# Maximum Principle for Forward-Backward Control System Driven by Itô-Lévy Processes under Initial-Terminal Constraints

Authors: Meijuan Liu; Xiangrong Wang; Hong Huang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper investigates a stochastic optimal control problem where the control system is driven by Itô-Lévy process. We prove the necessary condition about existence of optimal control for stochastic system by using traditional variational technique under the assumption that control domain is convex. We require that forward-backward stochastic differential equations (FBSDE) be fully coupled, and the control variable is allowed to enter both diffusion and jump coefficient. Moreover, we also require that the initial-terminal state be constrained. Finally, as an application to finance, we show an example of recursive consumption utility optimization problem to illustrate the practicability of our result.


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## Body

## 1. Introduction

The stochastic optimal control problem is very important in control theory. The sufficient and necessary conditions for the optimal control, called the maximum principle, are one of the control topics. Since 1960s, Pontryagin gave the necessary maximum principle under the deterministic control system; a lot of work has been done on this topic. Especially in nearly 20 years, many important results have been obtained; see literatures [1–4]. One can refer to Yong and Zhou [5], Wu [6], or Peng and Wu [7], for a complete account on the subject of maximum principle and a complete list of references.In recent years, motivated by studying the mathematical economics and mathematical finance, many scholars turned their sight on forward-backward stochastic control system (FBSCS in short). Initially, they studied the optimal conditions about the control system driven by Brownian motion and obtained some fundamental results in many literatures. Ji and Zhou [8] studied a kind of forward-backward stochastic control system where the forward state is constrained in a convex set at the terminal time, and a stochastic maximum principle is obtained. Liu et al. [9] considered one kind of linear quadratic optimal control with constraint for discrete-time stochastic systems with state and disturbance dependent noise, presenting a necessary condition under which the problem is well posed and a state feedback solution can be derived. Wu and Xu [10] obtained the maximum principle for fully coupled FBSCS with state constraints, in which the control domain needs to be convex and the diffusion coefficient does not contain control variables. Based on the results of Wu and Xu’s work in [10], Shi [11] studied a fully coupled forward-backward control system where the forward diffusion coefficient does not contain the control variable, but control domain is not necessarily convex. For the above-mentioned question, Meng [12] and Ji and Wei [13] have applied the terminal variation approach to obtain the maximum principle under other versions.Considering the complexity of finance market, as a consequence, it becomes natural to investigate control problems for systems driven by one kind of Itô-Lévy processes which is introduced by Nualart and Shoutens [14] and detailedly described by Applebaum [15]. Many researches have been done to extend the stochastic maximum principle of stochastic differential equations (SDEs in short) involving some Lévy jumps. Motivated by risk minimization, Øksendal and Sulem [16] studied the partial coupled FBSCS which need control domain to be convex and obtained maximum principle. Shi and Wu [17] obtained both necessary and sufficient maximum principle for optimal control of stochastic system with random jumps consisting of forward-backward state variables. The control variable is allowed to enter both diffusion and jump coefficients. Based on this literature, Shi [18] proved the global result of maximum principle where the control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficient of the forward equation. Zhang et al. [19] discussed the stochastic optimal control question of FBSDE driven by Teugels martingales and obtained the maximum principle under the assumption of convex control domain.In many classical cases as well as recent studies of forward-backward stochastic systems, introducing the model to formulate the process of financial derivative products has been a valid method to solve questions. As application of stochastic control theory, the stochastic maximum principle is usually used to solve investment strategy, risk control, and recursive utility; see examples of the literature: Ji and Zhou [8], Nualart and Schoutens [14], Shi and Wu [17], and Aase [20]. Similarly, we also apply the maximum principle to study the problem of maximizing consumption utility with initial-terminal state restrained, which is also discussed by Øksendal and Sulem [21].In this paper, we extend the result of Shi and Wu [17] to the case of fully coupled FBSCS, in which the control system is driven by Itô-Lévy process. At the same time, we demand that this forward-backward control system be constrained about initial-terminal state. We firstly use the traditional convex variational technique to prove the necessary condition about optimality. Then, using the result of this paper, we consider the problem of recursive consumption utility optimization problem with initial-terminal state constraints.This paper is organized as follows: In Section2, we formulate the control problem and list some preliminaries for fully coupled FBSCS. In Section 3, we establish variational equations about one kind of FBSCS with Itô-Lévy jumps in general form and then obtain variational inequality. In Section 4, we obtain the necessary condition of optimal control problem. In Section 5, we apply our result to test the application about the recursive utility problems.


## 2. Preliminaries and Notations

Let(Ω,F,Ft,P;t≥0) be a complete filtration space and {Ft;t≥0} be a filtration satisfying the usual conditions. On the above filtration space there exist two mutually independent stochastic processes:(1)
A d-dimensional Brownian motion{Bt;t≥0}(2)
A Poisson random measureN on R+×E, where E=R-{0} with the Borel σ-field B(E): λ is the intensity of N with the property that ∫E‍1∧e2λde<∝ and N¯ is the compensator of N, N¯(de,dt)=λ(de)dt; then N~0,t×E=N-N¯0,t×Et≥0  is a martingale for all A∈B(E) satisfying λA<∝.Also we introduce the following notations:τ: it is the transpose of a matrix.α,β: it is the inner product in Rn,∀α,β∈Rn.|α|=α,α: it is the norm in Rn,∀α∈Rn.A,B=tr(ABτ): it is the inner product in Rn×m,∀A,B∈Rn×m.|A|=tr(AAτ): it is the inner product in Rn×m,∀A∈Rn×m.L2Ω,FT,P≔{ξ∣ξisFT-measurable,Eξ2<∝}.M20,T≔{φt∣φtisFt-predictable,E∫0T‍φt2<∝}.FN20,T≔{φt∣φtisFt-predictable,E∫0T‍∫E‍φt,e2πdedt<∝}.Consider the following fully coupled forward-backward stochastic differential equation:(1)dxt=bt,xt,yt,zt,kt,edt+σt,xt,yt,zt,kt,edBt+∫Egt,xt,yt,zt,kt,eN~de,dt,-dyt=ft,xt,yt,zt,kt,edt-ztdBt-∫Ekt,eN~de,dt,0≤t≤T,(2)x0=a,(3)yT=hxT, where xt,yt,zt,kt,e take value in Rn×Rm×Rm×d×Rm, and(4)b:Ω×0,T×Rn×Rm×Rm×d×Rm⟶Rn,σ:Ω×0,T×Rn×Rm×Rm×d×Rm⟶Rn×d,g:Ω×0,T×Rn×Rm×Rm×d×Rm⟶Rn,f:Ω×0,T×Rn×Rm×Rm×d×Rm⟶Rm,h:Rn⟶Rm,k:Ω×0,T×E⟶Rm. We use the notations μ=xt,yt,zt,kt,eτ, A(t,μ)=-Gτft,μ,Gbt,μ,Gσt,μ,Ggt,μτ, where G is a given n × m full-rank matrix. Make the following assumptions:(5)H1:ib,σ,gandfisuniformlyLipschitzwithrespecttoμ,iihxisuniformlyLipschitzwithrespecttox∈Rn,iiiforeachx,hxisinL2Ω·FT·P,ivLω,t,0,0,0,0∈M20,T,L=b,σ,f,grespectively,andgω,t,0,0,0,0∈FN20,Tforω,t∈Ω×0,T,H2:iAt,μ-At,μ¯,μ-μ¯+∫EGτg^,k^λde≤-ν1Gx^2-ν2Gτy^2+Gτz^2+∫EGτk^2λde,iihx-hx¯,Gx-x¯≥ν3Gx^2,∀μ^=μ-μ¯=x^,y^,z^=x-x¯,y-y¯,z-z¯,g^=gt,μ,k-gt,μ¯,k¯, where ν1,ν2, and ν3 are given nonnegative constants with ν1+ν2>0,ν2+ν3>0.Lemma 1 (see [22]).
Assume(H1) and (H2) hold; then there is a unique adapted solution (x(t),y(t),z(t),k(t,e))∈M2(0,T;Rn+m+m×d)×FN2(0,T;Rm), and 0≤t≤T satisfied (1).Now we consider the following fully coupled FBSDE controlled system:(6)dxt=bt,xt,yt,zt,kt,e,utdt+σt,xt,yt,zt,kt,e,utdBt+∫Egt,xt,yt,zt,kt,e,utN~de,dt,-dyt=ft,xt,yt,zt,kt,e,utdt-ztdBt-∫Ekt,eN~de,dt,0≤t≤T,(7)x0=a,(8)yT=hxT,where xt,yt,zt,kt,e∈M20,T;Rn+m+m×d×FN2(0,T;Rm), u(t)∈U, and U⊂Rk is nonempty convex set, and mappings are(9)b:Ω×0,T×Rn×Rm×Rm×d×Rm×Uad⟶Rn,σ:Ω×0,T×Rn×Rm×Rm×d×Rm×Uad⟶Rn×d,g:Ω×0,T×Rn×Rm×Rm×d×Rm×Uad⟶Rn,f:Ω×0,T×Rn×Rm×Rm×d×Rm×Uad⟶Rm.Let Uad={v(·)∈M20,T;Rk;v(t)∈U, 0≤t≤T.a.e,a.s}; an element of Uad is called an admissible control.We define the following cost function:(10)Jv·=E∫0Tlt,xt,yt,zt,kt,·,vtdt+ΦxT+γy0, and the state variables of initial and terminal values satisfy the following constraint conditions:(11)EG1xxT=0,EG0yy0=0, where(12)l:Ω×0,T×Rn×Rm×Rm×d⟶R,Φ:Rn⟶R,G1:Rn⟶Rn1,n1≤n,γ:Rm⟶R,G0:Rm⟶Rm1,m1≤m.Problem 2.
Find an admissible controlu(·)∈Uad, such that(13)Ju·=infv·∈UadJv·.
We also assume that(14)H3:ib,σ,g,f,h,l,Φ,γ,G1andG0arecontinuousdifferentiableintheirarguments,iithederivativesofb,σ,g,f,h,l,G1andG0areuniformlybounded,iiilx,ly,lz,lk,lvareboundedbyc1+x+y+z+k·+v,ivγ≤k21+y2,Φ≤k21+x2,Φx≤k21+x,γy≤k21+y2,k2>0.Remark 3.
According to the conclusion of Lemma1, for the given admissible controller v(·), (6) has a unique adapted solution x·,y·,z·,k·,·,v·.


## 3. Variational Equations and Variational Inequality

Now we recall Ekeland’s variational principle.Lemma 4 (Ekeland’s variational principle; see [23]).
Let(V,d(·,·)) be a complete metric space and F(·):V→R be a proper lower semicontinuous function bounded from below, if for every ε>0, there exist u∈V, such that F(u)≤infv∈VF(v)+ε; then there exists uε∈V, such that(15)iFuε≤Fu,iidu,uε≤ε,iiiFv+εdv,uε≥Fuε,∀v∈V. Define the following distance on V, the space of admissible control; then (V,d(·,·)) is a complete metric space:(16)dv·,u·=E∫0Tvt-ut2dt1/2.
Letx·,y·,z·,k·,·,u· be the optimal solution of problem, for v(·)∈Uad; define(17)Fεv·=EG1xvT2+EG0yv02+EJv·-Ju·+ε21/2, where (xv(·),yv(·),zv(·),kv(·)) is the trajectory of (6) corresponding to v(·). And according to Ekeland’s variational principle, it is easy to get the following proposition.Proposition 5.
If(18)iFεv·:Uad⟶Riscontinuousfunction,iiFεv·>0,∀v·∈Uad,iiiFεu·≤infv·∈UadFεv·+ε, then there exists uε(·)∈Uad, such that(19)iFεuε·≤Fεu·=ε,iidu·,uε·≤ε,iiiFεv·+εdv·,uε·≥Fεuε·,∀v∈Uad.∀u¯(·)∈Uad, note u^(·)=u¯(·)-u(·),  u^ε(·)=u¯(·)-uε(·),  uδε(·)=uε(·)+δ(u¯(·)-uε(·)), due to the convexity of Uad,  for  0≤δ≤1, uδε(·)∈Uad.
Letxδε·,yδε·,zδε·,kδε· and (xε(·),yε(·),zε(·),kε(·)) be the trajectory of (6) corresponding to uδε(·) and uε(·), respectively; then Fε(uδε(·))-Fε(uε(·)+εd(uδε(·),uε(·))≥0.
Consider the following variational equation:(20)dxδ′t=bxtxδ′t+bytyδ′t+bztzδ′t+bktkδ′t,e+bvtu^εtdt+σxtxδ′t+σytyδ′t+σztzδ′t+σktkδ′t,e+σvtu^εtdBt+∫Egxtxδ′t+gytyδ′t+gztzδ′t+gktkδ′t,e+gvtu^εtN~de,dt,-dyδ′t=fxtxδ′t+fytyδ′t+fztzδ′t+fktkδ′t,e+fvtu^εtdt-zδ′tdBt-∫Ekδ′t,eN~de,dt,xδ′0=0,yδ′=hxxδTxδ′, and according to assumption (H3), one can verify (20) satisfying assumptions (H1) and (H2). Then it is easy to get, for a given u^ε, the fact that there exists a unique adapted solution (xδ′(·),yδ′(·),zδ′(·),kδ′(·,·)),0≤t≤T satisfying (20).Proposition 6.
Assume(H3) holds; then(21)limδ→0⁡xδεt-xεtδ=xδ′t,limδ→0⁡yδεt-yεtδ=yδ′t,limδ→0⁡zδεt-zεtδ=zδ′t,limδ→0⁡kδεt,·-kεt,·δ=kδ′t,·, where the limit is in M20,T;Rn+m+m×d×FN2(0,T;Rm).Proof.
Let(22)x^εt=xδεt-xεt,y^εt=yδεt-yεt,z^εt=zδεt-zεt,k^εt=kδεt,·-kεt,·, and the notations ⋀δε and ⋀ε denote (xδε(t),yδε(t),zδε(t),kδε(t,·)) and (xε(t),yε(t),zε(t),kε(t,·)), respectively; then the equation (for simplicity, we omit the time sign t)(23)dx^εt=bt,xδε,yδε,zδε,kδε·,uε+δu^ε-bt,xε,yε,zε,kε·,uεdt+σt,xδε,yδε,zδε,kδε·,uε+δu^ε-σt,xε,yε,zε,kε·,uεdBt+∫Egt,xδε,yδε,zδε,kδε·,uε+δu^ε-gt,xε,yε,zε,kε·,uεN~de,dt-dy^εt=ft,xδε,yδε,zδε,kδε·,uε+δu^ε-ft,xε,yε,zε,kε·,uεdt-z^εdBt-∫Ek^δε·N~de,dt,x^ε0=0,y^εT=0,can be rewritten into the following simple form:(24)dx^εt=bt,⋀δε,uε+δu^ε-bt,⋀ε,uεdt+σt,⋀δε,uε+δu^ε-σt,⋀ε,uεdBt+∫Egt,⋀δε,uε+δu^ε-gt,⋀ε,uεN~de,dt,-dy^εt=ft,⋀δε,uε+δu^ε-ft,⋀ε,uεdt-z^εdBt-∫Ek^δεeN~de,dt,x^ε0=0,y^εT=0.Sign(25)g^=gt,⋀δε,uε-gt,⋀ε,uε,g^δ=gt,⋀δε,uε+δu^ε-gt,⋀δε,uε,σ^δ=σt,⋀δε,uε+δu^ε-σt,⋀δε,uε,b^δ=bt,⋀δε,uε+δu^ε-bt,⋀δε,uε,f^δ=ft,⋀δε,uε+δu^ε-ft,⋀δε,uε.
By applying Itô formula tox^ε2, y^ε2, and Gx^ε,y^ε, we can prove that (x^ε,y^ε,z^ε,k^ε) converges to 0 in M2(0,T;Rn+m+m×d)×FN2(0,T;Rm) which is stated in the following part.(26)E∫0Tx^ε2dt≤C1E∫0Ty^ε2+z^ε2+∫Ek^εe2λdedt+C1δ2E∫0Tu^ε2dt,E∫0Ty^ε2+z^ε2+∫Ek^εe2λdedt≤C1E∫0Tx^ε2dt+C1δ2E∫0Tu^ε2dt,(27)0=E∫0TdGx^ε,y^ε=E∫0TAt,μδ,kδε-At,μ^δ,k^δε,μδ-μ^δdt+E∫0T∫EGg^,k^εeλdedt+E∫0TGb^δ,y^ε+Gx^ε,-f^δ+Gσ^δ,z^ε+∫EGg^δ,k^εeλdedt≤-ν1E∫0TGx^ε2dt-ν2E∫0TGτy^ε2+Gτz^ε2+∫EGτk^εe2λdedt+C1E∫0Tx^ε2+y^ε2+z^ε2+∫Ek^εe2λdedt+C2E∫0Tδ2u^ε2dt.That is,(28)ν1E∫0TGx^ε2dt+ν2E∫0TGy^ε2+Gz^ε2+∫EGk^εe2λdedt≤C1E∫0Tx^ε2+y^ε2+z^ε2+∫Ek^εe2λdedt+C2E∫0Tδ2u^ε2dt.
Ifm>n, ν1>0,ν2≥0, we can get(29)E∫0Tx^ε2dt≤C3δ2E∫0Tu^ε2dt. Let δ→0; we can get xδε→xε, and from the unique solution of (6), we have (yδε,zδε,kδε) which converges to (yε,zε,kε) in M2(0,T;Rm+m×d)×FN2(0,T;Rm).
Ifm<n,ν1≥0,ν2>0, we can get(30)E∫0Ty^ε2+z^ε2+∫Ek^εe2λdedt≤C4δ2E∫0Tu^ε2dt. Let δ→0; we also can get (yδε,zδε,kδε) which converges to (yε,zε,kε) in M2(0,T;Rm+m×d)×FN2(0,T;Rm), and then from the unique solution of (6), we have xδε which converge to xε.
DefineΔxε=xδε-xε/δ, Δyε=yδε-yε/δ, Δzε=(zδε-zε)/δ, and Δkε=(kδε-kε)/δ; then(31)dΔxε=bt,⋀δε,uε+δu^ε-bt,⋀ε,u^εδdt+σt,⋀δε,uε+δu^ε-σt,⋀ε,u^εδdBt+∫Egt,⋀δε,uε+δu^ε-gt,⋀ε,u^εδN~de,dt,-dΔyε=ft,⋀δε,uε+δu^ε-ft,⋀ε,u^εδdt-ΔzεdBt+∫Ek^δεeδN~de,dt,Δxε0=0,ΔyεT=0.
Sign(32)K1=Lt,⋀δε,uε+δu^ε-Lt,xε,yδε,zδε,kδε,uε+δu^εxδε-xε,K2=Lt,xε,yδε,zδε,kδε,uε+δu^ε-Lt,xε,yε,zδε,kδε,uε+δu^εyδε-yε,K3=Lt,xε,yε,zδε,kδε,uε+δu^ε-Lt,xε,yε,zε,kδε,uε+δu^εzδε-zε,K4=Lt,⋀δε,uε+δu^ε-Lt,xδε,yδε,zδε,kε,uε+δu^εkδε-kε,K5=Lt,⋀δε,u+δv-Lt,⋀δε,uεδu^ε, where L denotes b, σ, and f, respectively, and(33)K1Lt=K1t,xδε-xε≠0,0,otherelse,K2Lt=K2t,yδε-yε≠0,0,otherelse,K3Lt=K3t,zδε-zε≠0,0,otherelse,K4Lt=K4t,kδε-kε≠0,0,otherelse,K5Lt=K5t,δ×v≠0,0,otherelse.
Let(34)L¯t,x,y,z,k,v=K1tx+K2ty+K3tz+K4tk+K5tv, according to (H3) and the result of continuity:(35)limδ→0⁡L¯t,Δxε,Δyε,Δzε,Δkε,u^ε-Lxt,xε,yε,zε,kε,uΔxε-Lyt,xε,yε,zε,kε,uΔyε-Lzt,xε,yε,zε,kε,uΔzε-Lkt,xε,yε,zε,kε,uΔkε-Lvt,xε,yε,zε,kε,uu^ε=0. From the unique solution of (20) and the continuity, we have (Δxε,Δyε,Δzε,Δkε) which converges to (xδ′,yδ′,zδ′,kδ′) in M2(0,T;Rn+m+m×d)×FN2(0,T;Rm) as δ→0.Proposition 7 (variational inequality).
u
(
·
) is the optimal control, and (x(·),y(·),z(·),k(·,·)) are the trajectory of (6) corresponding to u(·). Assume condition (H3) holds; then there exist θT∈Rn1,  θ0∈Rm1, and θ∈R, which are not all for 0, such that the following variational inequality holds:(36)limδ→0⁡θT,EG1xxTx′T+θ0,EG0yy0y′0+θE∫0T∫Elxtx′t+lyty′t+lztz′t+lktk′t,·+luδεt-lutλdedt+EΦxxTx′T+γyy0y′0≥0, where (x′(t),y′(t),z′(t),k′(t,·)) are the solution of (20).Proof.
Based on (17), we can get(37)Fε2uδεt-Fε2uεt=EG1xuδεT2+EG0Yuδε02+EJuδεt-Jut+ε2-EG1xuεT2+EG0Yuε02+EJuεt-Jut+ε2=2EG1xuεT,G1xxuεTxδ′+2EG0Yuε0,G0yYuε0yδ′+2Juεt-Jut+εE∫0T∫Elxtx′t+lyty′t+lztz′t+lktk′t,e+luδεt-lutλdedt+EΦxxδTxδ′T+γyyδ0yδ′0. Due to limδ→0Fε(uδε(t))=Fε(uε(t)), we have(38)limδ→0⁡Fεuδεt-Fεuεtδ=limδ→0⁡Fε2uδεt-Fε2uεtδ×1Fεuδεt+Fεuεt=1Fεuεtlimδ→0⁡EG1xuεT,G1xxuεTxδ′t+EG0Yuε0,G0yYuε0yδ′t+Juεt-Jut+εE∫0T∫Elxtx′t+lyty′t+lztz′t+lktk′t,e+luδεt-lutλdedt+EΦxxδTxδ′T+γyyδ0yδ′0. Let(39)θε=1FεuεtJuεt-Jut+ε,θεT=1FεuεtEG1xuεT,θε0=1FεuεtEG0Yuε0, and according to Proposition 6, one can get(40)limδ→0⁡θεT,EG1xxuεTxδ′T+limδ→0⁡θε0,EG0yyuε0yδ′0+limδ→0⁡θεE∫0T∫Elxtx′t+lyty′t+lztz′t+lktk′t,e+(luδεt-lutλdedt+EΦxxδTxδ′T+γyyδ0yδ′0≥limδ→0⁡Fεuδεt-Fεuεtδ≥limδ→0⁡-εduδεt,uεtδ=limδ→0⁡-εE∫0Tuδϵt-uεt2dt1/2δ.
From the definition ofFε(·) and θε2+θεT2+θε02=1, we know that (θε,θεT,θε0) has convergence subsequence whose limitation is (θ,θT,θ0). Due to δ→0, uδε→uε, it is easy to get xδ′→x′, yδ′→y′, zδ′→z′, and kδ′→k′ in M2(0,T,Rn+m+m×d)×FN2(0,T;Rm). Let ε→0; then variational inequality (36) can be obtained.


## 4. The Adjoint Equation and Maximum Principle

In this paper, we consider the following adjoint FBSDE of system (6):(41)dPt=fyτt,∏,utPt-byτt,∏,utQt-σyτt,∏,utRt-gyτt,∏,utKt,e-lyt,∏,utdt+fzτt,∏,utPt-bzτt,∏,utQt-σzτt,∏,utRt-gzτt,∏,utKt,e-lzτt,∏,utdBt+∫E-fkτt,∏,utPt+bkτt,∏,utQt+σkτt,∏,utRt+gkτt,∏,utKt,e-lkN~de,dt,-dQt=fxτt,∏,utPt+bxτt,∏,kt,utQt+σxτt,∏,utRt+gxτt,∏,utKt,e+lxt,∏,utdt-RtdBt-∫EKt,eN~de,dt,0≤t≤T,P0=-γyy0-G0yτy0,QT=ΦxxT+G1xτxT-hxτxTPT,where ∏ denote (x(t),y(t),z(t),k(t,e)).From assumption(H3), We can verify that variational equation (20) and adjoint equation (41) all satisfy assumptions (H1),(H2). Thus it can be seen that there exists a unique adapted solution Pt,Qt,Rt,Kt,e in M2(0,T;Rn+m+m×d+m)×FN2(0,T;Rm) satisfying (41).Define the Hamiltonian functionH(·) as follows:(42)Ht,x,y,z,k·,v,P,Q,R,K=P·,-ft,x,y,z,k·,v+Q·,bt,x,y,z,k·,v+R·,σt,x,y,z,k·,v+∫EK·,e,gt,x,y,z,k·,vλde+lt,x,y,z,k·,v.DenoteHt≡Hxt,yt,zt,kt,·,Pt,Qt,Rt,Kt,·, and we can rewrite (26) in the following Hamiltonian system’s form:(43)dPt=-Hytdt-HztdBt-∫EHkt,eN~de,dt,-dQt=Hxtdt-RtdBt-∫EKt,eN~de,dt,0≤t≤T,P0=-γyy0-G0yτy0,QT=ΦxxT+G1xτxT-hxτxTPT.In the following part, we give the detailed proof about the necessary maximum principle which is given in [24].Theorem 8 (necessary maximum principle; see [24]).
Let(H1), (H2), and (H3) hold, and (x(·),y(·),z(·),k(·,·),u(·)) is the solution to the optimal control; then one has(44)Hvt,xt,yt,zt,kt,ut,Pt,Qt,Rt,Kt,·,vt-utdt≥0,∀v·∈Uad,a.e,a.s.Proof.
Apply Itô formula toQt,x′t  +  Pt,y′t, then one can get(45)E∫0TdQtx′t+Pty′t=EQT,x′T+PT,y′T-Q0,x′0-P0,y′0=E∫0TPt,-fvtu^εt+Qt,bvtu^εt+Rt,σvtu^εt+-lxtx′t-lyty′t-lztz′t-lktk′t,·dt; that is,(46)EQT,x′T+PT,y′T-Q0,x′0-P0,y′0=E∫0TPt,-fvtu^εt+Qt,bvtu^εt+Rt,σvtu^εt+Kt,·,gvtu^εt+-lxtx′t-lyty′t-lztz′t-lktk′t,·dt; that is,(47)EQT,x′T+PT,y′T-Q0,x′0-P0,y′0+E∫0Tlxtx′t+lyty′t+lztz′t+lktk′t,·+lvtu^εt=E∫0TPt,-fvtu^εt+Qt,bvtu^εt+Rt,σvtu^εt+Kt,·,gvtu^εt+E∫0Tlvt,u^εtdt=E∫0THvt,u^εtdt. Due to(48)EQT,x′T+PT,y′T-Q0,x′0-P0,y′0+E∫0Tlxtx′t+lyty′t+lztz′t+lktk′t,·+lvtu^εtdt≥0, we can get(49)E∫0THvt,xt,yt,zt,kt,ut,Pt,Qt,Rt,Kt,vt-utdt≥0,and then the following holds:(50)Hvt,xt,yt,zt,kt,ut,Pt,Qt,Rt,Kt,vt-utdt≥0,∀v·∈Uada.e,a.s. This finishes the proof of Theorem 8.Remark 9.
In this paper, we discuss the global fully coupled FBSCS where the diffusion coefficientsσ and g both contain the control variables v(·) and k(t,·). And under the constraint condition of [11], it is easy to check that if k(t,·)=0 and g(·)=0, the result of [11] can be obtained.


## 5. Example: Maximizing the Consumption Utility

Consider two assets in the financial market. One of the assets is a risk-free asset; the other one is a risky asset (e.g., stock). The price processes are described by the following dynamics equations:(51)dS0t=S0trtdt,S00=1,dSt=St-μtdt+σtdBt+∫Eηt,eN~de,dt,0≤t≤T,S0=s>0,where we assume the following:(i)
The risk-free interest rater(t) is a nonnegative predictable and bounded deterministic scalar-valued process.(ii)
μ(t), σ(t) are given Ft-adapted process, where μ(t)>r(t). And also η(t,e) is bounded deterministic process. To ensure that S(t)>0 for all t, let η(t,e)>-1 for all e∈E and ∫E‍η2t,eνde<∝.(iii)
The investor can borrow or lend risk-free interest rate.Assume that an investor whose initial wealth isa>0 decides to invest in stock with the amount of π(t). Denote x(t) and y(t) by the investor’s wealth process and consumption utility process, respectively. Suppose c(t) is the consumption process and c(t)∈Uad; by (6) and (51), we have(52)dxt=xt1-πtrt+πtμt-ctdt+xtπtσtdBt+πt∫Eηt,eN~de,dt,x0=a,-dyt=ft,xt,yt,zt,kt,·,ctdt-ztdBt-∫Ekt,eN~de,dt,0≤t≤T,yT=hxT.The investor’s problem is to solve(53)supc~yc~0,subjecttoE∫0Tπtc~tdt≤E∫0Tπtctdt,thebudgetconstraint,where y(t)=yc~(t) is the solution of the following BSDE:(54)-dyt=ft,xt,yt,zt,kt,·,ctdt-ztdBt-∫Ekt,eN~de,dt,0≤t≤T,yT=hxT=0.Leth(x(T))=G1x(x(T)); then the problem can be transformed into the following:(55)supc~⁡yc~0with E[G1x(x(T))]=0 subject to E∫0Tπtc~tdt≤E∫0Tπtctdt. Then this problem changes to be finding the optimal consumption process ct under the wealth terminal constraint and the budget constraint.Forε>0 define the Lagrangian(56)Lεc~=yc~0-εE∫0Tπtct~-ctdt.Suppose we, for eachε>0, can find an optimal c(t,ε)~ such that(57)supc~⁡Lεc~t=Lεc~t,εwith constraints. Also, suppose we can find ε^ such that(58)E∫0Tπtc~t,ε^-ctdt=0.Then(59)c~∗≔c~t,ε^ is optimal for the original constrained problem. To see this, note that for all c~(t) we have(60)yc~t,ε^0=yc~t,ε^0-ε^E∫0Tπtc~t,ε^-ctdt=Lε^c~t,ε^≥Lε^c~t=yc~0-ε^E∫0Tπtct~-ctdt≥yc~0. In view of this, we can solve the original constrained problem (54) in the following steps.Step 1.
Maximize over allL(c~) (without constraints), for each given ε>0. Call the maximum c~(t,ε).Step 2.
Findε^ such that E∫0Tπtc~t,ε^-ct=0. Then c~∗≔c~(t,ε^) can solve the original constrained problem.We now apply this to problem (54). Thus we fix ε and proceed to study the unconstrained problem (54) in the context of Section 4: consider the controlled FBSDE system consisting of(61)dxt=0,x0=a,-dyt=ft,xt,yt,zt,kt,·,ctdt-ztdBt-∫Ekt,eN~de,dt,0≤t≤T,yT=hxT=0.The performance function is(62)Jc~=-εE∫0Tπt2c~t-ctdt+yc~0, where ε>0 is the Lagrange multiplier. The Hamiltonian function about this problem is(63)Ht,yt,zt,kt,·,αt,ct=-επt2c~t-ct+αtft,yt,zt,kt,·,ct, and the adjoint equation is(64)dαt=αtfyt,xt,yt,zt,kt,·,ctdt+fzt,xt,yt,zt,kt,·,ctdBt+∫Ed∇kfdνt,xt,yt,zt,kt,e,cteN~de,dt,0≤t≤T,(65)α0=1.Let ∇kf denote the Frechet derivative of f with respect to k and d∇kf/dν(·,·)(e) denote its Radon-Nikodym derivative with respect to ν. If c^=c^(t,ε) is optimal for a given Lagrange multiplier ε with corresponding values (α^(t),y^(t),z^(t),k^(t,·)), we get by the Itô form the solution of (61); that is,(66)α^t=exp⁡∫0tf^ys-12f^zs2+∫R0ln⁡1+d∇kf^dνs,e-d∇kf^dνs,eνdeds+∫0tf^zsdBs+∫0t∫Eln⁡1+d∇kf^dνs,eN~ds,de,t≥0, where we use the simplified notation(67)f^ys=fyss,y^s,z^s,k^s,·,c^s,etc, and we assume that(68)d∇kfdνs,e>-1∀s,e,a.s. Maximizing the Hamiltonian with respect to c~ gives the first-order equation:(69)-επt2+α^tfc~t,y^t,z^t,k^t,·,c~t=0. Assume that for each t, y, and z, the function(70)c~⟶∂f∂ct,c~,y,z,k has an inverse, denoted by ∂f/∂c-1(t,y,z,k). Then the solution c~(t) of the first-order condition (75) can be written into(71)c~=c~t,α^t,y^t,z^t,k^t,·=∂f∂c~-1t,επ2tα^t,y^t,z^t,k^t,·≕c~t. Substituting this into the adjoint equation for α^(t), we get(72)dαt=αtf^yt,c~tdt+f^zt,c~tdBt+∫Ed∇kfdνt,c~teN~de,dt,α0=1,(73)0≤t≤T,where(74)f^t,c~t=fyt,c~α^t,y^t,z^t,k^t,·,y^t,z^t,k^t,·,etc. This is coupled to the following BSDE for (y^,z^,k^):(75)-dy^t=ft,c~α^t,y^t,z^t,k^t,·,y^t,z^t,k^t,·dt-z^tdBt-∫Ek^t,eN~de,dt,0≤t≤T,y^T=0.According to Lemma 1, (α^(t),y^(t),z^(t),k^(t,e)) can be found as the solution of the fully coupled system of FBSDE (72) and (75).Remark 10.
Assume the following.(i)
f has the form (76)ft,c~,y,z,k=f0t,c~,y-12Ayz2-12∫E0A1y,ek2eνde,where f0 does not depend on z and k.(ii)
Wealth state constraints(77)dxt=0,x0=0, and then the question of maximizing consumption utility has been solved in [20].


## 6. Conclusion

In this paper, the global fully coupled FBSCS driven by Itô-Lévy process is investigated, and the initial-terminal state needs to be restrained. By using the variational technique, we get the variational inequality. We also get the necessary condition of optimal solution about this stochastic control system. Finally, we use the result to discuss the problem of consumption recursive utility.



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*Source: 1868560-2017-06-20.xml*