# A Necessary Condition for Optimal Control of Forward-Backward Stochastic Control System with Lévy Process in Nonconvex Control Domain Case

Authors: Hong Huang; Xiangrong Wang; Ying Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper analyzes one kind of optimal control problem which is described by forward-backward stochastic differential equations with Lévy 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.


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## Body

## 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-backward 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. The fully coupled forward-backward stochastic 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 example, [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 observable 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, and lots of complicated stochastic calculi in infinite-dimensional spaces 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 obtained the maximum principle and sufficient conditions of an optimal control under some certain convexity assumptions. Wang et al. [16] proved the maximum principles for forward-backward stochastic control systems with correlated state and observation noises. More recent conclusions of the partially observed stochastic control problem can be seen from the studies conducted by Wang et al. [17], Zhang et al. [18], and Xiong et al. [19].In these years, through the study of mathematical economics and mathematical finance, many scholars turn their attention to the stochastic control system driven by Lévy process. In 2000, Nualart and Schoutens [20] built a pair of pairwise strongly orthonormal martingales which was called the Teugels martingale. Meanwhile, under some exponential moment conditions, they also obtained a martingale representation in that paper. Under these two important conclusions, for BSDE driven by the Teugels martingale, they [21] proved the existence and uniqueness theorem of its solution in the next year. From then on, a number of important results were proved: Meng and Tang [22] obtained the maximum principle of the forward stochastic control system driven by Lévy process. A necessary and sufficient condition for the existence of the optimal control of backward stochastic control systems driven by Lévy process was deduced by Tang and Zhang [23] through convex variation methods and duality techniques. For the forward-backward stochastic control system driven by Lévy process, there are also a lot of achievements: based on the existence and uniqueness theorem of FBSDEL [24], Zhang et al. [25] obtained a necessary condition of the optimal control and verification theorem, but in their control system, the backward state variables yt and zt did not enter the forward part. Wang and Huang [26] extended this result to the fully coupled control system and obtained the continuity result depending on parameters about FBSDEL and the local form maximum principle. Subsequently, Huang et al. [27] studied this control system with terminal state constraints and obtained the corresponding necessary maximum principle using Ekeland’s variational. For more recent conclusions about the stochastic control problem driven by Lévy process, please refer to [28–30].In this paper, we will study the optimal control problem for forward-backward stochastic control systems driven by Lévy process, which could be considered as a nonconvex control domain case that is extended from the result of [25]. With the technique of spike variation and Ekeland’s variational principle, the maximum principle of this type of control system and the control system with initial and final state constraints are obtained.The structure of this paper is as follows. Section2 describes some of the preparations used in this paper. The maximum principle and the one with initial and terminal state constraints as the major results of this paper will be shown in Sections 3 and 4. As an application of the maximum principle, Section 5 gives an optimal consumption problem in the financial market. Section 6 is the summary of this article.


## 2. Preliminary Statement

LetΩ,ℱt,P be a complete probability space which satisfied the usual conditions, and the information structure is given by ℱt which is generated by two processes: a standard Brownian motion Bt0≤t≤T valued in Rd and an independent 1-dimensional Lévy process Lt0≤t≤T of the form Lt=bt+lt; here, lt is a pure jump process. And assume that Lévy measure ν satisfies the following two conditions; thereby, Lévy process Lt0≤t≤T has moments in all orders.(i)
∫R1∧x2νdx<∞.(ii)
∫−ε,εceλxνdx<∞, ∀ε>0 and some λ>0.DenoteLt1=Lt, △Lt=Lt−Lt−, and Lti=∑0<s≤t△Lsi for i≥2. And let Yti=Lti−ELtii≥1 be the compensated power jump process of order i; then, Teugels martingale is defined by Hti=∑j=1icijYtj; the coefficients cij correspond to orthonormalization of the polynomials 1,x,x2,… with respect to the measure μdx=υdx+σ2δ0dx.Then,Htii=1∞ are pairwise strongly orthogonal martingales, and their predictable quadratic variation processes are Hti,Htj=δijt, δij is an indicator function here. And Hi,Hjt−Hti,Htj is an ℱt−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 spaceℋ,l2ℋ:=ϕℋ−valued,∑i=1∞ϕi2<∞.L2Ω,ℋ:=ξℋvalued,ℱT−measurable,Eξ2<∞.l20,T;ℋ:=ϕtil2ℋ−valued,ℱt−measurable,∑i=1∞E∫0Tϕti2dt<∞.M20,T;ℋ:=ϕ⋅ℋ−valued,ℱt−measurable,E∫0Tϕt2dt<∞.For the following FBSDEL,(1)dxt=bt,xt,yt,zt,rtdt+σt,xt,yt,zt,rtdBt+∑i=1∞git,xt−,yt−,zt,rtdHti,−dyt=ft,xt,yt,zt,rtdt−ztdBt−∑i=1∞rtidHti,x0=a,yT=ΦxT,where xt,yt,zt,rt take the value in Ω×0,T×Rn×Rm×Rm×d×l2Rm and mappings b,σ,g, and f take the value in Rn, Rn×d, l2Rn, and Rm, respectively. Convenient for writing, set column vector α=x,y,zT and At,α,r=−Mτft,α,r,Mbt,α,r,Mσt,α,rT, where M is a m×n full rank matrix.Assumption 1.
(i)
All mappings in equation (1) are uniformly Lipschitz continuous in their own arguments, respectively.(ii)
For allω,t∈Ω×0,T, lω,t,0,0,0,0∈M20,T;Rn+m+m×d×l20,T;Rm for l=b,f,σ, respectively, and gω,t,0,0,0,0∈l2Rn.(iii)
Φx¯−Φx,Mx¯−x≥βMx^2.(iv)
At,α¯,r¯−At,α,r,α¯−α+∑i=1∞Mgi^,ri^≤−μ1Mx^2−μ2Mτy^2+Mτz^2+∑i=1∞Mτri^2, where α¯=x¯,y¯,z¯,x^=x¯−x,y^=y¯−y,z^=z¯−z,ri^=r¯i−ri,gi^=git,α¯,r¯−git,α,r. μ1, μ2, and β are nonnegative constants which satisfied μ1+μ2>0, μ2+β>0, and μ1>0,β>0(resp. μ2>0) when m>n (resp. n>m).
Then, the following existence and uniqueness of the solution conclusion holds.Lemma 1.
There exists a unique solution inM20,T;ℋ satisfying FBSDEL (1) under Assumption 1.
The detailed certification process of this conclusion can be seen in [24].


## 3. Stochastic Maximum Principle

In this section, for any given admissible controlu⋅, we consider the following stochastic control system:(2)dxt=bt,xt,utdt+σt,xtdBt+∑i=1∞git,xt−dHti,−dyt=ft,xt,yt,zt,rt,,utdt−ztdBt−∑i=1∞rtidHti,x0=a,yT=ΦxT,t∈0,T,where a∈Rn is given. An admissible control u⋅∈M20,T;Rp is an ℱt-predictable process which takes values in a nonempty subset U of Rp; Uad is the set of all admissible controls.And the performance criterion is(3)Ju=Eγy0,where γ:Rm⟶R is a given Frechet differential function.Our optimal control problem amounts to determining an admissible controlu∗∈Uad such that(4)Ju∗⋅=infu⋅∈UadJu⋅.In order to get the necessary conditions for the optimal control, we assumeut∗ is the optimal control, and the corresponding solution of (2) is recorded as xt∗,yt∗,zt∗,rt∗ and introduce the “spike variational control” as follows:(5)utε=vt,τ≤t≤τ+ε,ut∗,otherwise,and xtε,ytε,ztε,rtε are the state trajectories of utε; here, vt is an arbitrary admissible control and ε is a sufficiently small constant.We also need the following assumption and variational equation (6).Assumption 2.
(i)
b,f,g,σ, Φ, and γ are continuously differentiable with respect to x,y,z,r,u, and the derivatives are all bounded.(ii)
There exists a constantC>0, and it holds that γy≤C1+y.(6)dXt=bxtXt+bt,utε−bt,ut∗dt+σxtXtdBt+∑i=1∞gxitXtdHti,−dYt=fxtXt+fytYt+fztZt+frtRt+ft,utε−ft,ut∗dt,−ZtdBt−∑i=1∞RtidHti,X0=0,YT=ΦxtXT.
Here,bxt=bxt,xt∗,ut∗, σxt=σxt,xt∗, gxit=gxit,xt∗, bt,utε=bt,xtε,utε, bt,ut∗=bt,xt∗,ut∗, fwt=fwt,xt∗,yt∗,zt∗,rt∗,ut∗, w=x,y,z⋅r, ft,utε=fxtε,ytε,ztε,rtε,utε, and ft,ut∗=fxt∗,yt∗,zt∗,rt∗,ut∗.Lemma 2.
Suppose Assumptions1 and 2 hold; for the first-order variation X,Y,Z,R, we have the following estimations:(7)sup0≤t≤TEXt2≤Cε2,(8)sup0≤t≤TEXt4≤Cε4,(9)sup0≤t≤TEYt2≤Cε2,(10)sup0≤t≤TEYt4≤Cε4,(11)sup0≤t≤TE∫0TZt2ds≤Cε2,(12)sup0≤t≤TE∫0TZt2ds2≤Cε4,(13)sup0≤t≤TE∫0TRt2ds≤Cε2,(14)sup0≤t≤TE∫0TRt2ds2≤Cε4.Proof.
We first prove inequations (7) and (8). For the forward part of the first-order variation equation, we have(15)EXt2=E∫0tbxsXs+bs,usε−bs,us∗ds+σxsXsdBs+∑i=1∞gxisXsdHsi2≤4E∫0tbxsXsds2+E∫0tbs,usε−bs,us∗ds2+E∫0tσxsXs2ds+E∫0t∑i=1∞gxisXs2ds2≤12C2TE∫0tXs2ds+4E∫0tbs,usε−bs,us∗ds2.
Applying Gronwall’s inequation, we have(16)EXt2≤Cε2,for allt∈0,T.
Similarly, (8) holds.
We next estimateYt, Zt, and Rt; the backward part of the first-order variation equation can be rewritten as(17)Yt+∫tTZsdBs+∫tT∑i=1∞RsidHsi=ΦxtXT+∫tTfxtXt+fytYt+fztZt+frtRt+ft,utε−ft,ut∗dt.
Squaring both sides of (17) and using the fact of(18)EYt∫tTZsdBsM=0,EYt∫tT∑i=1∞RsidHsi=0,E∫tTZsdBs∫tT∑i=1∞RsidHsi=0,we get(19)EYt2+E∫tTZs2ds+E∫tT∑i=1∞Rsi2dHsi=EΦxtXT+∫tTfxtXt+fytYt+fztZt+frtRt+ft,utε−ft,ut∗dt2≤6C2EXT2+6C2TE∫tTXs2ds+6C2TE∫tTYs2ds+6C2T−tE∫tTZs2ds+6C2T−tE∫tT∑i=1∞Rsi2ds+6E∫tTfs,usε−fs,us∗ds2.
Whent∈T−δ,T with δ=1/12C2, we have(20)EYt2+12E∫tTZs2ds+12E∫tT∑i=1∞Rsi2dHsi≤6C2EXT2+6C2TE∫tTXs2ds+6C2TE∫tTYs2ds+6E∫tTfs,usε−fs,us∗ds2.
Applying Gronwall’s inequation, we have(21)sup0≤t≤TEYt2≤Cε2,t∈T−δ,T;sup0≤t≤TE∫0TZt2ds≤Cε2,t∈T−δ,T;sup0≤t≤TE∫0TRt2ds≤Cε2,t∈T−δ,T.
Consider the BSDE of the first-order variation equation in the intervalt,T−δ:(22)Yt+∫tT−δZsdBs+∫tT−δ∑i=1∞RsidHsi=YT−δ+∫tT−δfxtXt+fytYt+fztZt+frtRt+ft,utε−ft,ut∗dt.
Thus,(23)EYt2+E∫tT−δZs2ds+E∫tT−δ∑i=1∞Rsi2dHsi,=EYT−δ+∫tT−δfxtXt+fytYt+fztZt+frtRt+ft,utε−ft,ut∗dt2≤6C2EXT2+6C2TE∫tT−δXs2ds+6C2TE∫tT−δYs2ds+6C2T−tE∫tT−δZs2ds+6C2T−tE∫tT−δ∑i=1∞Rsi2ds+6E∫tT−δfs,usε−fs,us∗ds2.
So, whent∈T−2δ,T with δ=1/12C2, we have(24)sup0≤t≤TEYt2≤Cε2,t∈T−2δ,T,sup0≤t≤TE∫0TZt2ds≤Cε2,t∈T−2δ,T,sup0≤t≤TE∫0TRt2ds≤Cε2,t∈T−2δ,T.
After a finite number of iterations, (9), (11), and (13) are obtained. And (10), (12), and (14) can be proved by using a similar method and the following inequalities: E∫tTZsdBs4≥β1E∫tTZs2ds2,β1>0; E∫tT∑i=1∞RsidHsi4≥β2E∫tT∑i=1∞Rsi2ds2,β2>0.Lemma 3.
Under hypothesis, Assumptions1 and 2, it holds the following four estimations:(25)sup0≤t≤TExtε−xt∗−Xt2≤Cεε2,Cε⟶0,whenε⟶0,(26)sup0≤t≤TEytε−yt∗−Yt2≤Cεε2,Cε⟶0,whenε⟶0,(27)E∫0Tztε−zt∗−Zt2ds≤Cεε2,Cε⟶0,whenε⟶0,(28)E∫tTrtε−rt∗−Rt2ds≤Cεε2,Cε⟶0,whenε⟶0.Proof.
To prove (25), we observe that(29)∫0tbs,xs∗+Xs,usεds+∫0tσs,xs∗+XsdBs+∫0t∑i=1∞gis,xs−∗+Xs−dHsi=∫0tbs,xs∗,usε+∫01bxs,xs∗+λXs,usεdλXsds+∫0tσs,xs∗+∫01σxs,xs∗+λXsdλXsdBs+∫0t∑i=1∞gis,xs∗+∫01∑i=1∞gxis,xs∗+λXsdλXsdHsi=∫0tbs,xs∗,us∗ds+∫0tσs,xs∗dBs+∫0t∑i=1∞gis,xs∗dHsi+∫0tbxs,xs∗,us∗Xsds+∫0tσxs,xs∗XsdBs+∫0t∑i=1∞gxis,xs−∗dHsi+∫0tbs,xs∗,usε−bs,xs∗,us∗ds+∫0tAεds+∫0tBεdBs+∫0tCεdHsi=xt∗−x0+Xt+∫0tAεds+∫0tBεdBs+∫0tCεdHsi,where(30)Aε=∫01bxs,xs∗+λXs,usε−bxs,xs∗,us∗dλXs,Bε=∫01σxs,xs∗+λXs−σxs,xs∗dλXs,Cε=∫01∑i=1∞gxis,xs∗+λXs−gxis,xs∗dλXs.
It follows easily from Lemma2 that(31)sup0≤t≤TE∫0tAεds2+∫0tBεdBs2+∫0tCεdHsi2=oε2.
Since(32)xtε−x0=∫0tbs,xsε,usεds+∫0tσs,xsεdBs+∫0t∑i=1∞gis,xs−εdHsi,then(33)xtε−xt∗−Xt=∫0tbs,xsε,usε−bs,xs∗+Xs,usεds+∫0tσs,xsε−σs,xs∗+XsdBs+∫0t∑i=1∞gis,xs−ε−gis,xs−∗+Xs−dHsi+∫0tAεds+∫0tBεdBs+∫0tCεdHsi=∫0tDεxsε−xs∗−Xsds+∫0tEεxsε−xs∗−XsdBs+∫0tFεxsε−xs∗−XsdHsi,with Dε=∫01bxs,xs∗+Xs+λxsε−xs∗−Xs,usεdλ, Eε=∫01σxs,xs∗+Xs+λxsε−xs∗−Xsdλ, and Fε=∫01∑i=1∞gxis,xs∗+Xs+λxsε−xs∗−Xsdλ.
By Gronwall’s inequation, we have(34)sup0≤t≤TExtε−xt∗−Xt2≤Cεε2,Cε⟶0,whenε⟶0.
Next, we prove (26), (27),, (28); it can be easily checked that(35)−∫tTfs,xs∗+Xs,ys∗+Ys,zs∗+Zs,rs∗+Rs,usεds+∫tTs,zs∗+ZsdBs+∫tT∑i=1∞gis,xs−∗+Xs−dHsi=ΦxT∗−yt∗+ΦxxT∗XT−Yt−∫tTGεds.
Here,(36)Gε=∫01fxs,xs∗+λXs,ys∗+λYs,zs∗+λZs,rs∗+λRs,usε−fxsdλXs+∫01fys,xs∗+λXs,ys∗+λYs,zs∗+λZs,rs∗+λRs,usε−fysdλYs+∫01fzs,xs∗+λXs,ys∗+λYs,zs∗+λZs,rs∗+λRs,usε−fzsdλZs+∫01frs,xs∗+λXs,ys∗+λYs,zs∗+λZs,rs∗+λRs,usε−frsdλRs.
Since(37)ytε=ΦxTε+∫tTfs,xsε,ysε,zsε,rsε,usεds−∫tTzsdBs−∫tT∑i=1∞rsi,εdHsi,then(38)ytε−yt∗−Yt=ΦxTε−ΦxT∗−ΦxxT∗XT+∫tTfs,xsε,ysε,zsε,rsε,usε−fs,xs∗+Xs,ys∗+Ys,zs∗+Zs,rs∗+Rs,usεds−∫tTzsε−zs∗−ZsdBs−∫tT∑i=1∞rsi,ε−rsi,∗−RsdHsi+∫tTGεds.
Squaring both sides of the equation above, we get(39)Eytε−yt∗−Yt2+E∫tTzsε−zs∗−Zs2ds−E∫tT∑i=1∞rsi,ε−rsi,∗−Rs2ds=Efs,xsε,ysε,zsε,rsε,usε−fs,xs∗+Xs,ys∗+Ys,zs∗+Zs,rs∗+Rs,usεds+ΦxTε−ΦxT∗−ΦxxT∗XT+∫tTGεds2.
From Lemma2 and equation (25), we have(40)sup0≤t≤TE∫tTGεds2=oε2,EΦxTε−ΦxT∗−ΦxxT∗XT2=oε2.
Then, we can get (26), (27), and (28) by applying the iterative method to the above relations.Lemma 4.
(variational inequality). Under the conditions that Assumptions1 and 2 are established, we can get the following variational inequality:(41)Eγyy0∗Y0≥oε.Proof.
From the four estimations in Lemma3, we have the following estimation:(42)Eγy0ε−γy0∗+Y0=oε.
Therefore,(43)0≤Eγy0∗+Y0−γy0∗+oε=Eγyy0∗Y0+oε.
We introduce the following Hamiltonian functionH:0,T×Rn×Rm×Rm×d×l2Rm×U×Rn×Rm×Rn×d×l2Rn as(44)Ht,x,y,z,r,u,p,q,w,k=p,bt,x,u+w,σt,x+k,gt,x−q,ft,x,y,z,r,u,and the following adjoint equation(45)dqt=Hyut∗dt+Hzut∗dBt+∑i=1∞Hriut∗dHti,−dpt=Hxut∗dt−wtdBt−∑i=1∞kt−idHti,q0=γyy0∗,pT=−ΦxxT∗qT,t∈0,T,where(46)Hxut∗=Hxt,xt∗,yt∗,zt∗,rt∗,ut∗,pt,qt,wt,kt,Hyut∗=Hyt,xt∗,yt∗,zt∗,rt∗,ut∗,pt,qt,wt,kt,Hzut∗=Hzt,xt∗,yt∗,zt∗,rt∗,ut∗,pt,qt,wt,kt,Hrut∗=Hrt,xt∗,yt∗,zt∗,rt∗,ut∗,pt,qt,wt,kt.
It is easily check that adjoint equation (45) has a unique solution quartet pt,qt,wt,kt∈M20,T;Rn+m+m×d×l20,T;Rm.
Then, we get the main result of this section.Theorem 1.
Let hypothesis, Assumptions1 and2, hold; ut∗ is an optimal control, and the corresponding optimal state trajectories are xt∗,yt∗,zt∗,rt∗; let pt,qt,wt,kt be the solution of adjoint equation (45); and then, for each admissible control ut∈Uad0,T, we have(47)Ht,xt∗,yt∗,zt∗,rt∗,ut,pt,qt,wt,kt≥Ht,xt∗,yt∗,zt∗,rt∗,ut∗,pt,qt,wt,kta.s.a.e..Proof.
Applying Ito^,s formula to p,X and q,Y, it follows from (6), (45), and the variational inequality that(48)E∫0THt,xt,yt,zt,rt,utε,pt,qt,wt,kt−Ht,xt,yt,zt,rt,ut∗,pt,qt,wt,ktdt=Eγyy0∗Y0≥oε.
By the definition ofutε, we know that, for any u∈Uad0,T, the following inequation holds:(49)EHt,xt,yt,zt,rt,utε,pt,qt,wt,kt−Ht,xt,yt,zt,rt,ut∗,pt,qt,wt,kt≥0.
Then, (47) can be easily checked.


## 4. Stochastic Control Problem with State Constraints

In this part, we are going to discuss stochastic control problems with state constraints in control system (2). Specifically, the initial state constraints and final state constraints are as follows:(50)EG1xT=0,EG0y0=0,where G1:Rn⟶Rn1n1<n,G0:Rm⟶Rm1m1<m. Our optimal control problem is to find u∗∈Uad such that(51)Ju∗⋅=infu⋅∈UadJu⋅,subject to the state constraints (50). In the following, we will apply Ekeland’s variational principle to solve this optimal control problem. Firstly, we need the following assumptions.Assumption 3.
Control domainU is assumed to be closed, the mappings in state constraints G1,G0 are continuously differentiable, and G1x,G0y are bounded.
For∀u1⋅,u2⋅∈Uad, let(52)du1⋅,u2⋅=Eu1⋅−u2⋅2>0;t∈0,T.
Same as Section3, we also assume ut∗ be the optimal control, and the corresponding optimal state trajectories are xt∗,yt∗,zt∗,rt∗. In order to solve the constraint problem, we need the following penalty cost functional, for any ρ>0:(53)Jρu=EG1xT2+EG0y02+Ju−Ju∗+ρ21/2.
It can be checked thatJρu:Uad⟶R1 is continuous, and for any u⋅∈Uad,(54)Jρu≥0,Jρu∗=ρ,Jρu∗≤infu⋅∈UadJρu+ρ.
It can be obtained by Ekeland’s variational principle that there existsutρ∈Uad such that(55)iJρuρ≤Jρu∗=ρ,iiduρ,u∗≤ρ,iiiJρv≥Jρuρ−ρduρ,v,for∀v⋅∈Uad.
For fixedρ and admissible control utρ, we define the spike variation as follows:(56)utρ,ε=vt,τ≤t≤τ+ε;utρ,otherwise,for any ε>0, and it is easy to check from iii of (55) that(57)Jρuρ,ε−Jρuρ+ρduρ,ε,uρ≥0.
Letxtρ,ytρ,ztρ,rtρ be the trajectories corresponding to utρ and xtρ,ε,ytρ,ε,ztρ,ε,rtρ,ε be the trajectories corresponding to utρ,ε. The variational equation we used in this section is the same as the one in Section 3, with ut∗=utρ and xt∗,yt∗,zt∗,rt∗=xtρ,ytρ,ztρ,rtρ. And we also assume that the solution of this variational equation is Xtρ,Ytρ,Ztρ,Rtρ. Similar to the approach in Lemmas 2 and 3, it can be shown that(58)sup0≤t≤TExtρ,ε−xtρ−Xtρ2≤Cεε2,Cε⟶0,whenε⟶0,sup0≤t≤TEytρ,ε−ytρ−Ytρ2≤Cεε2,Cε⟶0,whenε⟶0,E∫0Tztρ,ε−ztρ−Ztρ2ds≤Cεε2,Cε⟶0,whenε⟶0,E∫0Trtρ,ε−rtρ−Rtρ2ds≤Cεε2,Cε⟶0,whenε⟶0.
Then, by (57), the following variational inequality holds:(59)Jρuρ,ε−Jρuρ+ρduρ,ε,uρ=Jρ2uρ,ε−Jρ2uρJρuρ,ε+Jρuρ+ερ=h1ρ,ε,EG1xxTρXTρ+h0ρ,ε,EG0yy0ρY0ρ+hρ,εEγyy0ρY0ρ+ερ+oε,where(60)h1ρ,ε=2EG1xTρJρuρ,ε+Jρuρ,h0ρ,ε=2EG0y0ρJρuρ,ε+Jρuρ,hρ,ε=2Eγy0ρ−γy0+ρJρuρ,ε+Jρuρ.
Now, letptρ,ε,qtρ,ε,wtρ,ε,ktρ,ε be the solution of(61)−dptρ,ε=bxρtτptρ,ε−fxρtτqtρ,ε+σxρtτwtρ,ε+∑i=1∞gxρtτktρ,εidt,−wtρ,εdBt−∑i=1∞ktρ,εidHti,dqtρ,ε=fyρtτqtρ,εdt+fzρtτqtρ,εdBt+∑i=1∞frρ,itτqtρ,εdHti,q0=−G0,yy0ρh0ρ,ε+γyy0ρhρ,ε,pT=G1xxTρh1ρ,ε−ΦxxTρqTρ,ε,t∈0,T.
Here,bxρt=bxt,xtρ,utρ, σxρt=σxt,xtρ, fyρt=fyt,xtρ,ytρ,ztρ,rtρ,utρ, etc. Applying Ito^,s formula to Xtρ,qtρ,ε+Ytρ,ptρ,ε, variational inequality (59) can be rewritten as(62)Ht,xtρ,ytρ,ztρ,rtρ,utρ,ε,ptρ,ε,qtρ,ε,wtρ,ε,ktρ,ε−Ht,xtρ,ytρ,ztρ,rtρ,utρ,ptρ,ε,qtρ,ε,wtρ,ε,ktρ,ε+ερ+oε≥0,∀vt∈U,a.e.,a.s.,where the Hamiltonian function H is defined as (44). Since(63)limε⟶0h0ρ,ε2+h1ρ,ε2+hρ,ε2=1,there exists a convergent subsequence, still denoted by h0ρ,ε,h1ρ,ε,hρ,ε such that h0ρ,ε,h1ρ,ε,hρ,ε⟶h0ρ,h1ρ,hρ when ε⟶0 with h0ρ2+h1ρ2+hρ2=1.
Letptρ,qtρ,wtρ,ktρ be the solution of(64)−dptρ=bxρtτptρ−fxρtτqtρ+σxρtτwtρ+∑i=1∞gxρtτktρidt,−wtρdBt−∑i=1∞ktρidHti,dqtρ=fyρtτqtρdt+fzρtτqtρdBt+∑i=1∞frρ,itτqtρdHti,q0=−G0,yy0ρh0ρ+γyy0ρhρ,pT=G1xxTρh1ρ−ΦxxTρqTρ,t∈0,T.
From the continuous dependence of the solutions of FBSDEs with Lévy process on the parameters, we can prove that the following convergence holds:ptρ,ε,qtρ,ε,wtρ,ε,ktρ,ε⟶ptρ,qtρ,wtρ,ktρ; then, in equation (62), it implies that(65)Ht,xtρ,ytρ,ztρ,rtρ,vt,ptρ,qtρ,wtρ,ktρ−Ht,xtρ,ytρ,ztρ,rtρ,utρ,ptρ,qtρ,wtρ,ktρ+ρ≥0,∀vt∈U,a.e.,a.s..
Similarly, there exists a convergent subsequenceh0ρ,h1ρ,hρ such that h0ρ,h1ρ,hρ⟶h0,h1,h when ρ⟶0 with h02+h12+h2=1. Since utρ⟶ut∗, as ρ⟶0, we have xtρ,ytρ,ztρ,rtρ⟶xt∗,yt∗,zt∗,rt∗ and Xtρ,Ytρ,Ztρ,Rtρ⟶Xt,Yt,Zt,Rt, which is the solution of the variational equation as same as (6).
The following adjoint equation is introduced:(66)−dpt=bxtτpt−fxtτqt+σxtτwt+∑i=1∞gxtτktidt,−wtdBt−∑i=1∞ktidHti,dqt=fytτqtdt+fztτqtdBt+∑i=1∞frρ,itτqtdHti,q0=−G0,yy0h0+γyy0h,pT=G1xxTh1−ΦxxTqT,t∈0,T.
Similarly, it can be proved thatptρ,qtρ,wtρ,ktρ⟶pt,qt,wt,kt; then, inequation (65) implies(67)Ht,xt∗,yt∗,zt∗,rt∗,vt,pt,qt,wt,kt−Ht,xt∗,yt∗,zt∗,rt∗,ut∗,pt,qt,wt,kt≥0,∀vt∈U,a.e.,a.s..
Then, we get the following theorem:Theorem 2.
Let Assumptions1–3 hold; ut∗ is an optimal control, and the corresponding optimal state trajectories are xt∗,yt∗,zt∗,rt∗, and pt,qt,wt,kt is the solution of adjoint equation (66); then, there exists nonzero constant h1,h0,h∈Rn1×Rm1×R withh02+h12+h2=1 such that, for any admissible control vt∈Uad0,T, the maximum condition (67) holds.
This conclusion can be drawn from the above analysis directly.


## 5. A Financial Example

In this section, we will study the problem of optimal consumption rate selection in the financial market, which will naturally inspire our research to the forward-backward stochastic optimal control problem inSection 3.Assume the investor’s asset processxtt≥0 in the financial market is described by the following stochastic differential equation with the Teugels martingale:(68)dxt=μtxt−Ctdt+σtxtdBt+∑i=1∞gtixtdHti,x0=W>0,t∈0,T,where μt and σt≠0 are the expected return and volatility of the value process xt at time t, respectively, and Ct is the consumption rate process. Assume μt, σt, gt, and Ct are all uniformly bounded ℱt−measurable random processes. The purpose of investors is to select the optimal consumption strategy Ct∗ at time t≥0, minimizing the following recursive utility:(69)JC⋅=Ey0,where yt is the following backward stochastic process:(70)−dyt=Le−rtCt1−R1−R−rytdt−ztdBt−∑i=1∞rtidHti,yT=−xT>0,t∈0,T,with constant L>0, the discount factor r>0, and the Arrow–Pratt measure of risk aversion R∈0,1.If the consumption processCt is regarded as a control variable, then combining (68) with (70), we encounter the following control system:(71)dxt=μtxt−Ctdt+σtxtdBt+∑i=1∞gtixtdHti,−dyt=Le−rtCt1−R1−R−rytdt−ztdBt−∑i=1∞rtidHti,x0=W>0,yT=−xT>0,t∈0,T,which is obviously a special case of stochastic control system (2) with bt,x,u=μtxt−Ct,σt,x=σtxt,gt,x=gtixt,ΦxxT=−XT and ft,x,y,z,r,u=Le−rtCt1−R/1−R−ryt.We can check that both Assumptions1 and 2 are satisfied. Then, we can use our maximum principle (Theorem 1) to solve the above optimization problem. Let Ct∗ be an optimal consumption rate and xt∗,yt∗ be the corresponding wealth process and recursive utility process. In this case, the Hamiltonian function H reduces to(72)Ht,xt∗,yt∗,Ct,pt∗,qt∗,wt∗,kt∗=pt∗,μtxt∗−Ct+wt∗,σtxt∗+kt∗,gtixt∗−qt∗,Le−rtCt1−R1−R−ryt∗,and pt∗,qt∗,wt∗,kt∗ is the solution of the following adjoint equation:(73)dqt=rqtdt−dpt=μtpt+σtwt+∑i=1∞ktiτgtidt−wtdBt−∑i=1∞ktidHti,q0=1,pT=qT,t∈0,T.According to the maximum principle (47), we have(74)pt∗=−Le−rtCt∗−Rqt∗,and the optimal consumption rate(75)Ct∗=−ertLpt∗qt∗−1/R.By solving FBSDE (73), we get qt∗=ert,pt∗=erteμT−t, and wt∗=kt∗=0. Then, we get the optimal consumption rate of the investor which is(76)Ct∗=−erTLeμT−t−1/R,t∈0,T.


## 6. Conclusions

In this paper, a nonconvex control domain case of the forward-backward stochastic control driven by Lévy process is considered, and we obtain the global stochastic maximum principle for this stochastic control problem. And then, the problem of stochastic control with initial and final state constraints on the state variables is discussed, and a necessary condition about existence of the optimal control is also acquired. A financial example of optimal consumption is discussed to illustrate the application of the stochastic maximum principle.



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*Source: 1768507-2020-06-03.xml*