( ) for forallv(middot) isin Uad

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

For fixed ρ and admissible control uρt we define the spike

variation as follows

uρεt

vt τ le tle τ + ε

uρt otherwise

1113896 (56)

for any εgt 0 and it is easy to check from (iii) of (55) that

Jρ uρε( ) minus Jρ uρ( ) +ρradic

d uρε uρ( )ge 0 (57)

Let (xρt y

ρt z

ρt r

ρt ) be the trajectories corresponding to u

ρt

and (xρεt y

ρεt z

ρεt r

ρεt ) be the trajectories corresponding to

uρεt e variational equation we used in this section is the

same as the one in Section 3 with ulowastt uρt and

(xlowastt ylowastt zlowastt rlowastt ) (xρt y

ρt z

ρt r

ρt ) And we also assume that

the solution of this variational equation is (Xρt Y

ρt Z

ρt R

ρt )

Similar to the approach in Lemmas 2 and 3 it can be shownthat

sup0letleTE xρεt minus x

ρt minus X

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

sup0letleTE yρεt minus y

ρt minus Y

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0zρεt minus z

ρt minus Z

ρt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0rρεt minus r

ρt minus R

ρt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(58)

en by (57) the following variational inequality holds

Jρ uρε

( 1113857 minus Jρ uρ

( 1113857 +ρ

radicd u

ρε u

ρ( 1113857

J2ρ uρε( ) minus J2ρ uρ( )

Jρ uρε( ) + Jρ uρ( )+ ε

ρ

radic

langhρε1 E G1x x

ρT( 11138571113858 1113859X

ρTrang +langhρε

0 E G0y yρ0( 11138571113960 1113961Y

ρ0rang

+ hρε

E cy yρ0( 1113857Y

ρ01113960 1113961 + ε

ρ

radic+ o(ε)

(59)

where

hρε1

2E G1 xρT( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε0

2E G0 yρ0( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε

2E c y

ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859

Jρ uρε( ) + Jρ uρ( )

(60)

Now let (pρεt q

ρεt w

ρεt k

ρεt ) be the solution of

minus dpρεt b

ρx(t)( 1113857

τpρεt minus f

ρx(t)( 1113857

τqρεt + σρx(t)( 1113857

τw

ρεt + 1113936infin

i1gρx(t)( 1113857

τkρεt )i

1113890 1113891dt

minus wρεt dBt minus 1113936

infin

i1kρεt( 1113857

idHit

dqρεt f

ρy(t)( 1113857

τqρεt dt + f

ρz(t)( 1113857

τqρεt dBt + 1113936

infin

i1fρir (t)1113872 1113873

τ)q

ρεt dHi

t

q0 minus G0y yρ0( 1113857h

ρε0 + cy y

ρ0( 1113857hρε pT G1x x

ρT( 1113857h

ρε1 minus Φx x

ρT( 1113857q

ρεT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

8 Mathematical Problems in Engineering

Here bρx(t) bx(t x

ρt u

ρt ) σρx(t) σx(t x

ρt )

fρy(t) fy(t x

ρt y

ρt z

ρt r

ρt u

ρt ) etc Applying It 1113954os formula

to langXρt q

ρεt rang + langY

ρt p

ρεt rang variational inequality (59) can be

rewritten as

H t xρt y

ρt z

ρt r

ρt u

ρεt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

+ ερ

radic+ o(ε)ge 0 forallvt isin U ae as

(62)

where the Hamiltonian function H is defined as (44) Since

limε⟶0 hρε0

111386811138681113868111386811138681113868111386811138682

+ hρε1

111386811138681113868111386811138681113868111386811138682

+ hρε1113868111386811138681113868

11138681113868111386811138682

1113872 1113873 1 (63)

there exists a convergent subsequence still denoted by(h

ρε0 h

ρε1 hρε) such that (h

ρε0 h

ρε1 hρε)⟶ (h

ρ0 h

ρ1 hρ) when

ε⟶ 0 with |hρ0|2 + |h

ρ1|2 + |hρ|2 1

Let (pρt q

ρt w

ρt k

ρt ) be the solution of

minus dpρt b

ρx(t)( 1113857

τpρt minus f

ρx(t)( 1113857

τqρt + σρx(t)( 1113857

τw

ρt + 1113936infin

i1gρx(t)( 1113857

τkρt )i

1113890 1113891dt

minus wρtdBt minus 1113936

infin

i1kρt( 1113857

idHit

dqρt f

ρy(t)( 1113857

τqρtdt + f

ρz(t)( 1113857

τqρtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889q

ρtdHi

t

q0 minus G0y yρ0( 1113857h

ρ0 + cy y

ρ0( 1113857hρ pT G1x x

ρT( 1113857h

ρ1 minus Φx x

ρT( 1113857q

ρT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(64)

From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p

ρεt q

ρεt w

ρεt k

ρεt )⟶ (p

ρt q

ρt w

ρt k

ρt ) then in equation

(62) it implies that

H t xρt y

ρt z

ρt r

ρt vt p

ρt q

ρt w

ρt k

ρt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρt q

ρt w

ρt k

ρt( 1113857

+ρ

radic ge 0 forallvt isin U ae as

(65)

Similarly there exists a convergent subsequence(h

ρ0 h

ρ1 h

ρ) such that (h

ρ0 h

ρ1 h

ρ)⟶ (h0 h1 h) when

ρ⟶ 0 with |h0|2 + |h1|

2 + |h|2 1 Since uρt⟶ ulowastt as

ρ⟶ 0 we have (xρt y

ρt z

ρt r

ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and

(Xρt Y

ρt Z

ρt R

ρt )⟶ (Xt Yt Zt Rt) which is the solution of

the variational equation as same as (6)e following adjoint equation is introduced

minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857

τqt + σx(t)( 1113857

τwt + 1113936infin

i1gx(t)( 1113857

τkt)

i1113890 1113891dt

minus wtdBt minus 1113936infin

i1kt( 1113857

idHit

dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857

τqtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889qtdHi

t

q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(66)

Similarly it can be proved that(p

ρt q

ρt w

ρt k

ρt )⟶ (pt qt wt kt) then inequation (65)

impliesH t x

lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857

minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as

(67)

en we get the following theorem

Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint

equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|

2 + |h1|2 + |h|2 1

such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds

9is conclusion can be drawn from the above analysisdirectly

5 A Financial Example

In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3

Mathematical Problems in Engineering 9

Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

x0 Wgt 0 t isin [0 T]

⎧⎪⎪⎨

⎪⎪⎩(68)

where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility

J(C(middot)) E y01113858 1113859 (69)

where yt is the following backward stochastic process

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(70)

with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)

If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

x0 Wgt 0 yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(71)

which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)

gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R

t (1 minus R)) minus ryt

We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to

H t xlowastt ylowastt Ct p

lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx

lowastt minus Ctrang

+langwlowastt σtxlowastt rang +langklowastt g

itxlowastt rang minus langqlowastt Le

minus rt C1minus Rt

1 minus Rminus rylowastt rang

(72)

and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation

dqt rqtdt

minus dpt μtpt + σtwt + 1113936infin

i1ki

t( 1113857τgi

t1113890 1113891dt minus wtdBt minus 1113936infin

i1ki

tdHit

q0 1 pT qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(73)

According to the maximum principle (47) we have

plowastt minus Le

minus rtClowastt( 1113857

minus Rqlowastt (74)

and the optimal consumption rate

Clowastt minus

ert

L

plowasttqlowastt

1113888 1113889

minus 1R

(75)

By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is

Clowastt minus

erT

Leμ(Tminus t)

1113888 1113889

minus 1R

t isin [0 T] (76)

6 Conclusions

In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)

References

[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962

[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990

10 Mathematical Problems in Engineering

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11

Page 8: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

where G1 Rn⟶ Rn1(n1 lt n) G0 Rm⟶ Rm1(m1 ltm)Our optimal control problem is to find ulowast isin Uad such that

J ulowast(middot)( 1113857 inf

u(middot)isinUad

J(u(middot)) (51)

subject to the state constraints (50) In the following we willapply Ekelandrsquos variational principle to solve this optimalcontrol problem Firstly we need the following assumptions

Assumption 3 Control domain U is assumed to be closedthe mappings in state constraints G1 G0 are continuouslydifferentiable and G1x G0y are bounded

For forallu1(middot) u2(middot) isin Uad let

d u1(middot) u2(middot)( 1113857 E u1(middot) minus u2(middot)1113868111386811138681113868

11138681113868111386811138682 gt 0 t isin [0 T]1113966 1113967

11138681113868111386811138681113868

11138681113868111386811138681113868 (52)

Same as Section 3 we also assume ulowastt be the optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) In order to solve the constraint problemwe need the following penalty cost functional for any ρgt 0

Jρ(u) E G1 xT( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ E G0 y0( 11138571113858 11138591113868111386811138681113868

11138681113868111386811138682

+ J(u) minus J ulowast( ) + ρ1113868111386811138681113868

11138681113868111386811138682

1113966 111396712

(53)

It can be checked that Jρ(u) Uad⟶ R1 is continuousand for any u(middot) isin Uad

Jρ(u)ge 0 Jρ ulowast

( 1113857 ρ

Jρ ulowast

( 1113857le infu(middot)isinUad

Jρ(u) + ρ(54)

It can be obtained by Ekelandrsquos variational principle thatthere exists u

ρt isin Uad such that

(i) Jρ uρ( )le Jρ ulowast( ) ρ

(ii) d uρ ulowast( )le ρradic

(iii) Jρ(v)ge Jρ uρ( ) minusρradic

d uρ v( ) for forallv(middot) isin Uad

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

For fixed ρ and admissible control uρt we define the spike

variation as follows

uρεt

vt τ le tle τ + ε

uρt otherwise

1113896 (56)

for any εgt 0 and it is easy to check from (iii) of (55) that

Jρ uρε( ) minus Jρ uρ( ) +ρradic

d uρε uρ( )ge 0 (57)

Let (xρt y

ρt z

ρt r

ρt ) be the trajectories corresponding to u

ρt

and (xρεt y

ρεt z

ρεt r

ρεt ) be the trajectories corresponding to

uρεt e variational equation we used in this section is the

same as the one in Section 3 with ulowastt uρt and

(xlowastt ylowastt zlowastt rlowastt ) (xρt y

ρt z

ρt r

ρt ) And we also assume that

the solution of this variational equation is (Xρt Y

ρt Z

ρt R

ρt )

Similar to the approach in Lemmas 2 and 3 it can be shownthat

sup0letleTE xρεt minus x

ρt minus X

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

sup0letleTE yρεt minus y

ρt minus Y

ρt

111386811138681113868111386811138681113868111386811138682 leCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0zρεt minus z

ρt minus Z

ρt

111386811138681113868111386811138681113868111386811138682dsleCεε

2 Cε⟶ 0when ε⟶ 0

E 1113946T

0rρεt minus r

ρt minus R

ρt

2dsleCεε

2 Cε⟶ 0when ε⟶ 0

(58)

en by (57) the following variational inequality holds

Jρ uρε

( 1113857 minus Jρ uρ

( 1113857 +ρ

radicd u

ρε u

ρ( 1113857

J2ρ uρε( ) minus J2ρ uρ( )

Jρ uρε( ) + Jρ uρ( )+ ε

ρ

radic

langhρε1 E G1x x

ρT( 11138571113858 1113859X

ρTrang +langhρε

0 E G0y yρ0( 11138571113960 1113961Y

ρ0rang

+ hρε

E cy yρ0( 1113857Y

ρ01113960 1113961 + ε

ρ

radic+ o(ε)

(59)

where

hρε1

2E G1 xρT( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε0

2E G0 yρ0( 11138571113858 1113859

Jρ uρε( ) + Jρ uρ( )

hρε

2E c y

ρ0( 1113857 minus c y0( 1113857 + ρ1113858 1113859

Jρ uρε( ) + Jρ uρ( )

(60)

Now let (pρεt q

ρεt w

ρεt k

ρεt ) be the solution of

minus dpρεt b

ρx(t)( 1113857

τpρεt minus f

ρx(t)( 1113857

τqρεt + σρx(t)( 1113857

τw

ρεt + 1113936infin

i1gρx(t)( 1113857

τkρεt )i

1113890 1113891dt

minus wρεt dBt minus 1113936

infin

i1kρεt( 1113857

idHit

dqρεt f

ρy(t)( 1113857

τqρεt dt + f

ρz(t)( 1113857

τqρεt dBt + 1113936

infin

i1fρir (t)1113872 1113873

τ)q

ρεt dHi

t

q0 minus G0y yρ0( 1113857h

ρε0 + cy y

ρ0( 1113857hρε pT G1x x

ρT( 1113857h

ρε1 minus Φx x

ρT( 1113857q

ρεT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

8 Mathematical Problems in Engineering

Here bρx(t) bx(t x

ρt u

ρt ) σρx(t) σx(t x

ρt )

fρy(t) fy(t x

ρt y

ρt z

ρt r

ρt u

ρt ) etc Applying It 1113954os formula

to langXρt q

ρεt rang + langY

ρt p

ρεt rang variational inequality (59) can be

rewritten as

H t xρt y

ρt z

ρt r

ρt u

ρεt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

+ ερ

radic+ o(ε)ge 0 forallvt isin U ae as

(62)

where the Hamiltonian function H is defined as (44) Since

limε⟶0 hρε0

111386811138681113868111386811138681113868111386811138682

+ hρε1

111386811138681113868111386811138681113868111386811138682

+ hρε1113868111386811138681113868

11138681113868111386811138682

1113872 1113873 1 (63)

there exists a convergent subsequence still denoted by(h

ρε0 h

ρε1 hρε) such that (h

ρε0 h

ρε1 hρε)⟶ (h

ρ0 h

ρ1 hρ) when

ε⟶ 0 with |hρ0|2 + |h

ρ1|2 + |hρ|2 1

Let (pρt q

ρt w

ρt k

ρt ) be the solution of

minus dpρt b

ρx(t)( 1113857

τpρt minus f

ρx(t)( 1113857

τqρt + σρx(t)( 1113857

τw

ρt + 1113936infin

i1gρx(t)( 1113857

τkρt )i

1113890 1113891dt

minus wρtdBt minus 1113936

infin

i1kρt( 1113857

idHit

dqρt f

ρy(t)( 1113857

τqρtdt + f

ρz(t)( 1113857

τqρtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889q

ρtdHi

t

q0 minus G0y yρ0( 1113857h

ρ0 + cy y

ρ0( 1113857hρ pT G1x x

ρT( 1113857h

ρ1 minus Φx x

ρT( 1113857q

ρT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(64)

From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p

ρεt q

ρεt w

ρεt k

ρεt )⟶ (p

ρt q

ρt w

ρt k

ρt ) then in equation

(62) it implies that

H t xρt y

ρt z

ρt r

ρt vt p

ρt q

ρt w

ρt k

ρt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρt q

ρt w

ρt k

ρt( 1113857

+ρ

radic ge 0 forallvt isin U ae as

(65)

Similarly there exists a convergent subsequence(h

ρ0 h

ρ1 h

ρ) such that (h

ρ0 h

ρ1 h

ρ)⟶ (h0 h1 h) when

ρ⟶ 0 with |h0|2 + |h1|

2 + |h|2 1 Since uρt⟶ ulowastt as

ρ⟶ 0 we have (xρt y

ρt z

ρt r

ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and

(Xρt Y

ρt Z

ρt R

ρt )⟶ (Xt Yt Zt Rt) which is the solution of

the variational equation as same as (6)e following adjoint equation is introduced

minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857

τqt + σx(t)( 1113857

τwt + 1113936infin

i1gx(t)( 1113857

τkt)

i1113890 1113891dt

minus wtdBt minus 1113936infin

i1kt( 1113857

idHit

dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857

τqtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889qtdHi

t

q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(66)

Similarly it can be proved that(p

ρt q

ρt w

ρt k

ρt )⟶ (pt qt wt kt) then inequation (65)

impliesH t x

lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857

minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as

(67)

en we get the following theorem

Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint

equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|

2 + |h1|2 + |h|2 1

such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds

9is conclusion can be drawn from the above analysisdirectly

5 A Financial Example

In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3

Mathematical Problems in Engineering 9

Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

x0 Wgt 0 t isin [0 T]

⎧⎪⎪⎨

⎪⎪⎩(68)

where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility

J(C(middot)) E y01113858 1113859 (69)

where yt is the following backward stochastic process

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(70)

with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)

If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

x0 Wgt 0 yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(71)

which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)

gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R

t (1 minus R)) minus ryt

We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to

H t xlowastt ylowastt Ct p

lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx

lowastt minus Ctrang

+langwlowastt σtxlowastt rang +langklowastt g

itxlowastt rang minus langqlowastt Le

minus rt C1minus Rt

1 minus Rminus rylowastt rang

(72)

and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation

dqt rqtdt

minus dpt μtpt + σtwt + 1113936infin

i1ki

t( 1113857τgi

t1113890 1113891dt minus wtdBt minus 1113936infin

i1ki

tdHit

q0 1 pT qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(73)

According to the maximum principle (47) we have

plowastt minus Le

minus rtClowastt( 1113857

minus Rqlowastt (74)

and the optimal consumption rate

Clowastt minus

ert

L

plowasttqlowastt

1113888 1113889

minus 1R

(75)

By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is

Clowastt minus

erT

Leμ(Tminus t)

1113888 1113889

minus 1R

t isin [0 T] (76)

6 Conclusions

In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)

References

[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962

[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990

10 Mathematical Problems in Engineering

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11

Page 9: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

Here bρx(t) bx(t x

ρt u

ρt ) σρx(t) σx(t x

ρt )

fρy(t) fy(t x

ρt y

ρt z

ρt r

ρt u

ρt ) etc Applying It 1113954os formula

to langXρt q

ρεt rang + langY

ρt p

ρεt rang variational inequality (59) can be

rewritten as

H t xρt y

ρt z

ρt r

ρt u

ρεt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρεt q

ρεt w

ρεt k

ρεt( 1113857

+ ερ

radic+ o(ε)ge 0 forallvt isin U ae as

(62)

where the Hamiltonian function H is defined as (44) Since

limε⟶0 hρε0

111386811138681113868111386811138681113868111386811138682

+ hρε1

111386811138681113868111386811138681113868111386811138682

+ hρε1113868111386811138681113868

11138681113868111386811138682

1113872 1113873 1 (63)

there exists a convergent subsequence still denoted by(h

ρε0 h

ρε1 hρε) such that (h

ρε0 h

ρε1 hρε)⟶ (h

ρ0 h

ρ1 hρ) when

ε⟶ 0 with |hρ0|2 + |h

ρ1|2 + |hρ|2 1

Let (pρt q

ρt w

ρt k

ρt ) be the solution of

minus dpρt b

ρx(t)( 1113857

τpρt minus f

ρx(t)( 1113857

τqρt + σρx(t)( 1113857

τw

ρt + 1113936infin

i1gρx(t)( 1113857

τkρt )i

1113890 1113891dt

minus wρtdBt minus 1113936

infin

i1kρt( 1113857

idHit

dqρt f

ρy(t)( 1113857

τqρtdt + f

ρz(t)( 1113857

τqρtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889q

ρtdHi

t

q0 minus G0y yρ0( 1113857h

ρ0 + cy y

ρ0( 1113857hρ pT G1x x

ρT( 1113857h

ρ1 minus Φx x

ρT( 1113857q

ρT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(64)

From the continuous dependence of the solutions ofFBSDEs with Levy process on the parameters we can provethat the following convergence holds(p

ρεt q

ρεt w

ρεt k

ρεt )⟶ (p

ρt q

ρt w

ρt k

ρt ) then in equation

(62) it implies that

H t xρt y

ρt z

ρt r

ρt vt p

ρt q

ρt w

ρt k

ρt( 1113857

minus H t xρt y

ρt z

ρt r

ρt u

ρt p

ρt q

ρt w

ρt k

ρt( 1113857

+ρ

radic ge 0 forallvt isin U ae as

(65)

Similarly there exists a convergent subsequence(h

ρ0 h

ρ1 h

ρ) such that (h

ρ0 h

ρ1 h

ρ)⟶ (h0 h1 h) when

ρ⟶ 0 with |h0|2 + |h1|

2 + |h|2 1 Since uρt⟶ ulowastt as

ρ⟶ 0 we have (xρt y

ρt z

ρt r

ρt )⟶ (xlowastt ylowastt zlowastt rlowastt ) and

(Xρt Y

ρt Z

ρt R

ρt )⟶ (Xt Yt Zt Rt) which is the solution of

the variational equation as same as (6)e following adjoint equation is introduced

minus dpt bx(t)( 1113857τpt minus fx(t)( 1113857

τqt + σx(t)( 1113857

τwt + 1113936infin

i1gx(t)( 1113857

τkt)

i1113890 1113891dt

minus wtdBt minus 1113936infin

i1kt( 1113857

idHit

dqt fy(t)1113872 1113873τqtdt + fz(t)( 1113857

τqtdBt + 1113936

infin

i1fρir (t)1113872 1113873

τ1113888 1113889qtdHi

t

q0 minus G0y y0( 1113857h0 + cy y0( 1113857h pT G1x xT( 1113857h1 minus Φx xT( 1113857qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(66)

Similarly it can be proved that(p

ρt q

ρt w

ρt k

ρt )⟶ (pt qt wt kt) then inequation (65)

impliesH t x

lowastt ylowastt zlowastt rlowastt vt pt qt wt kt( 1113857

minus H t xlowastt ylowastt zlowastt rlowastt ulowastt pt qt wt kt( 1113857ge 0forallvt isin U ae as

(67)

en we get the following theorem

Theorem 2 Let Assumptions 1ndash3 hold ulowastt is an optimalcontrol and the corresponding optimal state trajectories are(xlowastt ylowastt zlowastt rlowastt ) and (pt qt wt kt) is the solution of adjoint

equation (66) then there exists nonzero constant(h1 h0 h) isin (Rn1 times Rm1 times R) with |h0|

2 + |h1|2 + |h|2 1

such that for any admissible control vt isin Uad[0 T] themaximum condition (67) holds

9is conclusion can be drawn from the above analysisdirectly

5 A Financial Example

In this section we will study the problem of optimal con-sumption rate selection in the financial market which willnaturally inspire our research to the forward-backwardstochastic optimal control problem in Section 3

Mathematical Problems in Engineering 9

Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

x0 Wgt 0 t isin [0 T]

⎧⎪⎪⎨

⎪⎪⎩(68)

where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility

J(C(middot)) E y01113858 1113859 (69)

where yt is the following backward stochastic process

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(70)

with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)

If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

x0 Wgt 0 yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(71)

which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)

gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R

t (1 minus R)) minus ryt

We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to

H t xlowastt ylowastt Ct p

lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx

lowastt minus Ctrang

+langwlowastt σtxlowastt rang +langklowastt g

itxlowastt rang minus langqlowastt Le

minus rt C1minus Rt

1 minus Rminus rylowastt rang

(72)

and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation

dqt rqtdt

minus dpt μtpt + σtwt + 1113936infin

i1ki

t( 1113857τgi

t1113890 1113891dt minus wtdBt minus 1113936infin

i1ki

tdHit

q0 1 pT qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(73)

According to the maximum principle (47) we have

plowastt minus Le

minus rtClowastt( 1113857

minus Rqlowastt (74)

and the optimal consumption rate

Clowastt minus

ert

L

plowasttqlowastt

1113888 1113889

minus 1R

(75)

By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is

Clowastt minus

erT

Leμ(Tminus t)

1113888 1113889

minus 1R

t isin [0 T] (76)

6 Conclusions

In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)

References

[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962

[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990

10 Mathematical Problems in Engineering

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11

Page 10: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

Assume the investorrsquos asset process xt (tge 0) in the fi-nancial market is described by the following stochasticdifferential equation with the Teugels martingale

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

x0 Wgt 0 t isin [0 T]

⎧⎪⎪⎨

⎪⎪⎩(68)

where μt and σt ne 0 are the expected return and volatility ofthe value process xt at time t respectively and Ct is theconsumption rate process Assume μt σt gt and Ct are alluniformly bounded Ftminus measurable random processes epurpose of investors is to select the optimal consumptionstrategy Clowastt at time tge 0 minimizing the following recursiveutility

J(C(middot)) E y01113858 1113859 (69)

where yt is the following backward stochastic process

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(70)

with constant Lgt 0 the discount factor rgt 0 and theArrowndashPratt measure of risk aversion R isin (0 1)

If the consumption process Ct is regarded as a controlvariable then combining (68) with (70) we encounter thefollowing control system

dxt μtxt minus Ct1113858 1113859dt + σtxtdBt + 1113936infin

i1gi

txtdHit

minus dyt Leminus rtC1minus R

t

1 minus Rminus ryt1113890 1113891dt minus ztdBt minus 1113936

infin

i1ri

tdHit

x0 Wgt 0 yT minus xT gt 0 t isin [0 T]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(71)

which is obviously a special case of stochastic control system(2) with b(t x u) μtxt minus Ct σ(t x) σtxt g(t x)

gitxt Φx(xT) minus XT and f(t x y z r u) Leminus rt(C1minus R

t (1 minus R)) minus ryt

We can check that both Assumptions 1 and 2 are sat-isfieden we can use ourmaximum principle (eorem 1)to solve the above optimization problem Let Clowastt be anoptimal consumption rate and xlowastt ylowastt be the correspondingwealth process and recursive utility process In this case theHamiltonian function H reduces to

H t xlowastt ylowastt Ct p

lowastt qlowastt wlowastt klowastt( 1113857 langplowastt μtx

lowastt minus Ctrang

+langwlowastt σtxlowastt rang +langklowastt g

itxlowastt rang minus langqlowastt Le

minus rt C1minus Rt

1 minus Rminus rylowastt rang

(72)

and (plowastt qlowastt wlowastt klowastt ) is the solution of the following adjointequation

dqt rqtdt

minus dpt μtpt + σtwt + 1113936infin

i1ki

t( 1113857τgi

t1113890 1113891dt minus wtdBt minus 1113936infin

i1ki

tdHit

q0 1 pT qT t isin [0 T]

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(73)

According to the maximum principle (47) we have

plowastt minus Le

minus rtClowastt( 1113857

minus Rqlowastt (74)

and the optimal consumption rate

Clowastt minus

ert

L

plowasttqlowastt

1113888 1113889

minus 1R

(75)

By solving FBSDE (73) we get qlowastt ert plowastt erteμ(Tminus t)and wlowastt klowastt 0 en we get the optimal consumptionrate of the investor which is

Clowastt minus

erT

Leμ(Tminus t)

1113888 1113889

minus 1R

t isin [0 T] (76)

6 Conclusions

In this paper a nonconvex control domain case of theforward-backward stochastic control driven by Levy processis considered and we obtain the global stochastic maximumprinciple for this stochastic control problem And then theproblem of stochastic control with initial and final stateconstraints on the state variables is discussed and a nec-essary condition about existence of the optimal control isalso acquired A financial example of optimal consumptionis discussed to illustrate the application of the stochasticmaximum principle

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the NNSF (11701335) Shan-dong NSF (ZR2019BG015) NSSF (17BGL058) ShandongProvincial Higher Education Science and Technology PlanProject (J18KA236) and Shandong Womenrsquos College High-level Research Project Cultivation Foundation(2018GSPGJ07)

References

[1] L S Pontryakin V G Boltyanskti R V Gamkrelidze andE F Mischenko9eMathematical9eory of Optimal ControlProcesses John Wiley New York NY USA 1962

[2] S Peng ldquoA general stochastic maximum principle for optimalcontrol problemsrdquo SIAM Journal on Control and Optimiza-tion vol 28 no 4 pp 966ndash979 1990

10 Mathematical Problems in Engineering

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11

Page 11: ANecessaryConditionforOptimalControlofForward-Backward ...downloads.hindawi.com/journals/mpe/2020/1768507.pdf · where(xt,y t,z t,r t)takethevalueinΩ×[0,T]×Rn×Rm× Rm×d×l2(Rm)andmappingsb,σ,g,andftakethevaluein

[3] S G Peng ldquoBackward stochastic differrntial equations andapplication to optimal controlrdquo Applied Mathematics ampOptimization vol 27 no 4 pp 125ndash144 1993

[4] W S Xu ldquoStochastic maximum principle for optimal controlproblem of forward and backward systemrdquo Journal of Aus-tralian Mathematical Sciences vol 37 pp 249ndash259 1998B

[5] Z Wu ldquoMaximum principle for optimal control problem offully coupled forward-backward stochastic control systemrdquoSystem Science and Mathematical Sciences vol 11 no 3pp 249ndash259 1998

[6] J T Shi and ZWu ldquoemaximum principle for fully coupledforward-backward stochastic control systemrdquo Acta Auto-matica Sinica vol 32 no 2 pp 161ndash169 2006

[7] R Situ ldquoA maximum principle for optimal controls of sto-chastic systems with random jumpsrdquo in Proceedings of theNational Conference on Control 9eory and its ApplicationsQingdao China 1991

[8] J Shi and Z Wu ldquoMaximum principle for forward-backwardstochastic control system with random jumps and applica-tions to financerdquo Journal of Systems Science and Complexityvol 23 no 2 pp 219ndash231 2010

[9] J T Shi ldquoGlobal maximum principle for forward-backwardstochastic control system with Poisson jumpsrdquo Asian Journalof Control vol 16 no 1 pp 1ndash11 2014

[10] M Liu X Wang and H Huang ldquoMaximum principle forforward-backward control system driven by Ito-Levy pro-cesses under initial-terminal constraintsrdquo MathematicalProblems in Engineering vol 2017 pp 1ndash13 2017

[11] S Tang ldquoe maximum principle for partially observedoptimal control of stochastic differential equationsrdquo SIAMJournal on Control and Optimization vol 36 no 5pp 1596ndash1617 1998

[12] Q Meng ldquoA maximum principle for optimal control problemof fully coupled forward-backward stochastic systems withpartial informationrdquo Science in China Series A Mathematicsvol 52 no 7 pp 1579ndash1588 2009

[13] B Oslashksendal and A Sulem ldquoMaximum principles for optimalcontrol of forward backward stichastic differential equationswith jumpsrdquo SIAM J Control Optimenleadertwodots vol 48pp 2845ndash2976 2009

[14] G Wang and Z Wu ldquoKalman-Bucy filtering equations offorward and backward stochastic systems and applications torecursive optimal control problemsrdquo Journal of MathematicalAnalysis and Applications vol 342 no 2 pp 1280ndash12962008

[15] H Xiao ldquoe maximum principle for partially observedoptimal control of forward-backward stochastic systems withrandom jumpsrdquo Journal of Systems Science and Complexityvol 24 no 6 pp 1083ndash1099 2011

[16] G Wang Z Wu and J Xiong ldquoMaximum principles forforward-backward stochastic control systems with correlatedstate and observation noisesrdquo SIAM Journal on Control andOptimization vol 51 no 1 pp 491ndash524 2013

[17] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Controlvol 59 no 2 pp 522ndash528 2014

[18] S Q Zhang J Xiong and X D Liu ldquoStochastic maximumprinciple for partially observed forward backward stochasticdifferential equations with jumps and regime switchingrdquoScience China Information Sciences vol 61 no 7 pp 1ndash132018

[19] J Xiong S Q Zhang S Zhang and Y Zhuang ldquoA partiallyobserved non-zero sum differential game of forward-

backward stochastic differential equations and its applicationin financerdquo Mathematical Control amp Related Fields vol 9no 2 pp 257ndash276 2019

[20] D Nualart and W Schoutens ldquoChaotic and predictablerepresentations for Levy processesrdquo Stochastic Processes and9eir Applications vol 90 no 1 pp 109ndash122 2000

[21] D Nualart and W Schoutens ldquoBackward stochastic differ-ential equations and Feynman-Kac formula for Levy pro-cesses with applications in financerdquo Bernoulli vol 7 no 5pp 761ndash776 2001

[22] Q Meng and M Tang ldquoNecessary and sufficient conditionsfor optimal control of stochastic systems associated with Levyprocessesrdquo Science in China Series F Information Sciencesvol 52 no 11 pp 1982ndash1992 2009

[23] M Tang and Q Zhang ldquoOptimal variational principle forbackward stochastic control systems associated with Levyprocessesrdquo Science China Mathematics vol 55 no 4pp 745ndash761 2012

[24] F Baghery N Khelfallah B Mezerdi and I Turpin ldquoFullycoupled forward backward stochastic differential equationsdriven by Levy processes and application to differentialgamesrdquo De Gruyter vol 22 no 3 pp 151ndash161 2014

[25] F Zhang M N Tang and Q X Meng ldquoSochastic maximumprinciple for forward-backward stochastic control systemsassociated with Levy processesrdquo Chinese Annals of Mathe-matics vol 35A no 1 pp 83ndash100 2014

[26] X R Wang and H Huang ldquoMaximum principle for forward-backward stochastic control system driven by Levy processrdquoMathematical Problems in Engineering vol 2015 Article ID702802 12 pages 2015

[27] H Huang X Wang and M Liu ldquoA maximum principle forfully coupled forward-backward stochastic control systemdriven by Levy process with terminal state constraintsrdquoJournal of Systems Science and Complexity vol 31 no 4pp 859ndash874 2018

[28] P Muthukumar and R Deepa ldquoInfinite horizon optimalcontrol of forward backward stochastic system driven byTeugels martingales with Levy processrdquo Stochastic and Dy-namics vol 3 no 17 pp 1ndash17 2017

[29] J B Wu and Z M Liu ldquoOptimal control of mean-fieldbackward doubly stochastic systems driven by Ito-Levyprocessesrdquo International Journal of Control vol 93 no 4pp 953ndash970 2018

[30] W Wang J Wu and Z Liu ldquoe optimal control of fully-coupled forward-backward doubly stochastic systems drivenby Ito-Levy processesrdquo Journal of Systems Science andComplexity vol 32 no 4 pp 997ndash1018 2019

Mathematical Problems in Engineering 11


Free Volume of Polymers using Positron Annihilation ...positron.physik.uni-halle.de/talks/Qasim_in_Kiel.pdfV h and σ h vs T(K) The mean, , and the mean dispersion, σ

Free Volume of Polymers using Positron Annihilation ...positron.physik.uni-halle.de/talks/Qasim_in_Kiel.pdfV h and σ h vs T(K) The mean, , and the mean dispersion, σ

Dynamic nonlinear elasticity in geomaterials€¦ · DYNAMIC NONLINEAR ELASTICITY IN GEOMATERIALS 3 x σ = 0 σ = 1-π-π π π x σ = _ σ = 4-π-π π π 2 π x σ >> 1-π π Amplitude

Dynamic nonlinear elasticity in geomaterials€¦ · DYNAMIC NONLINEAR ELASTICITY IN GEOMATERIALS 3 x σ = 0 σ = 1-π-π π π x σ = _ σ = 4-π-π π π 2 π x σ >> 1-π π Amplitude

Measurement of F 2 and R= σ L / σ T in Nuclei at Low Q 2 Phase I

Measurement of F 2 and R= σ L / σ T in Nuclei at Low Q 2 Phase I

Σ −Σ cations anions 100 Σ +Σ cations anions · According to the WHO, the range of desirable pH values of water prescribed for drinking purposes is 6.5 – 8.5 (WHO, 1984), while

Σ −Σ cations anions 100 Σ +Σ cations anions · According to the WHO, the range of desirable pH values of water prescribed for drinking purposes is 6.5 – 8.5 (WHO, 1984), while

Assessing Biases in Recent in situ SST and Marine Air ...εε εε εε εε εε where: εε σ σ εε εε σ εε σ σ xx air eri xy y xeri yy bucket eri =+ == =+ 22 2 22 to give

Assessing Biases in Recent in situ SST and Marine Air ...εε εε εε εε εε where: εε σ σ εε εε σ εε σ σ xx air eri xy y xeri yy bucket eri =+ == =+ 22 2 22 to give

MONTGOMERY COUNTY PUBLIC SCHOOLS EARLE B WOOD …...Jun 29, 2017  · rm cr accomm. rm cr relo. t t t t gen st relo. mech mech s el cr cr cr cr cr team wkrm team wkrm math int. rm

MONTGOMERY COUNTY PUBLIC SCHOOLS EARLE B WOOD …...Jun 29, 2017  · rm cr accomm. rm cr relo. t t t t gen st relo. mech mech s el cr cr cr cr cr team wkrm team wkrm math int. rm

Security Analysis of HyENA Authenticated Encryption Mode · Security Statement for HyENA. Main Theorem σ Adv. AE,σ 0. e nq e nσ v. e,σ v, t) ≤ Adv prp (q , t 0) + O + + . HyENA

Security Analysis of HyENA Authenticated Encryption Mode · Security Statement for HyENA. Main Theorem σ Adv. AE,σ 0. e nq e nσ v. e,σ v, t) ≤ Adv prp (q , t 0) + O + + . HyENA

Hydrostatic Pressure (fluids) · 2018. 5. 15. · (σ 1 −σ 2)/2is the largest possible value of τ σ 1 −σ 2is called the “differential stress” If you know the two principal

Hydrostatic Pressure (fluids) · 2018. 5. 15. · (σ 1 −σ 2)/2is the largest possible value of τ σ 1 −σ 2is called the “differential stress” If you know the two principal

WCCI 2016-2018. Results in section C - Moremovers. · 4Mihalco, Oto SK 2.117 5Efrosinin, Oleg RU 1.784 Ranking: №1 №2 №3 №4 №5 №6 Σ Σ Σ Σ Σ Σ AF 1 Marandyuk, ...

WCCI 2016-2018. Results in section C - Moremovers. · 4Mihalco, Oto SK 2.117 5Efrosinin, Oleg RU 1.784 Ranking: №1 №2 №3 №4 №5 №6 Σ Σ Σ Σ Σ Σ AF 1 Marandyuk, ...

Braid-based cryptography - Patrick Dehornoy · BRAID-BASED CRYPTOGRAPHY 3 = = = = σ−1 i σ i+1 σ i σ i 1 σ i i σ i σ i σ i σ−1 σ i+1 +1 i+2 i+2 Figure 2. Geometric interpretation

Braid-based cryptography - Patrick Dehornoy · BRAID-BASED CRYPTOGRAPHY 3 = = = = σ−1 i σ i+1 σ i σ i 1 σ i i σ i σ i σ i σ−1 σ i+1 +1 i+2 i+2 Figure 2. Geometric interpretation

Introduction - Princeton University · of F–stopping times with values in [0,T], and S σ, {τ∈ S; τ≥ σ} for every σ∈ S. The classical optimal stopping problem is to compute

Introduction - Princeton University · of F–stopping times with values in [0,T], and S σ, {τ∈ S; τ≥ σ} for every σ∈ S. The classical optimal stopping problem is to compute

NAU8820 Low Power 24-bit Stereo Audio Codec with High Current Outputs … · 2016. 8. 18. · hpf alc notch filter limiter 5 band eq 3d Σ Σ Σ Σ alc control Σ Σ rinmix ldac rdac

NAU8820 Low Power 24-bit Stereo Audio Codec with High Current Outputs … · 2016. 8. 18. · hpf alc notch filter limiter 5 band eq 3d Σ Σ Σ Σ alc control Σ Σ rinmix ldac rdac

fave-img.s3.amazonaws.com · thaihousek Jl Apple J2 Orange J3 Watermelon J4 Coconut T HS TS2 RM 12 RM 12 RM 12 RM 12 thai hou sek VHS _ c, T _ DEC _ 2017 RM6 RM4 JUICES SOFT

fave-img.s3.amazonaws.com · thaihousek Jl Apple J2 Orange J3 Watermelon J4 Coconut T HS TS2 RM 12 RM 12 RM 12 RM 12 thai hou sek VHS _ c, T _ DEC _ 2017 RM6 RM4 JUICES SOFT

24 CV MeanShift-GCuts 0605users.ics.forth.gr/.../24_CV_MeanShift-GCuts_0605_01.pdf · 2020. 5. 6. · σ=.1 σ=.2 σ=1 σ=.2 Data points Affinity matrices Slide credit: Kristen Grauman.

24 CV MeanShift-GCuts 0605users.ics.forth.gr/.../24_CV_MeanShift-GCuts_0605_01.pdf · 2020. 5. 6. · σ=.1 σ=.2 σ=1 σ=.2 Data points Affinity matrices Slide credit: Kristen Grauman.

Yip Biotechnology for Biofuels - sklbe.ecust.edu.cnsklbe.ecust.edu.cn/BiomassLab/kindeditor/Upload... · Yipet al. Biotechnol Biofuels Pageq3qofq18 andσwaterσ[1210,σ,σ13,σ 15,σ

Yip Biotechnology for Biofuels - sklbe.ecust.edu.cnsklbe.ecust.edu.cn/BiomassLab/kindeditor/Upload... · Yipet al. Biotechnol Biofuels Pageq3qofq18 andσwaterσ[1210,σ,σ13,σ 15,σ

t-Static t-test is used to test hypothesis about an unknown population mean, µ, when the value of σ or σ² is unknown. t-test is used to test hypothesis.

t-Static t-test is used to test hypothesis about an unknown population mean, µ, when the value of σ or σ² is unknown. t-test is used to test hypothesis.

VolatilityEstimation - NTU Singapore | NTU · VolatilityEstimation totheobservedvalue. M. ofagivenmarketpriceallowsonetoinferavalue of. σ. when. t, S. t, r, T. areknown,asin. e.g.

VolatilityEstimation - NTU Singapore | NTU · VolatilityEstimation totheobservedvalue. M. ofagivenmarketpriceallowsonetoinferavalue of. σ. when. t, S. t, r, T. areknown,asin. e.g.

TheComplexityofModelChecking(Collapsible) Higher ... · Anorder-kPDS isatuple(P,R,Σ,p0,⊥) wherePisafinitesetofcontrol states,R⊆P×Σ×Ok×Pisafinitesetofrules,Σ isafinitestackalphabet,p0

TheComplexityofModelChecking(Collapsible) Higher ... · Anorder-kPDS isatuple(P,R,Σ,p0,⊥) wherePisafinitesetofcontrol states,R⊆P×Σ×Ok×Pisafinitesetofrules,Σ isafinitestackalphabet,p0

Curves, circles, and spheres · However, typically ptp(s,t,σ) 6= ppt(s,t,σ) 6= tpp(s,t,σ). We use c (for circle) as the mnemonic notation for second-order contact, so that cp(s,t)

Curves, circles, and spheres · However, typically ptp(s,t,σ) 6= ppt(s,t,σ) 6= tpp(s,t,σ). We use c (for circle) as the mnemonic notation for second-order contact, so that cp(s,t)

Σ πανιε Σ γ αιε Σ - PublicΣ πανιε Σ γ αιε Σ 15 τραγικό τρόπο– κι ένα κεφάλαιο μετά την Ντάρια Πλα-τόνοβα που

Σ πανιε Σ γ αιε Σ - PublicΣ πανιε Σ γ αιε Σ 15 τραγικό τρόπο– κι ένα κεφάλαιο μετά την Ντάρια Πλα-τόνοβα που