# Linear Quadratic Stochastic Optimal Control of Forward Backward Stochastic Control System Associated with Lévy Process

Authors: Hong Huang; Xiangrong Wang; Ting Hou; Lu Xu
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

This paper analyzes one kind of linear quadratic (LQ) stochastic control problem of forward backward stochastic control system associated with Lévy process. We obtain the explicit form of the optimal control, then prove it to be unique, and get the linear feedback regulator by introducing one kind of generalized Riccati equation. Finally, we discuss the solvability of the generalized Riccati equation, and its existence and uniqueness of the solutions are proved in a special case.


---

## Body

## 1. Introduction

LQ stochastic optimal control is a kind of special optimal control problem, which not only can be used to model many linear optimal problems practically, but also can reasonably be used to approach and solve many nonlinear problems. In 1962, Kushner [1] firstly established a forward random stochastic LQ model with a dynamic programming method and Wonham [2] firstly studied a LQ stochastic optimal control problem by introducing a Riccati equation in 1968. Then a lot of works have been done for forward or backward stochastic LQ control problems, the corresponding Riccati equation, and its application in finance, such as Li and Zhang [3], Ma and Hou [4], Liu et al. [5], Wang et al. [6], and Shen and Wang [7]. In 2003, Wang et al. [8] discussed a special kind of forward backward stochastic LQ problem and got the existence and uniqueness of the optimal control for the control system. Subsequently, Wu [9] extended this conclusion to the fully coupled forward backward stochastic LQ problem.The optimal control problem with random jumps was first considered by Boel and Varaiya [10]; in this case, the control system is often described by Brownian motion and Poisson processes. On the basis of proving the existence and uniqueness of solutions of a kind of forward backward stochastic differential equation with Poisson jumps (FBSDEP), Wu and Wang [11] got the explicit form of the optimal control for LQ stochastic control problem where the state variable was described by a stochastic differential equation with a Poisson process (SDEP). In 2009, Shi and Wu [12] extended Wu and Wang’s results in [11] to a fully coupled LQ stochastic control problem of forward backward stochastic control system with Poisson jumps. Moreover, Lin and Zhang [13] considered the H∞ control problem for linear stochastic systems driven by both Brownian motion and Poisson jumps. In 2016, Li et al. [14] studied a stochastic differential equations driven by G-Brownian motion and got the existence and uniqueness of the solution for these equations.In 2000, Nualart and Schoutens [15] introduced a class of Lévy processes with exponential moments satisfying some conditions. Using these exponential moments and the standard orthogonalization process, they constructed a series of orthogonal normal martingales called Teugels martingale. And they also proved a martingale representation theorem associated with Teugels martingale. In the next year, Nualart and Schoutens [16] considered a backward stochastic differential equation (BSDE) driven by Teugels martingale and proved the existence and uniqueness theory of this BSDE. In 2003, Bahlali et al. [17] studied a BSDE driven by Teugels martingale and an independent Brownian motion; they got the existence, uniqueness, and comparison of solutions for these equations, having a Lipschitz or locally Lipschitz coefficient. El Otmani [18] considered a kind of generalized BSDE (GBSDE) associated with Teugels martingale and Brownian motion associated with a pure jump-independent Lévy process. They got the existence and uniqueness theory of this GBSDE when the coefficient verifies some conditions of Lipschitz. More results about BSDE associated with Teugels martingale can be found in the theses of El Otmani [19], Ren and Fan [20], Tang and Zhang [21], and Huang and Wang [22]. On the basis of these results, in 2008, Mitsui and Tabata [23] studied a LQ regulation stochastic control problem with Lévy process and obtained the optimal control for the nonhomogeneous case. In [24], Tang and Wu considered the following LQ stochastic control problem in a given finite horizon [s,T] with Lévy process:(1)dxt=Aω,txt+Bω,tutdt+∑i=1∞Ciω,txt-+Diω,tutdHitxs=ξ,and the cost function was(2)Js,ξ,u·=Es∫sTxτtQω,txt+uτtRω,tutdt+xτTHωxT.They show that the solvability of one kind of generalized Riccati equation is sufficient to the well-posedness of this LQ problem and proved the existence of the optimal control.In this paper, we consider one kind of LQ stochastic control problem where the controlled system is driven by a fully coupled linear forward backward stochastic differential equation associated with Lévy process (FBSDEL).(3)dxt=Aω,txt+Bω,tut-Lω,tytdt+Cω,txt+Dω,tutdBt+∑i=1∞Eiω,txt-+Fiω,tutdHti-dyt=Aτω,tyt+Cτω,tzt+∑i=1∞Ei,τω,trti+Rω,txtdt-ztdBt-∑i=1∞rtidHtix0=a,yT=QωxT,where (xt,yt,zt,rti) are Ft-adapted stochastic processes taking values in Rn×Rn×Rn×l2(Rn) and u(·) is Ft-adapted stochastic process called admissible control process. Assume the control process set U=Rk and define the admissible control set as follows:(4)Uad=u·∈M20,T;Rk;ut∈U,0≤t≤T,a.e.,a.s..The cost functional we considered is(5)Ju=12E∫0TRω,txt,xt+Nω,tut,ut+Lω,tyt,ytdt+QωxT,xT.And the optimal control problem is to find u¯t∈Uad, such that(6)Ju¯·=infu·⁡Ju·.Note that (3) is a fully coupled FBSDEL. In 2012, Pereira and Shamarova [25] firstly considered this kind of FBSDEL, obtained a solution to this FBSDEL via a partial integrodifferential equation, and proved the uniqueness. Under some monotonicity assumptions, Baghery et al. [26] proved the existence and uniqueness of solutions of fully coupled FBSDEL and then obtained the existence of an open-loop Nash equilibrium point for nonzero sum stochastic differential games by using this result. Based on [25], Wang and Huang [27] got the maximum principle for forward backward stochastic control system driven by Lévy process; then they discussed a kind of LQ stochastic control problem of forward backward stochastic control system and got a necessary condition for the optimal control.We extend the result of Shi and Wu [12] to the fully coupled linear forward backward stochastic control system driven by Brownian motion and an independent Teugels martingale. Since Teugels martingale is more complex than the Poisson process, we also need more general formula aboutcàdlàg semimartingale. The rest of this paper is organized as follows. In Section 2, we provide a list of notations and results of the existence and uniqueness of solutions of fully coupled FBSDEL. In Section 3, we prove the existence and uniqueness of the optimal control of LQ stochastic control problem (6) and give the linear feedback regulator for the optimal control by the solution of a kind of generalized matrix-valued Riccati equation when assuming the coefficient matrices are deterministic. In Section 4, the solvability of this kind of matrix-valued Riccati equation is discussed.


## 2. Preliminaries and Notations

Let(Ω,Ft,P) be a complete probability space satisfying the usual conditions; Ft is a right continuous increasing family of complete sub-σ-algebra which is generated by the following two mutually independent processes: a one-dimensional standard Brownian motion {Bt}0≤t≤T and an R-valued Lévy process {Lt}0≤t≤T with a standard Lévy measure ν satisfy(i)
∫R(1∧x2)ν(dx)<∞,(ii)
∫(-ε,ε)ceλ|x|ν(dx)<∞, for every ε>0 and for some λ>0.Naluart and Schoutens denoted Teugels martingale associated with the Lévy process{Lt}0≤t≤T by {Hti}i=1∞, and Hti is given by(7)Hti=ci,iYti+ci,i-1Yti-1+ci,i-2Yti-2+⋯+ci,1Yt1,where Yti=Lti-E[Lti] is the compensated power-jump process of order i and Lti is power-jump processes: (8)Lti=Lt,i=1,∑0<s≤tΔLsi,i≥2,Coefficients ci,k correspond to orthonormalization of the polynomials 1,x,x2,… with respect the measure μ(dx)=υ(dx)+σ2δ0(dx). Please refer to Naluart and Schoutens [15] for more details about Teugels martingale.Introduce the following notations adopted in this paper:〈A,B〉=tr(ABT): the inner product in Rn×m, ∀A,B∈Rn×m|α|=〈α,α〉: the norm in Rn, ∀α∈RnL2(Ω,H): the space of H-valued FT-measurable random variable ξ satisfies E|ξ|2<∞M2(0,T;H): the space of H-valued Ft-measurable process ϕ(·)={ϕ(t,ω):(t,ω)∈[0,T]×Ω} satisfies E∫0T|ϕt|2dt<∞l2(H): the space of H-valued {fi}i≥1 satisfies ∑i=1∞|fi|2<∞l2(0,T;H): the space of l2(H)-valued Ft-measurable processes satisfies E∫0T∑i=1∞|fti|2dt<∞S2(0,T;H): the space of H-valued Ft-measurable càdlàg process f(·)={f(t,ω):(t,ω)∈[0,T]×Ω} satisfies Esup0≤t≤T⁡|ft|2dt<∞.For notational brevity, we set(9)M20,T=M20,T;Rn×M20,T;Rn×M20,T;Rn×l20,T;Rn.Next, consider the following fully coupled FBSDEL(10)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∞rtidHtix0=a,yT=ΦxT,where b:Ω×[0,T]×Rn×Rm×Rm×d×l2(Rm)→Rn, σ:Ω×[0,T]×Rn×Rm×Rm×d×l2(Rm)→Rn×d, g:Ω×[0,T]×Rn×Rm×Rm×d×l2(Rm)→l2(Rn), f:Ω×[0,T]×Rn×Rm×Rm×d×l2(Rm)→Rm.For a givenm×n full rank matrix G, set (11)λ=xyz,At,λ,r=-Gτft,λ,rGbt,λ,rGσt,λ,r.Assumption 1.
(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.
(12)
A
t
,
λ
1
,
r
1
-
A
t
,
λ
2
,
r
2
,
λ
1
-
λ
2
+
∑
i
=
1
∞
G
g
i
^
,
r
i
^
≤
-
β
1
G
x
^
2
-
β
2
G
τ
y
^
2
+
G
τ
z
^
2
+
∑
i
=
1
∞
G
τ
r
i
^
2
;
Φ
x
1
-
Φ
x
2
,
G
x
1
-
x
2
≥
μ
1
G
x
^
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, μ1 are nonnegative constants with β1+β2>0 and β2+μ1>0. Moreover, we have β1>0,μ1>0 (resp., β2>0) when m>n (resp., n>m).Lemma 3 (existence and uniqueness theorem of FBSDEL [25]).
Under Assumptions1 and 2, FBSDEL (10) admits a unique solution in M2(0,T).In the following sections we also need the more general Ito’s formula about a càdlàg semimartingales.Lemma 4 (Ito’s formula [27]).
LetX={Xt:t∈[0,T]} be càdlàg semimartingales, denote [X]={[X]t:t∈[0,T]} as the quadratic variation process, F is a C2 real valued function, then F(X) is also a semimartingales, and the following Ito’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 [X].
In particular, whenF(X)=X2 and F(X)=XtYt, where X,Y are two càdlàg 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.


## 3. Linear Quadratic Stochastic Optimal Control Problem

Let us consider the LQ stochastic optimal control problem (6). First of all, we give some necessary explanations for the coefficients in the system:A
(
ω
,
t
)
,
C
(
ω
,
t
)
,
E
i
(
ω
,
t
)
,
(
i
=
1,2
,
3
,
…
)
∈
R
n
×
n, B(ω,t),D(ω,t), and Fi(ω,t),(i=1,2,3,…)∈Rn×k are all bounded progressively measurable matrix-valued processes. R(ω,t),L(ω,t)∈Rn×n are nonnegative symmetric bounded progressively measurable matrix-valued processes, and N(ω,t) is a positive bounded k×k progressively measurable matrix-valued process; the inverse is N-1(ω,t), which is also bounded. Q(ω) is a Ft-adapted nonnegative symmetric bounded matrix-valued random variable.For a given admissible controlu(·)∈Uad, under assumptions of the coefficients above, we can verify that FBSDEL (3) satisfies Assumptions 1 and 2. Therefore, there exists a unique solution (xtu,ytu,ztu,rtu)∈M2(0,T) satisfying the control system (3) from Lemma 3.Then we get the explicit form of the optimal controlut for the LQ stochastic optimal control problem (6).Theorem 5.
There exists a unique optimal controlut for LQ stochastic optimal control problem (6), and ut is given by the following equation.(15)ut=-N-1ω,tBτω,tyt+Dτω,tzt+∑i=1∞Fi,τω,trti.Proof.
As we know, for a given admissible controlut, the control system (15) has a unique solution (xt,yt,zt,rt)∈M2(0,T).
Existence. For any admissible control vt, assume the corresponding trajectory is (xtv,ytv,ztv,rtv)∈M2(0,T); then(16)Jvt-Jut=12E∫0TRω,txtv-xt,xtv-xt+2Rω,txt,xtv-xt+Nω,tvt-ut,vt-ut+2Nω,tut,vt-ut+Lω,tytv-yt,ytv-yt+2Lω,tyt,ytv-ytdt+QωxTv-xT,xTv-xT+2QωxT,xTv-xT.Applying Ito’s formula to 〈xtv-xt,yt〉 we have (17)ExTv-xT,yT=E∫0T-Rω,txt,xtv-xt+Bτω,tyt,vt-ut-Lω,tyt,ytv-yt+Dτω,tzt,vt-ut+∑i=1∞Fi,τω,trti,vt-utdt.Since R(ω,t),L(ω,t), and Q(ω) are nonnegative and N(ω,t) is positive, we can get (18)Jvt-Jut≥E∫0TBτω,tyt+Dτω,tzt+∑i=1∞Fi,τω,trti,vt-ut+Nω,tut,vt-ut=E∫0TBτω,tyt+Dτω,tzt+∑i=1∞Fi,τω,trti,vt-ut-Nω,tN-1ω,tBτω,tyt+Dτω,tzt+∑i=1∞Fi,τω,trti,vt-ut=0.
Then the admissible controlut defined by (15) is the optimal control of LQ stochastic control problem (6).
Unique. Assume admissible control ut1 is an optimal control; the corresponding trajectories are (xt1,yt1,zt1,rt1) and ut2 is another optimal control; the corresponding trajectories are (xt2,yt2,zt2,rt2). So the trajectories corresponding to (ut1+ut2)/2 are(19)xt1+xt22,yt1+yt22,zt1+zt22,rt1+rt22,and the trajectories corresponding to ut1-ut2/2 are(20)xt1-xt22,yt1-yt22,zt1-zt22,rt1-rt22.
Sinceut1 and ut2 are both optimal controls, N(ω,t) is positive, and R(ω,t),L(ω,t), Q(ω) are nonnegative, we have(21)Jut1=Jut2=α≥0,2α=Jut1+Jut2=2Jut1+ut22+E∫0TRω,txt1-xt22,xt1-xt22+Nω,tut1-ut22,ut1-ut22+Lω,tyt1-yt22,yt1-yt22dt+EQωxT1-xT22,xT1-xT22≥2Jut1+ut22+E∫0TNω,tut1-ut22,ut1-ut22dt≥2α+δ2E∫0Tut1-ut22dt.Here δ is a constant and δ>0; then(22)E∫0Tut1-ut22dt≤0,hence, ut1=ut2 in M2(0,T;Rk).AssumeA(ω,t),B(ω,t),C(ω,t),D(ω,t),E(ω,t),F(ω,t), R(ω,t),N(ω,t),L(ω,t), and Q(ω) are all deterministic matrices, denoted as At,Bt,Ct,Dt,Et,Ft,Rt,Nt,Lt, and Q for convenience. Introducing the following generalized n×n matrix-valued Riccati equation (23), t∈0,T,i=1,2,….(23)-K˙t=AtτKt+KtAt+CtτMt+∑i=1∞EtiτYti-KtLtτ+BtNt-1BtτKt-KtBtNt-1DtτMt-∑i=1∞KtBtNt-1FtiτYti+RtMt=KtCt-KtDtNt-1BtτKt-KtDtNt-1DtMt-∑i=1∞KtDtNt-1FtiτYtiYti=KtEti-KtFtiNt-1BtτKt-KtFtiNt-1DtτMt-∑i=1∞KtFtiNt-1FtiτYtiKT=Q;Then we can get the following conclusions.Theorem 6.
Suppose the generalized matrix-valued Riccati equation (23) has solution (Kt,Mt,Yti) for all t∈[0,T]; then the optimal linear feedback regulator for LQ stochastic optimal control problem (6) is(24)ut=-Nt-1BtτKt+DtτMt+∑i=1∞FtiτYtixt, and the optimal value function is(25)Jut=12K0a,a.Proof.
If(Kt,Mt,Yti) is the solution of the matrix-valued Riccati equation (23), then we can check that the solution of (6) (xt,yt,zt,rti) satisfies(26)yt=Ktxt,zt=Mtxt,rti=Ytixt.As we have proved that the optimal control has the form of (15), take (26) into (15); then the optimal control can be written by(27)ut=-Nt-1BtτKt+DtτMt+∑i=1∞FtiτYtixt.
For the optimal value function, using Ito’s formula to〈xt,yt〉, then (28)E∫0TRtxt,xtdt+E∫0TLtyt,ytdt+QxT,xT-Ka,a=E∫0Tyt,Btutdt+E∫0Tzt,Dtutdt+E∫0T∑i=1∞rti,Ftiutdt.On the other hand, from the relationship of u and (xt,yt,zt,rti), we can verify that (29)E∫0Tyt,Btutdt+E∫0Tzt,Dtutdt+E∫0T∑i=1∞rti,Ftiutdt=-E∫0TNtut,utdtand then (30)E∫0TRtxt,xtdt+E∫0TLtyt,ytdt+E∫0TNtut,utdt+QxT,xT=Ka,a.By the definition of cost function J(·) (5), we prove that the optimal value function is (31)Jut=12K0a,a.Now consider a special case of stochastic LQ control problem whenL(ω,t)=0, and the control system is reduced to(32)dxt=Aω,txt+Bω,tutdt+Cω,txt+Dω,tutdBt+∑i=1∞Eiω,txt-+Fiω,tutdHtix0=a.The cost functional now is(33)Ju=12E∫0TRω,txt,xt+Nω,tut,ut+QωxT,xT.Remark 7.
Comparing the LQ stochastic optimal control system (32) and control system (1) which was considered in [22] by Tang and Wu, we know that control system (1) is a special case of control system (32) when C(ω,t)=D(ω,t)=0.We can get the following Corollary8 easily from Theorem 5.Corollary 8.
There exists a unique optimal control for LQ stochastic optimal control problem (32)-(33), and(34)ut=-N-1ω,tBτω,tyt+Dτω,tzt+∑i=1∞Fi,τω,trti,where the (yt,zt,rt) is the solution of the following BSDE driven by Lévy process.(35)-dyt=Aτω,tyt+Cτω,tzt+∑i=1∞Ei,τω,trti+Rω,txtdt-ztdBt-∑i=1∞rtidHtiyT=QωxT.AssumeA(ω,t),B(ω,t),C(ω,t),D(ω,t),E(ω,t),F(ω,t), R(ω,t), N(ω,t), and Q(ω) are all deterministic; then Riccati equation (23) changes to(36)-K˙t=AtτKt+KtAt+CtτMt+∑i=1∞EtiτYti-KtBtNt-1BtτKt-KtBtNt-1DtτMt-∑i=1∞KtBtNt-1FtiτYti+RtMt=KtCt-KtDtNt-1BtτKt-KtDtNt-1DtMt-∑i=1∞KtDtNt-1FtiτYtiYti=KtEti-KtFtiNt-1BtτKt-KtFtiNt-1DtτMt-∑i=1∞KtFtiNt-1FtiτYtiKT=Q.Then from Theorem 6 we can get Corollary 9.Corollary 9.
For LQ stochastic optimal control problem (32)-(33), if, for all t∈[0,T], there exist matrices (Kt,Mt,Yti) satisfying (36), then the optimal linear feedback regulator is(37)ut=-Nt-1BtτKt+DtτMt+∑i=1∞FtiτYtixt,and the optimal value function is(38)Jut=12K0a,a.


## 4. Solvability of the Generalized Riccati Equation

From the discussion of the previous section, we can see that the key to get the optimal linear feedback regulator for LQ stochastic optimal control problem is the solvability of the generalized Riccati equation (23). But (23) is so complicated that we cannot prove its existence and uniqueness at this moment. Using technique introduced by Shi and Wu [12], we only discuss a special case: Dt=0; in this case Riccati equation (23) becomes(39)-K˙t=AtτKt+KtAt+CtτMt+∑i=1∞EtiτYti-KtLtτ+BtNt-1BtτKt-∑i=1∞KtBtNt-1FtiτYti+RtYti=KtEti-KtFtiNt-1BtτKt-∑i=1∞KtFtiNt-1FtiτYti,Mt=KtCt,KT=Q,i=1,2,3,….Equivalently, consider the following equation:(40)-K˙t=AtτKt+KtAt+CtτKtCt-KtLtτ+BtNt-1BtτKt+Rt+∑i=1∞EtiτIn+KtFtiNt-1Ftiτ-1KtEti-KtFtiNt-1BtτKt-∑i=1∞KtBtNt-1FtiτIn+∑i=1∞KtFtiNt-1Ftiτ-1KtEti+∑i=1∞KtBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1KtFtiNt-1BtτKtKT=Q,In+∑i=1∞KtFtiNt-1Ftiτ>0,i=1,2,3,….Compare (39) and (40); we can find that if we can prove Kt the solution of (40), then (41)Mt=KtCt,Yti=In+KtFtiNt-1Ftiτ-1KtEti-KtFtiNt-1BtτKtis the solution of the Riccati equation (39).In the following, we will focus on the existence and uniqueness of solutions of (40). Firstly, let S+n denote the space of all n×n nonnegative symmetric matrices, and C([0,T];S+n) is a Banach space of S+n-valued continuous functions on [0,T]. We have the following uniqueness result.Theorem 10.
The Riccati equation (40) admits at most one solution Kt∈C[0,T;S+n].Proof.
SupposeK~t∈C[0,T;S+n] satisfying In+∑i=1∞K~tFtiNt-1(Fti)τ>0 is another solution of (40). Let K^t=Kt-K~t; then(42)-K^˙t=AtτK^t+K^tAt+CtτK^tCt-K^tLtτ+BtNt-1BtτKt-K~tLtτ+BtNt-1BtτK^t+I1+I2+I3+I4K^T=0,In+∑i=1∞K^tFtiNt-1Ftiτ>0,i=1,2,3,…,where(43)I1=∑i=1∞EtiτIn+KtFtiNt-1Ftiτ-1K^tEti-∑i=1∞EtiτIn+KtFtiNt-1Ftiτ-1K^tFtiNt-1FtiτIn+K~tFtiNt-1Ftiτ-1K~tEtiI2=∑i=1∞K^tBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1KtFtiNt-1BtτKt+∑i=1∞K~tBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1K^tFtiNt-1BtτKt+∑i=1∞K~tBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1K~tFtiNt-1BtτK^t-∑i=1∞K~tBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1K^tFtiNt-1FtiτIn+K~tFtiNt-1Ftiτ-1K~tFtiNt-1BtτK~tI3=-∑i=1∞EtiτIn+KtFtiNt-1Ftiτ-1K^tFtiNt-1BtτKt-∑i=1∞EtiτIn+KtFtiNt-1Ftiτ-1K~tFtiNt-1BtτK^t-∑i=1∞K^tBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1KtEti-∑i=1∞K~tBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1K^tEti+∑i=1∞EtiτIn+KtFtiNt-1Ftiτ-1K^tFtiNt-1FtiτIn+K~tFtiNt-1Ftiτ-1K~tFtiNt-1BtτK~t+∑i=1∞K~tBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1K^tFtiNt-1FtiτIn+K~tFtiNt-1Ftiτ-1K~tEtiI4=-∑i=1∞K^tBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1KtEti-∑i=1∞K~tBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1K^tEti+∑i=1∞K~tBtNt-1FtiτIn+KtFtiNt-1Ftiτ-1K^tFtiNt-1FtiτIn+K~tFtiNt-1Ftiτ-1K~tEti.
[
I
n
+
K
t
F
t
i
N
t
-
1
(
F
t
i
)
τ
]
-
1 and In+K~tFtiNt-1Ftiτ-1 are uniformly bounded as they are continuously in [0,T]; apply Gronwall’s inequality; we can get, for all t∈[0,T], K^t=0. Then we prove the uniqueness of solution.For the existence part, first of all, if we let(44)Φt=ΛK=In+∑i=1∞KFiN-1Fiτ-1K,then from the conventional Riccati equation theory, for all Φt∈C([0,T];S+n), the following conventional Riccati equation(45)-K˙t=At-∑i=1∞BtNt-1FtiτΦtEtiτKt+KtAt-∑i=1∞BtNt-1FtiτΦtEti-KtLtτ+BtNt-1Btτ-∑i=1∞BtNt-1FtiτΦtFtiNt-1BtτKt+CtτKtCt+∑i=1∞EtiτΦtEti+RtKT=Q,In+∑i=1∞KtFtiNt-1Ftiτ>0,i=1,2,3,….has a unique solution K(·)∈C([0,T];S+n), when(46)Ltτ+BtNt-1Btτ-∑i=1∞BtNt-1FtiτΦFtiNt-1Bt∈C0,T;S+n.LetSsn be the subspace of S+n which is formed by the symmetric matrices satisfying (46). Obviously, as Kt≡0∈Ssn the definition of Ssn is reasonable. Define a mapping Ψ:C([0,T];Ssn)→C([0,T];S+n); we can get Lemma 11 about Φ and Ψ.Lemma 11.
The operatorsΦ=Λ(K) are monotonously increasing when K>0, and the operator Ψ is continuous and monotonously increasing.Proof.
WhenK>0, from the definition of Λ(K), we have(47)ΛK=In+∑i=1∞KFiN-1Fiτ-1K=K-1In+∑i=1∞KFiN-1Fiτ-1=K-1+∑i=1∞FiN-1Fiτ-1.So if K1≥K2, then Λ(K1)≥Λ(K2); that is, Λ(K) is monotonously increasing when K>0.
AsK=Ψ(Φ), set K¯=Ψ(Φ¯); then the conventional Riccati equation (45) can be rewritten(48)-K˙t=AtτKt+KtAt-KtLtτ+BtNt-1BtτKt+Rt+CtτKtCt+∑i=1∞Eti-FtiNt-1BtτKtτΦtEti-FtiNt-1BtτKtKT=Q,i=1,2,3,….
From the conclusion ofΛ(K) above in this lemma and Lemma 8.2 in [28], if Φ≥Φ¯ then K≥K¯; the operator Ψ is monotonously increasing. On the other hand, by Gronwall’s inequality, we know that if Φ→Φ¯, then K-K¯→0, so the operator Ψ is also continuous.For (45), it is easy to know that if there exists Φ(·)∈C([0,T];S+n) satisfying(49)Φ=In+∑i=1∞ΨΦFiN-1Fiτ-1ΨΦ,then Riccati equation (40) admits a unique solution. So the following task is to find the suitable Φ(·)∈C([0,T];S+n) satisfying (49). We need the following lemma.Lemma 12.
If there existΦ+,Φ-∈C([0,T];Ssn) which satisfy(50)Φ+≥In+∑i=1∞ΨΦ+FiN-1Fiτ-1ΨΦ+≥In+∑i=1∞ΨΦ-FiN-1Fiτ-1ΨΦ-≥Φ-,then Riccati equation (40) admits a solution K(·)∈C([0,T];S+n).Proof.
For givenΦ+,Φ- which satisfied (49), define the sequences Φj+,Φj-,Kj+,Kj- as follows: (51)Φ0+=Φ+∈Ssn,Φ0-=Φ-∈Ssn,K0+=ΨΦ0+,K0-=ΨΦ0-,Φj+1+=In+∑i=1∞KFiN-1Fiτ-1Kj+,Φj+1-=In+∑i=1∞KFiN-1Fiτ-1Kj-,Kj+1+=ΨΦj+1+,Kj+1-=ΨΦj+1-,j=1,2,3,….From (50) and Lemma 11, by induction, we obtain (52)K0+≥Kj+≥Kj+1+≥Kj+1-≥Kj-≥K0-≥0,Φ0+≥Φj+≥Φj+1+≥Φj+1-≥Φj-≥Φ0-≥0,and Φj+,Φj-∈Ssn; we have (53)limj→∞⁡Φj+=Φ+∈Ssn,limj→∞⁡Kj+=K+∈S+n,K+=limj→∞⁡Kj+=limj→∞⁡ΨΦj+=Ψlimj→∞⁡Φj+=ΨΦj+.So K+ is a solution of (45) corresponding to Φ=Φ+; then (54)Φ+=In+∑i=1∞KFiN-1Fiτ-1K+,where K+ is a solution of Riccati equation (40). By the same step, we can get (55)limj→∞⁡Φj-=Φ-∈Ssn,limj→∞⁡Kj-=K-∈S+n.and K- is also a solution of Riccati equation (40). From Theorem 10, K+=K-.From Lemma12, in order to get the existence of solution for Riccati equation (40), we only need to find Φ+ and Φ- satisfying (50). Obviously, we can let Φ-=0 and for the existence of Φ+, we need the following Assumption 13.Assumption 13.
There existsΦ¯(·)∈Ssn, such that(56)∑i=1∞FiτtΦ¯tFit=Nt,∑i=1∞In+K¯tFitN-1Fiτt-1K¯t≤Φ¯t,where K¯(t) is the unique solution of the following equation:(57)-K¯˙t=At-∑i=1∞BtNt-1FtiτΦ¯tEtiτK¯t+K¯tAt-∑i=1∞BtNt-1FtiτΦ¯tEti-K¯tLtτK¯t+CtτK¯tCt+∑i=1∞EtiτΦ¯tEti+RtK¯T=Q.It is easy to known that when matrixFt is invertible and k=n, Assumption 13 is satisfied. Then we get the main result of this section.Theorem 14.
LetD=0 and Assumption 13 holds; the Riccati equation (40) has a unique solution (K,M,Y)∈C1([0,T];S+n)×L∞([0,T];Rn×n)×L∞([0,T];Rn×n).At last, we give a simple example of the Riccati equation which has a unique solution.Example 15.
Assume the dimensions of the state and control are the same; that is,k=n, when D=0,Fi=In,(i=1,2,3,…) in Riccati equation (39); let Φ¯(·)=N(·); then check Assumption 13.In fact,Φ¯(·)=N(·)≥0, so ∑i=1∞[Φ¯tFti+K¯tFtiNt-1(Fti)τΦ¯tFti]≥∑i=1∞K¯tFti; here K¯t is the solution of(58)-K¯˙t=At-∑i=1∞BtEtiτK¯t+K¯tAt-∑i=1∞BtEti-K¯tLtτK¯t+CtτK¯tCt+∑i=1∞EtiτNtEti+RtK¯T=Q.Φ¯t+∑i=1∞K¯tFtiNt-1(Fti)τΦ¯t≥K¯t, and then [In+∑i=1∞K¯tFtiNt-1(Fti)τ]-1K¯t≤Φ¯t. From Theorem 14, the Riccati equation (39) has a unique solution when k=n, D=0,Fi=In,(i=1,2,3,…).


## 5. Conclusion

In this paper, we discussed one kind of LQ stochastic control problem with Lévy process as noise source where the control system is described by a linear FBSDEL. Explicit form of optimal control is obtained, and it can be proved to be unique. When assuming that all the coefficient matrices in this control problem are deterministic, it has been shown that the linear feedback regulator for this LQ problem has a close relation to the solutions of a kind of generalized Riccati equation. Finally, we discuss the solvability of the generalized Riccati equation and prove the existence and uniqueness of the solution for it in a special case.



---

*Source: 2541687-2017-09-25.xml*