# Maximum Principle for Forward-Backward Stochastic Control System Driven by Lévy Process

Authors: Xiangrong Wang; Hong Huang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We study a stochastic optimal control problem where the controlled system is described by a forward-backward stochastic differential equation driven by Lévy process. In order to get our main result of this paper, the maximum principle, we prove the continuity result depending on parameters about fully coupled forward-backward stochastic differential equations driven by Lévy process. Under some additional convexity conditions, the maximum principle is also proved to be sufficient. Finally, the result is applied to the linear quadratic problem.


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## Body

## 1. Introduction

The stochastic optimal control problem is one of the central themes of modern control science. Forward-backward stochastic control systems which the controlled systems described by forward-backward stochastic differential equations (FBSDEs) are widely used in mathematics and finance. Peng and Wu [1] firstly used a probabilistic method to get the existence and uniqueness results of fully coupled FBSDEs; then Peng [2] considered one kind of forward-backward stochastic control systems with economic background when the control domain is convex and obtained the maximum principle; since then a number of developments in this direction were reported in Wu [3] and Shi and Wu [4]. Wu [5] firstly proved the existence and uniqueness results of the solutions to fully coupled FBSDEs with Brownian motion and Poisson process; then Shi and Wu [6] got the stochastic maximum principle for fully coupled FBSDEs with random jumps. More conclusions about stochastic maximum principle about forward-backward stochastic control systems driven by Brownian motion and Poisson process can be seen in [7–9].It is natural to extend the stochastic differential equations (SDEs) with Brownian motion and Poisson process to the case of Lévy process with independent and stationary increments. Baghery et al. [10] firstly considered the following fully coupled forward-backward stochastic differential equation driven by Lévy process (FBSDEL):(1)dxt=bt,xt,yt,zt,rtdt+∑i=1dσit,xt,yt,zt,rtdBti+∑i=1∞git,xt-,yt-,zt,rtdHti,-dyt=ft,xt,yt,zt,rtdt-∑i=1dztidBti-∑i=1∞rtidHti,x0=a,yT=ΦxT,and, under some monotonicity assumptions, they got the existence and uniqueness of solutions for this equation. Zhu [11] had proposed the asymptotic stability in the Pth moment for SDE with Lévy noise. Nualart and Schoutens [12] constructed a set of pairwise strongly orthonormal martingales called Teugels martingale and they also proved a martingale representation theorem for Lévy processes satisfying some exponential moment condition. Using the martingale representation theorem they [13] had proved the existence and uniqueness of a solution for backward stochastic differential equations (BSDEs) driven by Teugels martingale. Bahlali et al. [14] extended this conclusion to the BSDEs driven by Teugels martingale and an independent Brownian motion; they got the existence, uniqueness, and comparison of solutions for these equations under Lipschitz and locally Lipschitz conditions on the coefficient. Based on these consequences, Mitsui and Tabata [15] established the closeness property of the solution of the multidimensional backward stochastic Riccati differential equation with Lévy process; then they used this solution to study a linear quadratic regulation problem with Lévy process. After the foundation of the existence and uniqueness of the solutions of SDEs and multidimensional BSDEs driven by Lévy process, Tang and Wu [16] proceed to study a stochastic linear quadratic optimal control problem with a Lévy process, where the cost weighting matrices of the state and control were allowed to be indefinite.These consequences are important for the researching of maximum principle for forward stochastic control system driven by Lévy process, as the adjoint equation for forward stochastic control system is a BSDE. Meng and Tang [17] firstly were concerned with optimal control for forward stochastic control system driven by Teugels martingale; they got the maximum principle and verification theorem for this system. In 2012, Tang and Zhang [18] were concerned with optimal control of BSDE driven by Teugels martingale and an independent multidimensional Brownian motion; they derived the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. When the control domain was nonconcave and the control variable was allowed to enter the coefficients of the Teugels martingales, Lin [19] got the necessary maximum principle for optimal control of stochastic system driven by multidimensional Teugels martingales. Zhang et al. [20] firstly studied the forward-backward stochastic control system where the system was driven by Teugels martingale and an independent multidimensional Brownian motion as follows:(2)dxt=bt,xt,utdt+∑i=1dσit,xt,utdBti+∑i=1∞git,xt-,utdHti,-dyt=ft,xt,yt,zt,rt,,utdt-∑i=1dztidBti-∑i=1∞rtidHti,x0=a,yT=ΦxT,and they had got the maximum principle and verification theorem in the condition of the SDE part did not contain the backward state variables; the forward-backward stochastic control system they studied was not fully coupled.In this paper, we extend the result of Zhang et al. [20] to the fully coupled forward-backward stochastic control system. Here the state variables are described by fully coupled FBSDEs driven by Brownian motion and an independent Teugels martingale. Before applying the convex variation and duality technique to obtain the stochastic maximum principle, we use the same method in [5] to get the continuity result depending on parameters, as the continuity result is not only important for us to get the stochastic maximum principle but also important property of FBSDEL especially in practice. Different from the Wu [3] and Shi and Wu [6] about maximum principles to Brownian motions and Poisson process, we also need more generalIto’s formula about ca`dla`g semimartingale.This paper is organized as follows. In Section2, we will give some preliminaries used in this paper. Section 3 presents the continuity result depending on parameters about fully coupled FBSDEs driven by Lévy process. In Section 4, we obtain the main result of this paper, the maximum principle. We also prove that, under some additional convexity conditions, the maximum principle can be a sufficient condition for optimal control. And, in Section 5, an application of our stochastic maximum principle to the linear quadratic problem which the linear control system described by fully coupled FBSDEL is proved.


## 2. Preliminaries and Notations

Let(Ω,F,P,Ft,Bt,Lt)(t∈[0,T]) be a complete space driving by Brownian motion and Lévy process in Rm×R∖{0}, with Lévy measure ν; that is, Bt0≤t≤T is a standard Brownian motion. Lt0≤t≤T is R-valued Lévy process of the form Lt=bt+lt independent of Bt0≤t≤T, corresponding to a standard Lévy measure ν satisfying the following conditions:(i)
∫R‍(1∧x2)ν(dx)<∞,(ii)
∫(-ɛ,ɛ)c‍eλ|x|ν(dx)<∞, for every ɛ>0 and for some λ>0, and (3)Ft=σLs,s≤t∨σBs,s≤t∨ℵ.Here ℵ is the totality of P-null sets and g1∨g2 denotes the σ-field generated by g1∪g2.Letx be a Lévy process and denote the left limit process by xt-=lims→t,s≤t⁡xs and the jump size at time t by Δxt=xt-xt-. Set(4)xti=∑0<s≤tΔxsi,i≥2xt,i=1and we denote the compensated power jump process of order i by Yti=xti-E[xti]; then Teugels martingale (Hti)0≤t≤T can be defined as follows:(5)Hti=ci,iYti+ci,i-1Yti-1+ci,i-2Yti-2+⋯+ci,1Yt1.Here the coefficients ci,k correspond to orthonormalization of the polynomials 1,x,x2,… with respect to the measure μ(dx)=υ(dx)+σ2δ0(dx).Now we introduce some notations adopted in this paper as follows:(1)
H: Hilbert space,(2)
〈α,β〉: the inner product in Rn, ∀α,β∈Rn,(3)
|α|=〈α,α〉: the norm in Rn, ∀α∈Rn,(4)
〈A,B〉=tr(ABT): the inner product in Rn×m, ∀A,B∈Rn×m,(5)
|A|=tr(AAT): the inner product in Rn×m, ∀A∈Rn×m,(6)
l2: the space of real valued sequences X=(xn)n≥0 such that (6)xl22=∑i=1∞xi2<∞,(7)
l2(H): the space of H-valued sequences ϕ=φii≥1 such that (7)ϕl2H2=∑i=1∞ϕi2<∞,(8)
l2(0,T;H): the corresponding spaces of l2(H) valued Ft-measurable processes equipped with the norm(8)ϕl20,T;H2=E∫0T∑i=1∞ϕtiH2dt<∞,(9)
L2(Ω,Ft,P;H): the space of H-valued random variable ξ with the norm (9)ξ2=EξL2Ω,Ft,P;H2<∞,(10)
M2(0,T;H): the space of H-valued Ft-measurable process ϕ(·)=ϕ(t,ω):(t,ω)∈[0,T]×Ω with the norm (10)ϕ·M20,T;H2=E∫0TϕtM20,T;H2dt<∞,(11)
S2(0,T;H): the space of H-valued Ft-measurable ca`dla`g process f(·)=f(t,ω):(t,ω)∈[0,T]×Ω with the norm (11)f·S20,T;H2=Esup0≤t≤T⁡ftH2dt<∞,(12)
for notational brevity:(12)M20,T=M20,T;Rn×M20,T;Rm×M20,T;Rm×d×l20,T;Rm.Let us recall more generalIto’s formula about ca`dla`g semimartingales which is important for us to get the maximum principle. Let X=Xt:t∈[0,T] be ca`dla`g semimartingales, and we denoted the quadratic variation by [X]=[X]t:t∈[0,T]; F is a C2 real valued function; then F(X) is also semimartingales and followingIto’s formula holds:(13)FXt=FX0+∫0tF′Xs-dXs+12∫0tF′′XsdXsC+∑0<s≤tFXs-FXs--F′Xs-ΔXs,where [X]C is the continuous part of the quadratic variation [X].WhenF(X)=X2 and F(X)=XtYt, where X, Y are two ca`dla`g semimartingales, we get(14)Xt2=X02+∫0t2Xs-dXs+∫0tdXs,XtYt=X0Y0+∫0tXs-dYs+∫0tYs-dXs+∫0tdX,Ys.Here [X,Y] is the quadratic covariation of X, Y. We can refer to Protter [21] for a complete survey in this topic.Next, we introduce the existence and uniqueness results for fully coupled FBSDEL (1):(15)dxt=bt,xt,yt,zt,rtdt+∑i=1dσit,xt,yt,zt,rtdBti+∑i=1∞git,xt-,yt-,zt,rtdHti,-dyt=ft,xt,yt,zt,rtdt-∑i=1dztidBti-∑i=1∞rtidHti,x0=a,yT=ΦxT,where (16)b:Ω×0,T×Rn×Rm×Rm×d×l2Rm⟶Rn,σ:Ω×0,T×Rn×Rm×Rm×d×l2Rm⟶Rn×d,g:Ω×0,T×Rn×Rm×Rm×d×l2Rm⟶l2Rn,f:Ω×0,T×Rn×Rm×Rm×d×l2Rm⟶Rm.For a givenm×n full rank matrix G, we set(17)λ=xyz,At,λ,r=-GTft,λ,rGbt,λ,rGσt,λ,r.Assumption 1.
Assume the following.(i)
b, σ, g, and f are uniformly Lipschitz continuous with respect to (x,y,z,r).(ii)
For each(ω,t)∈Ω×[0,T], l(ω,t,0,0,0,0)∈M2(0,T) and g(ω,t,0,0,0,0)∈H2(l2), where l=b,σ,f, respectively.(iii)
Φ(·) is uniformly Lipschitz continuous with respect to x and ∀x,Φ(x)∈L2(Ω,FT,P).Assumption 2.
We also assume that(18)At,λ1,r1-At,λ2,r2,λ1-λ2+∑i=1∞Ggi^,ri^≤-β1Gx^2-β2GTy^2+GTz^2+∑i=1∞GTri^2;Φx1-Φx2,Gx1-x2≥μ1Gx^2,where λ1=(x1,y1,z1), λ2=(x2,y2,z2), x^=x1-x2, y^=y1-y2, z^=z1-z2, gi^=gi(t,λ1,r1)-gi(t,λ2,r2), ri^=r1i-r2i, and β1, β2 and μ1 are nonnegative constants with β1+β2>0, β2+μ1>0. Moreover, we have β1>0, μ1>0 (resp., β2>0) when m>n (resp., n>m). Under Assumptions 1 and 2, in [10], they have got the following lemma.Lemma 3 (existence and uniqueness theorem of FBSDEL [10]).
Under Assumptions1 and 2, FBSDEL (15) has a unique solution.


## 3. Continuity Result Depending on Parameters about FBSDEL

Next, we are going to get the continuity result depending on parameters about FBSDEL.Let(bα,σα,gα,fα,Φα), α∈R be a family of FBSDEL:(19)dxtα=bαt,xtα,ytα,ztα,rtαdt+∑i=1dσαit,xtα,ytα,ztα,rtαdBti+∑i=1∞gαit,xt-α,yt-α,ztα,rtαdHti,-dyt=fαt,xtα,ytα,ztα,rtαdt-∑i=1dzti,αdBti-∑i=1∞rti,αdHti,x0α=a,yTα=ΦαxTα.Assumption 4.
(i) The family(bα,σα,gα,fα,Φα), α∈R are equi-Lipschitz with respect to (x,y,z,r) and x separately.
(ii) The functionα→(bα,σα,gα,fα,Φα) is continuous in their existing space norm sense, respectively.Then we can get the following continuity result depending on parameters of forward-backward stochastic differential equation driven by Lévy processes.Theorem 5.
Let(bα,σα,gα,fα,Φα), α∈R be a family of FBSDEL satisfying Assumptions 1, 2, and 4 with solutions denoted by (xα,yα,zα,rα). Thus, the function(20)α⟶xα,yα,zα,rα,xTα:R⟶M20,T×L2Ω,Ft,P,Rnis continuous.Proof.
For notational brevity, we only prove the continuity of FBSDEL (19) at α=0. Set λ^t=λtα-λt0=(xtα-xt0,ytα-yt0,ztα-zt0) and r^t=rtα-rt0. From Assumptions 1, 2, and 4, applying the usual technique to x^t ofItô’s SDE with Lévy process, we can get (21)sup0≤t≤T⁡Ex^t2≤K1E∫0Ty^t2+z^t2+r^ti2dt+K1E∫0Tb^t2+σ^t2+g^t2dt.
Applying the same technique toy^t of BSDE with Lévy process, then (22)E∫0Ty^t2+z^t2+r^ti2dt≤K1E∫0Tx^t2dt+Ex^T2+E∫0Tf^t2dt+EΦαxT0-Φ0xT02.Here K1, K2 depend on the Lipschitz constants and T, and (23)f^t=fαt,λt0,rt0-f0t,λt0,rt0,b^t=bαt,λt0,rt0-b0t,λt0,rt0,σ^t=σαt,λt0,rt0-σ0t,λt0,rt0,g^t=gαt,λt0,rt0-g0t,λt0,rt0.
Set(24)Aαt,λ,r=-GTfαt,λ,rGbαt,λ,rGσαt,λ,r,and applyingItô’s formula to 〈Gx^t,y^t〉 yields (25)EΦαxTα-ΦαxT0,Gx^T+EΦ0xTα-Φ0xT0,Gx^T=E∫0TAαt,λtα,rtα-Aαt,λt0,rt0,λ^t+∑i=1∞Ggαit,λtα,rtα-gαit,λt0,rt0,r^tidt+E∫0Tx^t,-GTf^t+GTy^t,b^t+∑i=1dz^ti,Gσ^ti+∑i=1∞GTr^ti,g^tidt.
From the above three estimates, we get(26)β1E∫0TGx^t2dt+μ1EGx^T2+β2E∫0TGTy^t2+GTz^t2+GTr^t2dt≤C1EΦαxT0-Φ0xT02+E∫0Tf^t2+b^t2+σ^t2+g^t2dt+δEx^T2+δE∫0Tx^t2+y^t2+z^t2+r^t2dt,where the constant C1 depends on the Lipschitz constants T and δ. When m≥n, β1>0, and μ1>0, then (27)δ=min⁡13,13K1,β1GTG3,μ1GTG3.
Ifm≤n, β2>0, and μ1≥0, then (28)δ=min⁡13,13K1,13K1T,β2GGT3.
Thus, it is clear whateverβ1>0, β2≥0, and μ1>0 or β1≥0, β2>0, and μ1≥0 we always have(29)Ex^T2+E∫0Tλ^t2+r^t2dt≤CEΦαxT0-Φ0xT02+E∫0Tf^t2+b^t2+σ^t2+g^t2.
The proof is completed.


## 4. Maximum Principle

Let us consider the following full coupled forward-backward stochastic control system:(30)dxt=bt,xt,yt,zt,rt,utdt+∑i=1dσit,xt,yt,zt,utdBti+∑i=1∞git,xt-,yt-,rt,utdHti,-dyt=ft,xt,yt,zt,rt,,utdt-∑i=1dztidBti-∑i=1∞rtidHti,x0=a,yT=ΦxT,where (xt,yt,zt,rt) take values in Rn×Rm×Rm×d×l2(Rm); a∈Rn is given.LetU be a nonempty convex subset of Rk. We define the admissible control set Uad=u(·)∈M2(0,T;Rk);ut∈U,0≤t≤T,a.e.,a.s. and the cost functional: (31)Ju=E∫0TLxt,yt,zt,rt,utdt+hxT+γy0.The optimal control problem is to find u¯∈Uad, such that (32)Ju¯·=infu·∈U0,T⁡Ju·.Assumption 6.
Now we introduce the basic assumptions of this section as follows.(i)
b, f, and L are continuously differentiable with respect to (x,y,z,r,u); σ is continuously differentiable with respect to (x,y,z,u); g is continuously differentiable with respect to (x,y,r,u); Φ and h are continuously differentiable with respect to x; γ is continuously differentiable with respect to y. And the derivatives of each function are all bounded.(ii)
For each(x,y,z,r,u)∈Rn×Rm×Rm×d×l2(Rm)×Uad, there exists a constant C>0, such that (33)L≤C1+x2+y2+z2+r2+u2,h≤C1+x2,γ≤1+y2,Lx+Ly+Lz+Lr+Lu≤C1+x+y+z+r+u,hx≤C1+x,γy≤C1+y.(iii)
For any given admissible controlu(·), (30) satisfies Assumptions 1 and 2.Then, for a given admissible control, from Lemma3, there exists a unique solution satisfying control system (30).Letu¯t be an optimal control and let (x¯t,y¯t,z¯t,r¯t) be the corresponding trajectory. For any given admissible control ut and 0≤ɛ≤1, we define (34)utɛ=u¯t+ɛut-u¯t.Since Uad is convex, then utɛ is in Uad; that is, utɛ is an admissible control and (xtε,ytε,ztε,rtε) is the corresponding trajectory.We introduce the following variational equation:(35)dXt=bxtXt+bytYt+bztZt+brtRt+butut-u¯tdt+∑i=1dσxitXt+σyitYt+σzitZt+σuitut-u¯tdBti+∑i=1∞gxitXt-+gyitYt-+gritRt+guitut-u¯tdHti,-dyt=fxtXt+fytYt+fztZt+frtRt+fuitut-u¯tdt-∑i=1dZtidBti-∑i=1∞RtidHti,X0=0,YT=Φxx¯tXT.From Assumption6, we can verify that variational equation (35) satisfies Lemma 3. Thus, there exists a unique solution (Xt,Yt,Zt,Rt) satisfying variational equation. In order to get the maximum principle, we also need the following lemma.Lemma 7.
Assume that Assumption6 holds. We have (36)Esup0≤t≤T⁡xtε-x¯t-ɛXt2=oε2,Esup0≤t≤T⁡ytε-y¯t-ɛYt2+E∫0Tztε-z¯t-ɛZt2dt+E∫0Trtε-r¯t-ɛRt2dt=oε2,where (Xt,Yt,Zt,Rt) is the solution of variational equation (35).Proof.
Set(37)Δxt=ε-1xtɛ-x¯t,Δyt=ε-1ytɛ-y¯t,Δzt=ε-1ztɛ-z¯t,Δrt=ε-1rtɛ-r¯t,and then(38)dΔxt=ε-1bt,xtε,ytε,ztε,rtε,utε-bt,x¯t,y¯t,z¯t,r¯t,u¯tdt+ε-1∑i=1dσit,xtε,ytε,ztε,rtε,utε-∑i=1dσit,x¯t,y¯t,z¯t,r¯t,u¯tdBti+ε-1∑i=1∞git,xt-ε,yt-ε,ztε,rtε,utε-∑i=1∞git,x¯t-,y¯t-,z¯t,r¯t,u¯tdHti,-dΔyt=ε-1ft,xtε,ytε,ztε,rtε,utε-ft,x¯t,y¯t,z¯t,r¯t,u¯tdt-∑i=1dΔztidBti-∑i=1∞ΔrtidHti,Δx0=0,ΔyT=ε-1ΦxTε-Φx¯T.
We can transform (38) into(39)dΔxt=b^t,Δxt,Δyt,Δzt,Δrt,ut-u¯tdt+∑i=1dσ^it,Δxt,Δyt,Δzt,Δrt,ut-u¯tdBti+∑i=1∞g^it,Δxt-,Δyt-,Δzt,Δrt,ut-u¯tdHti,-dΔyt=f^t,Δxt,Δyt,Δzt,Δrt,ut-u¯tdt-∑i=1dΔztidBti-∑i=1∞ΔrtidHti,Δx0=0,ΔyT=ε-1ΦxTε-Φx¯T,where (40)l^t,x¯t,y¯t,z¯t,r¯t,ut-u¯t=Altx¯t+Blty¯t+Cltz¯t+Dltr¯t+Eltut-u¯t,for l=b,σ,g,f, respectively, and(41)Alt=At,xtε-x¯t≠00,otherwise,Blt=Bt,ytε-y¯t≠00,otherwise,Clt=Ct,ztε-z¯t≠00,otherwise,Dlt=Dt,rtε-r¯t≠00,otherwise,Elt=Et,ut-u¯t≠00,otherwise,where (42)At=xtε-x¯t-1·lt,xtε,ytε,ztε,rtε,utε-lt,x¯t,ytε,ztε,rtε,utε,Bt=ytε-y¯t-1·lt,x¯t,ytε,ztε,rtε,utε-lt,x¯t,y¯t,ztε,rtε,utε,Ct=ztε-z¯t-1·lt,x¯t,y¯t,ztε,rtε,utε-lt,x¯t,y¯t,z¯t,rtε,utε,Dt=rtε-r¯t-1·lt,x¯t,y¯t,z¯t,rtε,utε-lt,x¯t,y¯t,z¯t,r¯t,utε,Et=ɛut-u¯t-1·lt,x¯t,y¯t,z¯t,r¯t,utε-lt,x¯t,y¯t,z¯t,r¯t,u¯t.
From the continuity result depending on parameters we have got in Section3, we know that(43)Esup0≤t≤T⁡xtε-x¯t2+Esup0≤t≤T⁡ytε-y¯t2+E∫0tztε-z¯t2dt+E∫0trtε-r¯t2dt≤Kε2;that is, (xtε-x¯t,ytε-y¯t,ztε-z¯t,rtε-r¯t)→0 in M2(0,T), as ɛ→0. Together with Assumption 6, we can get (44)limɛ→0⁡Alt=lxt,x¯t,y¯t,z¯t,r¯t,u¯t,limɛ→0⁡Blt=lyt,x¯t,y¯t,z¯t,r¯t,u¯t,limɛ→0⁡Clt=lzt,x¯t,y¯t,z¯t,r¯t,u¯t,limɛ→0⁡Dlt=lrt,x¯t,y¯t,z¯t,r¯t,u¯t,limɛ→0⁡Elt=lut,x¯t,y¯t,z¯t,r¯t,u¯t,limɛ→0⁡l^t,Δxt,Δyt,Δzt,Δrt,ut-u¯t=lxt,x¯t,y¯t,z¯t,r¯t,u¯tΔxt+lyt,x¯t,y¯t,z¯t,r¯t,u¯tΔyt+lzt,x¯t,y¯t,z¯t,r¯t,u¯tΔzt+lrt,x¯t,y¯t,z¯t,r¯t,u¯tΔrt+lut,x¯t,y¯t,z¯t,r¯t,u¯tut-u¯t.
As FBSDEL (35) has a unique solution (Xt,Yt,Zt,Rt) under Assumption 6, from the continuity and uniqueness result by Lemma 3, we know that (Δxt,Δyt,Δzt,Δrt) converges to (Xt,Yt,Zt,Rt) in M2([0,T]) as ɛ→0.
The proof is complete.Notice thatlimɛ→0+(J(utɛ-J(u¯t))/ɛ)≥0 and we can get the following variational inequality.Lemma 8.
Assume that Assumption6 holds; then(45)E∫0TLxtXtdt+E∫0TLytYtdt+E∫0TLztZtdt+E∫0TLrtRtdt+E∫0TLutut-u¯tdt+EhxxTXT+Eγyy0Y0≥0.Proof.
First we have(46)Jutɛ-Ju¯tɛ=1ɛE∫0TLt,xtε,ytε,ztε,rtε,utε-Lt,x¯t,y¯t,z¯t,r¯t,u¯tdt+EhxTε-hx¯T+Eγy0ε-γy¯0.
Under Assumption6, from Lemma 7, we can get (47)limɛ→0⁡1ɛE∫0TLt,xtε,ytε,ztε,rtε,utε-Lt,x¯t,y¯t,z¯t,r¯t,u¯tdt⟶E∫0TLxt,x¯t,y¯t,z¯t,r¯t,u¯tXt+Lyt,x¯t,y¯t,z¯t,r¯t,u¯tYt+Lzt,x¯t,y¯t,z¯t,r¯t,u¯tZt+Lrt,x¯t,y¯t,z¯t,r¯t,u¯tRt+Lut,x¯t,y¯t,z¯t,r¯t,u¯tut-u¯tdt,limɛ→0⁡1ɛEhxTε-hx¯T⟶Ehxx¯TXT,limɛ→0⁡1ɛEγy0ε-γy¯0⟶Eγyy¯0Y0.Then (45) is proved.
The proof is complete.We define the Hamiltonian functionH as follows:(48)Ht,x,y,z,r,u,p,q,k,ρ=p,-ft,x,y,z,r,u+q,bt,x,y,z,r,u+k,σt,x,y,z,u+ρ,gt,x,y,r,u+Lt,x,y,z,r,u,and the following adjoint forward-backward equation to variational equation (35):(49)dpt=-Hyt,xt,yt,zt,rt,ut,pt,qt,kt,ρtdt-∑i=1dHzit,xt,yt,zt,rt,ut,pt,qt,kt,ρtdBti-∑i=1∞Hrit,xt-,yt-,zt,rt,ut,pt,qt,kt,ρtdHti,-dqt=Hxt,xt,yt,zt,rt,ut,pt,qt,kt,ρtdt-∑i=1dktidBti-∑i=1∞ρtidHti,p0=-γyy0,qT=hxxT-ΦxxTpT.It is easy to verify that (49) satisfies Assumptions 1 and 2; then there exists a unique quarter (pt,qt,kt,ρt) satisfying (49).Then we have the main result of this paper which is the following theorem.Theorem 9.
Supposing that Assumptions1, 2, 4, and 6 hold, (x¯t,y¯t,z¯t,r¯t,u¯t) is an optimal pair for our optimal control problem and (pt,qt,kt,ρt) is the solution to corresponding adjoint equation (49). Then for each admissible control ut∈U(0,T) we have(50)Hut,x¯t,y¯t,z¯t,r¯t,u¯t,pt,qt,kt,ρt,ut-u¯t≥0a.e.,a.s.,where H is defined by (48).Proof.
ApplyingIto’s formula to 〈Xt,qt〉+〈Yt,pt〉, we can obtain(51)Ehxx¯TXT+Eγyy¯0Y0+E∫0TLxtXt+LytYt+LztZt+LrtRt+Lut·ut-u¯tdt=E∫0THut,x¯t,y¯t,z¯t,r¯t,u¯t,pt,qt,kt,ρt,ut-u¯tdt.The variational inequality implies for each ut∈U[0,T] that (52)E∫0THut,x¯t,y¯t,z¯t,r¯t,u¯t,pt,qt,kt,ρt,ut-u¯tdt≥0.
The proof is completed.Next, under some additional convexity conditions, we prove that the maximum principle can be a sufficient condition for optimal control.Theorem 10.
For stochastic control system (30) and the cost functional J(u), if Assumptions 1, 2, 4, and 6 hold, and yT=MxT, M∈L2(Ω,FT,P;Rm×n), h is convex in x, and γ is convex in y. Let ut be an admissible control and let (xt,yt,zt,rt) be the corresponding trajectory. Let (pt,qt,kt,ρt) be the solution of corresponding adjoint equation (49). Suppose that the Hamiltonian function H is convex in (x,y,z,r,u) and inequality (50) holds; then ut is an optimal control.Proof.
Letvt be an arbitrary admissible control and the corresponding trajectory is (xtv,ytv,ztv,rtv); then(53)Jv·-Ju·=E∫0TLxtv,ytv,ztv,rtv,vt-Lxt,yt,zt,rt,utdt+EhxTv-hxT+Eγy0v-γy0=I1+I2,where (54)I1=E∫0TLxtv,ytv,ztv,rtv,vt-Lxt,yt,zt,rt,utdt,I2=EhxTv-hxT+Eγy0v-γy0.From the definition of Hamiltonian function H, we get(55)I1=E∫0THxtv,ytv,ztv,rtv,vt,ptv,qtv,ktv,ρtv-Hxt,yt,zt,rt,ut,pt,qt,kt,ρtdt+E∫0Tfxtv,ytv,ztv,rtv,vt-fxt,yt,zt,rt,ut,ptdt-E∫0Tbxtv,ytv,ztv,rtv,vt-bxt,yt,zt,rt,ut,qtdt-∑i=1dE·∫0Tσixtv,ytv,ztv,vt-σixt,yt,zt,ut,ktidt-∑i=1∞E∫0Tgixtv,ytv,rtv,vt-gixt,yt,rt,ut,ρtidt.By convexity of h,γ and usingItô’s formula to 〈qt,xtv-xt〉+〈pt,ytv-yt〉 we can get(56)I2≥EhxxT,xTv-xT+Eγyy0,y0v-y0=-E∫0THxxt,yt,zt,rt,ut,pt,qt,kt,ρt,xtv-xtdt-E∫0THyxt,yt,zt,rt,ut,pt,qt,kt,ρt,ytv-ytdt-∑i=1dE·∫0THzixt,yt,zt,rt,ut,pt,qt,kt,ρt,ztiv-ztidt-∑i=1∞E∫0THrixt,yt,zt,rt,ut,pt,qt,kt,ρt,rtiv-rtidt-E∫0Tfxtv,ytv,ztv,rtv,vt-fxt,yt,zt,rt,ut,ptdt+E∫0Tbxtv,ytv,ztv,rtv,vt-bxt,yt,zt,rt,ut,qtdt+∑i=1dE·∫0Tσixtv,ytv,ztv,vt-σixt,yt,zt,ut,ktidt+∑i=1∞E∫0Tgixtv,ytv,rtv,vt-gixt,yt,rt,ut,ρtidt,and from (53) to (56) we have(57)Jv·-Ju·=E∫0THxtv,ytv,ztv,rtv,vt,ptv,qtv,ktv,ρtv-Hxt,yt,zt,rt,ut,pt,qt,kt,ρtdt-E∫0THxxt,yt,zt,rt,ut,pt,qt,kt,ρt,xtv-xtdt-E∫0THyxt,yt,zt,rt,ut,pt,qt,kt,ρt,ytv-ytdt-∑i=1dE·∫0THzixt,yt,zt,rt,ut,pt,qt,kt,ρt,ztiv-ztidt-∑i=1∞E∫0THrixt,yt,zt,rt,ut,pt,qt,kt,ρt,rtiv-rtidt.
Moreover, as the Hamiltonian functionH is convex in (x,y,z,r,u), the following inequality holds:(58)Hxtv,ytv,ztv,rtv,vt,ptv,qtv,ktv,ρtv-Hxt,yt,zt,rt,ut,pt,qt,kt,ρt≥Hxxt,yt,zt,rt,ut,pt,qt,kt,ρt,xtv-xt+Hyxt,yt,zt,rt,ut,pt,qt,kt,ρt,ytv-yt+∑i=1dHzixt,yt,zt,rt,ut,pt,qt,kt,ρt,ztiv-zti+∑i=1∞Hrixt,yt,zt,rt,ut,pt,qt,kt,ρt,rtiv-rti+Huxt,yt,zt,rt,ut,pt,qt,kt,ρt,vt-ut,and from (57) to (58) and together with (50) for arbitrary admissible control vt we have (59)Jv·-Ju·≥0,then admissible control ut is an optimal control.
The proof is completed.


## 5. Applications in Linear Quadratic Problem

In this section, we will apply our stochastic maximum principle to the linear quadratic problem which the linear control system described by fully coupled forward-backward stochastic differential equation driven by Lévy processes:(60)dxt=Atxt+Btyt+∑i=1dCtizti+∑i=1∞Dtirti+Etutdt+∑i=1dFtixt+Gtiyt+Vtizti+WtiutdBti+∑i=1∞Jtixt-+Ktiyt-+Ltirti+MtiutdHti,-dyt=Ntxt+Ptyt+∑i=1dQtizti+∑i=1∞Rtirti+Stutdt-∑i=1dztidBt-∑i=1∞rtidHti,x0=a,yT=hxT+ξand the cost functional:(61)Ju=EQ~xT,xT+EP~y0,y0+E∫0TR~txt,xt+N~tut,ut+L~tyt,ytdt+∑i=1dE∫0TM~tizti,ztidt+∑i=1∞E∫0TK~tirti,rtidt,where the Ft-predictable matrix processes (62)A,Fi,Jj,R~:0,T×Ω⟶Rn×n,i=1,2,…,d,j=1,2,…B,Ci,Dj,Gi,Vi,Kj,Lj:0,T×Ω⟶Rn×m,i=1,2,…,d,j=1,2,…E,Wi,Mj:0,T×Ω⟶Rn×k,i=1,2,…,d,j=1,2,…P,Qi,Rj,L~,M~i,K~j:0,T×Ω⟶Rm×m,i=1,2,…,d,j=1,2,…N:0,T×Ω⟶Rm×n,S:0,T×Ω⟶Rm×k,N~:0,T×Ω⟶Rk×k;the FT-predictable random matrix Q~:Ω→Rn×n and the F0-predictable stochastic matrix P~:Ω→Rm×m are all uniformly bounded. And a∈Rn×n, h∈L2(Ω,FT,P;Rm×n), and ξ∈L2(Ω,FT,P;Rm).To study this problem, we need the assumptions on the coefficients as follows.Assumption 11.
The state weighting matrix processesR~, L~, M~i, and K~j, i=1,2,…,d,j=1,2,…, and the control weighting matrix process N~ and random matrixes Q~ and P~ are almost everywhere almost surely symmetric and nonnegative. Furthermore, N~ is almost everywhere almost surely uniformly positive; that is, N≥δI, for some positive constant δ almost everywhere almost surely.Assumption 12.
For the control processes there is no further constraint:(63)Uad=u·:u·isFt-predictable with values inRk,E∫0Tutdt<∞.If we denote the norm of Uad by (64)u·Uad=E∫0Tut2dt;then Uad is a Hilbert space. And by Lemma 3 we also know that the linear FBSDEL (60) has a unique solution; that is, the linear quadratic problem is well defined.Theorem 13.
Under Assumptions11 and 12, LQ problems (60) and (61) have a unique optimal control, and the optimal control is (65)ut=-12N~t-1EtTqt-StTpt+∑i=1dWtiTkti+∑i=1∞MtiTρti;here p, q, k, and ρ are the solution of the following adjoint FBSDEL:(66)dpt=Ptpt-BtTqt-∑i=1dGtiTkti-∑i=1∞KtiTρti-2L~tytdt-∑i=1dQtipt-CtiTqt-VtiTkti-2M~tiztidBti-∑i=1∞Rtipti-DtiTqt-LtiTρt-2K~tirtidHti,-dqt=NtTpt-Atqt-∑i=1dFtikti-∑i=1∞Jtiρti-2R~txtdt-∑i=1dktidBt-∑i=1∞ρtidHti,p0=-2P~y0,qT=2Q~xT-hpT.Proof.
From Assumptions11 and 12 and inequality (29), we can verify that the cost function J(u(·)) (61) is strictly convex and continuous over Uad and (67)limu·Uad→∞⁡Ju·=∞;then, from Lemma 5.3 in [18], the cost function has a unique minimal value and, together with Lemma 3 (existence and uniqueness theorem of FBSDEL), the LQ problem has a unique optimal control. Next, we will prove that the optimal control ut has an expression as (65).
Let(xt,yt,zt,rt) be the optimal state process corresponding to the optimal control ut and let (pt,qt,kt,ρt) be the unique solution to adjoint equation (66) corresponding to the optimal pair (ut;xt,yt,zt,rt); then the Hamiltonian function(68)Ht,x,y,z,r,u,p,q,k,ρ=-p,Ntxt+Ptyt+∑i=1dQtizti+∑i=1∞Rtirti+Stut+q,Atxt+Btyt+∑i=1dCtizti+∑i=1∞Dtirti+Etut+k,∑i=1dFtixt+Gtiyt+Vtizti+Wtiut+ρ,∑i=1∞Jtixt-+Ktiyt-+Ltirti+Mtiut+R~txt,xt+N~tut,ut+L~tyt,yt+∑i=1dM~tizti,zti+∑i=1∞K~tirti,rti.
From Theorem9 and Assumption 12 we have (69)Hu=-StTpt+EtTqt+∑i=1dWtiTkti+∑i=1∞MtiTρti+2Ntut=0;that is (70)ut=-12N~t-1EtTqt-StTpt+∑i=1dWtiTkti+∑i=1∞MtiTρti;then (65) holds.
The proof is completed.


## 6. Conclusion

In this paper, the continuity result depending on parameters of forward-backward stochastic differential equation driven by Lévy process is proved. Based on this result, we get the stochastic maximum principle for fully coupled forward-backward stochastic control system driven by Lévy process. And then we use the stochastic maximum principle to solve LQ problem.



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*Source: 702802-2015-10-19.xml*