ISRN.APPLIED.MATHEMATICS ISRN Applied Mathematics 2090-5572 Hindawi Publishing Corporation 903912 10.1155/2014/903912 903912 Research Article On Optimal Control Problem for Backward Stochastic Doubly Systems Chala Adel Jauberteau F. Sartoretto F. Laboratory of Applied Mathematics University Mouhamed Khider P.O. Box 145, 07000 Biskra Algeria univ-biskra.dz 2014 432014 2014 15 12 2013 16 01 2014 4 3 2014 2014 Copyright © 2014 Adel Chala. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We are going to study an approach of optimal control problems where the state equation is a backward doubly stochastic differential equation, and the set of strict (classical) controls need not be convex and the diffusion coefficient and the generator coefficient depend on the terms control. The main result is necessary conditions as well as a sufficient condition for optimality in the form of a relaxed maximum principle.

1. Introduction

In 1994, Pardoux and Peng  considered a new kind of BSDE, that is a class of backward doubly stochastic differential equations (BDSDEs in short) with two different directions of stochastic integrals, that is, the equations involve both a standard (forward) stochastic Itô integral dWt and a backward stochastic Itô integral dBt. More precisely, they dealt with the following BDSDE: (1)dYt=f(t,Yt,Zt)dt+g(t,Yt,Zt)dBt-ZtdWt,YT=ξ.

They proved that if f and g are uniform Lipschitz, then (1), for any square integrable terminal value ξ, has a unique solution (Yt,Zt) in the interval [0,T]. They also showed that BDSDEs can produce a probabilistic representation for solutions to some quasi-linear stochastic partial differential equations. Since this first existence and uniqueness result, many papers have been devoted to existence and/or uniqueness result under weaker assumptions. Among these papers, we can distinguish two different classes: Scalar BDSDEs and multidimensional BDSDEs. In the first case, one can take advantage of the comparison theorem: we refer to Shi et al.  who weakened the uniform Lipschitz assumptions to linear growth and continuous conditions by virtue of a comparison theorem introduced by them. They obtained the existence of solutions to BDSDEs but without uniqueness. In this spirit, let us mention the contributions of N’zi and Owo , which dealt with discontinuous coefficients. For multidimensional BDSDE, there is no comparison theorem, and to overcome this difficulty, a monotonicity assumption on the generator f in the variable y is used. This appears in the works of Peng and Shi  which have introduced a class of forward backward doubly stochastic differential equations, under the Lipschitz condition and monotonicity assumption. Unfortunately, the uniform Lipschitz condition cannot be satisfied in many applications. More recently, N’zi and Owo  established existence and uniqueness result under non-Lipschitz assumptions. Moreover the authors apply his theory to solve the financial model of cash flow valuation.

In this paper, we study a stochastic control problem where the system is governed by a nonlinear backward doubly stochastic differential equation (BDSDE) of the type (2)dytν=-f(t,ytν,ztν,νt)dt+g(t,ytν,ztν,νt)dBt+ztνdWt,yTν=ξ.

The control variable ν=(νt), called strict control, is an t-adapted process with values in some sets U of k. We denote by 𝒰 the class of all strict controls.

The criteria to be minimized, over the set 𝒰, has the following form: (3)J(ν)=𝔼[Ψ(y0ν)+0Tl(t,ytν,ztv,νt)dt], where Ψ and l are given maps and (yt,zt) is the trajectory of the system controlled by ν.

A control u𝒰 is called optimal if it satisfies (4)J(u)=infν𝒰J(ν).

Our objective in this paper is to establish necessary as well sufficient optimality conditions, of the Pontryagin maximum principle type, for relaxed models.

In this paper, we solve this problem by using an approach developed by Bahlali , and it was developed by Chala [7, 8]. We introduce then a bigger new class of processes by replacing the U-valued process (vt) by a P(U)-valued process (qt), where P(U) is the space of probability measures on U equipped with the topology of stable convergence. This new class of processes is called relaxed controls, and it has a richer structure of convexity, for which the control problem becomes solvable.

The main idea is to use the property of convexity of the set of relaxed controls and treat the problem with the method of convex perturbation on relaxed controls (instead of that of the spike variation on strict one). We establish then necessary and sufficient optimality conditions for relaxed controls.

In the relaxed model, the system is governed by the BDSDE as follows: (5)dytq=-Uf(t,ytq,ztq,a)qt(da)dt+Ug(t,ytq,ztq,a)qt(da)dBt+ztqdWt,yTq=ξ.

The expected cost to be minimized, the relaxed model, is defined from into by (6)𝒥(q)=𝔼[Ψ(y0q)+0TUl(t,ytq,ztq,a)qt(da)dt].

A relaxed control μ is called optimal if it solves (7)𝒥(μ)=infq𝒥(q).

Existence of an optimal solution for this problem has been solved to achieve the objective of this paper and establish necessary and sufficient optimality conditions for these two models, we proceed as follows.

Firstly, we give the optimality conditions for relaxed controls. The idea is to use the fact that the set of relaxed controls is convex. Then, we establish necessary optimality conditions by using the classical way of the convex perturbation method. More precisely, if we denote by μ an optimal relaxed control, and q is an arbitrary element of , then with a sufficiently small θ>0 and for each t[0,T], we can define a perturbed control as follows μtθ=μt+θ(qt-μt).

By using the fact that the coefficients f, g, and l are linear with respect to the relaxed control variable, necessary optimality conditions are obtained directly in the global form.

We note that necessary optimality conditions for relaxed controls, where the systems are governed by a stochastic differential equation, were studied by Mezerdi and Bahlali  and Bahlali , and we note also that necessary optimality conditions for Stochastic controls, where the systems are governed by forward-backward doubly stochastic differential equation, were studied by Bahlali and Gherbal  and Han, Peng, and Wu .

The paper is organized as follows. In Section 2, we give the precise formulation of the problem and introduce the relaxed model. We formulate the problem and give the various assumptions used throughout the paper. In Section 3, we give our first main result, the necessary optimality conditions for control problem and under additional hypothesis. In Section 4, we derive our second main result in this paper, which is sufficient conditions of optimality for relaxed controls.

Along this paper, we denote by C some positive constants and for simplicity, we need the following matrix notations. We denote by n×d() the space of n×d real matrix and n×n() the linear space of vectors M=(M1,...,Md) where Min×n(). For any M,Nn×nd(), L,Sn×d(), Qn×d(), α,βn, and γd, we use the following notations:

αβ=i=1nαiβi is the product scalar in n;  LS=i=1dLiSi, where Li and Si are the ith column of L and S;  ML=i=1dMiLin;  M(αγ)=i=1d(Miα)γin;  MN=i=1d(MiNi)n×n();  MQN=i=1dMiQNin×n();  MQγ=i=1dMiQγγin×n().

We denote by L* the transpose of the matrix L and M*=(M1*,...,Md*).

2. Formulation of the Problem

Let (Ω,,(t(B,W))t0,) be a probability space, on which a d-dimensional Brownian motions W=(Wt:0tT) and B=(Bt:0tT) are defined. For each t[0,T], we define t(B,W)t,TBtW, where for any process {pt}, s,tp=σ{p(r)-p(s);srt}𝒩, tp=0,tp. Note that the collection {t,t[0,T]} is neither increasing nor decreasing, and it does not constitute a classical filtration, where 𝒩 denotes the total of σ-null sets and σ1σ2 denotes the σ-fields generated by σ1σ2.

Note that the collection {(t(B,W))t,t[0,T]} is neither increasing nor decreasing, and it does not constitute a filtration.

Let T be a positive real number and U a nonempty subset of k.

2.1. The Strict Control Problem

For any n, let 2(0,T;n) denote the set of n-dimensional jointly measurable random processes {φt,t[0,T]} which satisfy (8)(i):𝔼[0T|φt|2dt]<,(ii):φt  is  (t(B,W))measurable,for  a.e.t[0,T].

We denote similarly by 𝒮2([0,T];n) the set of continuous n-dimensional random processes which satisfy (9)(i):𝔼[sup0tT|φt|2]<,(ii):φt  is  (t(B,W))measurable,foranyt[0,T].

Let T be a strictly positive real number and U is a nonempty subset of k.

Definition 1.

( y t , z t ) is said to be a solution of (1), if and only if, (yt,zt)2(n×d)×𝒮2(n×m) and it satisfies (1).

Definition 2.

An admissible strict control is an t-adapted process ν=(νt) with values in U such that 𝔼[supt[0,T]|νt|2]<.

We denote by 𝒰 the set of all admissible controls.

For any ν𝒰, we consider the following BDSDE: (10)dytν=-f(t,ytν,ztν,νt)dt+g(t,ytν,ztν,νt)dBt+ztνdWt,yTν=ξ, where f:[0,T]×m×m×d()×Um,  g:[0,T]×m×m×d()×Um×k, and ξ is an n-dimensional 0-measurable random variable such that 𝔼|ξ|2<.

The expected cost is defined from 𝒰 into by (11)J(ν)=𝔼[Ψ(y0ν)+0Tl(t,ytν,ztν,νt)dt], where Ψ:m,  l:[0,T]×m×m×d()×U.

The control problem is to minimize the functional J over 𝒰, if u𝒰 is an optimal solution; that is, (12)J(u)=infν𝒰J(ν).

A control that solves this problem is called optimal. Our goal is to establish a necessary condition of optimality for controls in the form of stochastic maximum principle.

The following assumptions will be in force throughout this paper: (13)f,g,Ψ,andlarecontinuouslydifferentiablewithrespectto(y,z).

They and all their derivatives with respect to (y,z) are continuous in (y,z,ν) and uniformly bounded by C>0;  Ψ,  l and are bounded by C>0.

We assume moreover that there exist constants C>0 and 0ζ<1 such that for any (w,t)Ω×[0,T],  (Y1,Z1);  (Y2,Z2)m×m×d(), (Hypo02)(i):|f(t,Y1,Z1)-f(t,Y2,Z2)|2C(|Y1-Y2|2+Z1-Z22),(ii):g(t,Y1,Z1)-g(t,Y2,Z2)2C|Y1-Y2|2+ζ(Z1-Z22).

Under the above assumptions, for every ν𝒰, (10) has a unique strong solution and the functional cost J is well defined from 𝒰 into .

2.2. The Relaxed Model

The idea for the relaxed strict control problem defined above is to embed the set U of strict controls into a wider class which gives a more suitable topological structure. In the relaxed model, the U-valued process ν is replaced by a (U)-valued process q, where (U) denotes the space of probability measure on U equipped with the topology of stable convergence.

Definition 3.

A relaxed control (qt)t is a (U)-valued process, progressively measurable with respect to (t)t and such that for each t,  1]0,t[qt is t-measurable.

We denote by the set of all admissible relaxed controls.

Remark 4.

Every relaxed control q may be disintegrated as q(dt,da)=q(t,da)dt=qt(da)dt, where qt(da) is a progressively measurable process with value in the set of probability measures (U). The set U is embedded into the set of relaxed process by the mapping f:ν𝒰fυ(dt,da)=δυt(da)dt, where δυ is the atomic measure concentrated at a single point υ.

For any q, we consider the following relaxed BDSDE: (14)dytq=-Uf(t,ytq,ztq,a)qt(da)dt+Ug(t,ytq,ztq,a)qt(da)dBt+ztqdWt,yTq=ξ.

The expected cost to be minimized, the relaxed model, is defined from into by (15)𝒥(q)=𝔼[Ψ(y0q)+0TUl(t,ytq,ztq,a)qt(da)dt].

A relaxed control ρ is called optimal if it solves (16)𝒥(μ)=infq𝒥(q).

Existence of an optimal solution for the problem {(14), (15), (16)} has been solved.

3. Optimality Conditions for Relaxed Controls

In this section, we study the problem {(14), (15), (16)} and we establish necessary condition of optimality for relaxed controls.

3.1. Preliminary Results

Since the set of relaxed controls is convex, then the classical way of treating such a problem is to use the convex perturbation method. More precisely, let μ be an optimal relaxed control and (ytμ,ztμ) the solution of (14) controlled by μ. Then, we can define a perturbation relaxed as follows: (17)μtθ=μt+θ(qt-μt), where θ>0 is sufficiently small and q is an arbitrary element of .

Denote by (ytθ,ztθ) the solution of (14) associated with μθ.

From the optimality of μ, the variational inequality will be derived from the fact that (18)0𝒥(μθ)-𝒥(μ).

To this end, we need the following classical Lemmas.

Lemma 5.

Under the assumption (13) and (Hypo02), we have (19)limθ0[sup0tT𝔼|ytθ-ytμ|2]=0,(20)limθ0𝔼[0T|ztθ-ztμ|2dt]=0.

Proof.

Let us prove (19) and (20).

Applying Itô’s formula to (ytθ-ytμ)2, and since (ytθ-ytμ)(zsθ-zsμ)dWs is martingale; then taking the expectation for any sT, we have (21)𝔼|ytθ-ytμ|2+𝔼tT|zsθ-zsμ|2ds=2𝔼tT|(ysθ-ysμ)[Uf(s,ysθ,zsθ,a)μsθ(da)-Uf(s,ysμ,zsμ,a)μs(da)]|ds+𝔼tT|[Ug(s,ysθ,zsθ,a)μsθ(da)-Ug(s,ysμ,zsμ,a)μs(da)]|ds.

From the Young formula, for every ε>0, ab(1/(2(1+α)))a2+((1+α)/2)b2, and by using the definition of μsθ, we have (22)𝔼|ytθ-ytμ|2+𝔼tT|zsθ-zsμ|2ds(1+α)𝔼tT|(ysθ-ysμ)|2ds+θ2C(1+α)𝔼tT|Uf(s,ysθ,zsθ,a)qs(da)-Uf(s,ysθ,zsθ,a)μs(da)|2ds+θ2C𝔼tT|Ug(s,ysθ,zsθ,a)qs(da)-Ug(s,ysθ,zsθ,a)μs(da)|2ds+C(1+α)𝔼tT|Uf(s,ysθ,zsθ,a)μs(da)-Uf(s,ysμ,zsθ,a)μs(da)|2ds+C(1+α)𝔼tT|Uf(s,ysμ,zsθ,a)μs(da)-Uf(s,ysμ,zsμ,a)μs(da)|2ds+C𝔼tT|Ug(s,ysθ,zsθ,a)μs(da)-Ug(s,ysμ,zsθ,a)μs(da)|2ds+C𝔼tT|Ug(s,ysμ,zsμ,a)μs(da)-Ug(s,ysμ,zsμ,a)μs(da)|2ds.

Since g and f are uniformly Lipschitz with respect to y and z and from (Hypo02), where 0ζ<1, then (23)𝔼|ytθ-ytμ|2+(1-C(1+α)-ζ)𝔼tT|zsθ-zsμ|2ds(C(1+α)+(1+α)+C)𝔼tT|(ysθ-ysμ)|2ds+Mθ, where Mθ is given by Mθ=C((1+α)-1+1)θ2, we have (24)Mθ=o(θ).

Let (25)α={0,C1-ζ2,2c+ζ-11-ζ,C>1-ζ2.

Then, 1-C/(1+α)-ζ(1-ζ)/2>0.

Choose c1=1-C/(1+α)-ζ,  c2=C/(1+α)+(1+α)+C, then we have (26)𝔼|ytθ-ytμ|2+c1𝔼tT|zsθ-zsμ|2dsc2𝔼tT|(ysθ-ysμ)|2ds+Mθ.

From the above inequality, we derive the following two inequalities: (27)𝔼|ytθ-ytμ|c2𝔼tT|(ysθ-ysμ)|2ds+Mθ,(28)𝔼tT|zsθ-zsμ|2dsc2c1𝔼tT|(ysθ-ysμ)|2ds+1c1Mθ.

By using (24), Gronwall’s lemma, and Burkholder-Davis-Gundy inequality in (27), we obtain (19). Finally, (20) is derived from (27), (28), and (24).

Lemma 6.

Let y~t be the solution of the following linear equation (called variational equation): (29)dy~t=-U[fy(t,ytμ,ztμ,a)·y~t]μ(da)dt-U[fz(t,ytμ,ztμ,a)·z~t]μ(da)dt+[Uf(t,ytμ,ztμ,a)μs(da)-Uf(t,ytμ,ztμ,a)qt(da)]dt+U[gy(t,xtμ,ytμ,ztμ,a)·y~t]μ(da)dt+U[gz(t,ytμ,ztμ,a)·z~t]μ(da)dt+[Ug(t,ytμ,ztμ,a)qs(da)-Ug(t,ytμ,ztμ,a)μt(da)]dt+z~tdWt,y~0=0.

Then, we have (30)limθ0[sup0tT𝔼|ytθ-ytμθ-y~t|2]=0,(31)limθ0𝔼[0T|ztθ-ztμθ-z~t|2dt]=0.

Proof.

For simplicity, we put Ytθ=(ytθ-ytμ)/θ-y~t,  Ztθ=(ztθ-ztμ)/θ-z~t.

(i) Proof of (30): applying Itô’s formula to (ytθ-ytμ)2, and from the Young formula, for every ε>0, we have (32)𝔼|Ytθ|2+𝔼tT|Zsθ|2ds1ε𝔼tT|Ysθ|2ds+ε𝔼tT|[Uf(s,ysθ,zsθ,a)μsθ(da)-Uf(s,ysμ,zsμ,a)μs(da)+U[fy(s,ysμ,zsμ,a)·y~s+fz(s,ysμ,zsμ,a)·z~sfy(s,ysμ,zsμ,a)]μ(da)-(Uf(s,ysμ,zsμ,a)μs(da)-Uf(s,ysμ,zsμ,a)qs(da))]|2ds+ε𝔼tT|[Ug(s,ysθ,zsθ,a)μsθ(da)-Ug(s,ysμ,zsμ,a)μs(da)+U[gy(s,ysμ,zsμ,a)·y~s+gz(s,ysμ,zsμ,a)·z~sgy(s,ysμ,zsμ,a)]μ(da)-(Ug(s,ysμ,zsμ,a)μs(da)-Ug(s,ysμ,zsμ,a)qs(da))]|2ds.

For simplicity, we put Λtθ(a)=(t,ytμ+αθ(Ytθ+y~t),ztμ+αθ(Ztθ+z~t),a), we have the following inequality: (33)𝔼|Ytθ|2+𝔼tT|Zsθ|2ds1ε𝔼tT|Ysθ|2ds+ε𝔼tT|FsyYsθ+FszZsθ-ηsθ|2ds+ε𝔼tT|GsyYsθ+GszZsθ-ηsθ|2ds, where (34)Fty=-01Ufy(Λtθ(a))μt(da)dα,Gty=01Ugy(Λtθ(a))μt(da)dα,Ftz=-01Ufz(Λtθ(a))μt(da)dα,Gtz=01Ugz(Λtθ(a))μt(da)dα, and ηtθ, ηtθ are given by (35)ηtθ=tTU[fy(Λsθ(a))(ysθ-ysμ)+fz(Λsθ(a))(zsθ-zsμ)]qs(da)ds-tTU[fy(Λsθ(a))(ysθ-ysμ)+fz(Λsθ(a))(zsθ-zsμ)]μs(da)ds,ηtθ=tTU[gy(Λsθ(a))(ysθ-ysμ)+gz(Λsθ(a))(zsθ-zsμ)]qs(da)ds-tTU[gy(Λsθ(a))(ysθ-ysμ)+gz(Λsθ(a))(zsθ-zsμ)]μs(da)ds.

Since Fty, Ftz, Gty, and Gtz are bounded, then (36)𝔼|Ytθ|2+𝔼tT|Zsθ|2ds1ε𝔼tT|Ysθ|2ds+εC𝔼tT|Ysθ|2ds+εC𝔼Γ|Zsθ|2ds+Πtθ, where Πtθ=εC𝔼tT|ηsθ|2ds+εC𝔼tT|ηsθ|2ds. Choose ε=1/2C, then we have (37)𝔼|Ytθ|2+12𝔼tT|Zsθ|2ds(2C+12)𝔼tT|Ysθ|2ds+Πtθ.

From the above inequality, we deduce the following two inequalities: (38)𝔼|Ytθ|2(2C+12)𝔼tT|Ysθ|2ds+Πtθ,(39)𝔼tT|Zsθ|2ds2(2C+12)𝔼tT|Ysθ|2ds+2Πtθ.

Since fy and fz and gy and gz are continuous and bounded, then from (19) and (20), we have (40)limθ0(𝔼tT|ηsθ|2ds+𝔼tT|ηsθ|2ds)=0.

From (40), we deduce that (41)limθ0Πtθ=0.

By using (33), (41), Gronwall’s Lemma, and Burkholder-Davis-Gundy inequality, we obtain (30). Finally, (31) is derived from (39), (41), and (30).

Lemma 7.

Let μ be an optimal relaxed control minimizing the cost 𝒥 over and (ytμ,ztμ) the associated optimal trajectory. Then, for any q, we have (42)0𝔼[Ψy(y0μ)·y~0]+𝔼0T[Ul(t,ytμ,ztμ,a)qt(da)-Ul(t,ytμ,ztμ,a)μs(da)]dt+𝔼0T[Uly(t,ytμ,ztμ,a)μs(da)·y~t]dt+𝔼0T[Ulz(t,ytμ,ztμ,a)μt(da)·z~t]dt.

Proof.

Let μ be an optimal relaxed control minimizing the cost 𝒥 over , then from (18), and by using the definition of μtθ, we have (43)0𝔼[Ψ(y0θ)-Ψ(y0μ)]+θ𝔼0T[Ul(t,ytθ,ztθ,a)qt(da)-Ul(t,ytθ,ztθ,a)μt(da)]dt+𝔼0TU[l(t,ytθ,ztθ,a)-l(t,ytμ,ztμ,a)l(t,ytθ,ztθ,a)]μt(da)dt.

Then, (44)0𝔼01[Ψy(y0μ+αθ(y~0+Y0θ))y~0]dα+𝔼0T01U[ly(Λ¯tθ(a))·y~t+lz(Λ¯tθ(a))·z~t]μt(da)dαdt+𝔼0T[Ul(t,ytμ,ztμ,a)qt(da)-Ul(t,ytμ,ztμ,a)μt(da)]dt+βtθ, where βtθ is given by (45)βtθ=𝔼01[Ψy(y0μ+αθ(y~0+Y0θ))·Y0θ]dα+𝔼0T01U[ly(Λ¯tθ(a))·Ytθ+lz(Λ¯tθ(a))·Ztθ]μt(da)dt+𝔼0T01U[ly(Λ¯tθ(a))(ytθ-ytμ)+lz(Λ¯tθ(a))(ztθ-ztμ)]qt(da)dt-𝔼0T01U[ly(Λ¯tθ(a))(ytθ-ytμ)+lz(Λ¯tθ(a))(ztθ-ztμ)]μt(da)dt.

For simplicity, we put Λ¯tθ(a)=(t,ytμ+αθ(Ytθ+y~t),ztμ+αθ(Ztθ+z~t),a). Since the derivatives Ψy, ly, lz are continuous and bounded, then by using (30), (31), (19), and (20) and the Cauchy-Schwartz inequality, we have limθ0βtθ=0. By letting θ go to 0 in (44), the proof is completed.

3.2. Necessary Optimality Conditions for Relaxed Controls

Starting from the variational inequality (42), we can now state necessary optimality conditions for the relaxed control problem {(14), (15), (16)} in the global form.

The Hamiltonian is defined from [0,T]×m×n×m()×(U)×n×d() into by (46)(t,y,z,μ,Q,R)=Ul(t,y,z,a)μt(da)+Q·Uf(t,y,z,a)μt(da)+R·Ug(t,yt,zt,a)μt(da).

Theorem 8 (necessary optimality conditions for relaxed controls).

Let μ be an optimal relaxed control minimizing the functional 𝒥 over and (ytμ,ztμ) the solution of (14) associated with μ. Then, there exists unique adapted process (Qμ)L2([0,T];m) of the following FDSDE system (called adjoint equation): (47)dQt=-[Uly(t,ytμ,ztμ,a)μt(da)+Ufy(t,ytμ,ztμ,a)μt(da)Qtμ-Ugy(t,ytμ,ztμ,a)μt(da)Rtμ]dt-[Ulz(t,ytμ,ztμ,a)μt(da)+Ufz(t,ytμ,ztμ,a)μt(da)Qtμ-Ugz(t,ytμ,ztμ,a)μt(da)Rtμ]dWt-RtμdBt,Q0μ=Ψy(yμ(0)),

such that for every qt(U), (48)(t,ytμ,ztμ,qt,Qtμ,Rtμ)(t,ytμ,ztμ,μt,Qtμ,Rtμ),μt(U),  ae,  as.

Proof.

Since Q0μ=Ψy(y0μ), then (42) becomes (49)0𝔼[Q0μy~0]+𝔼0T[Ulμ(t,ytμ,ztμ,a)qt(da)-Ulμ(t,ytμ,ztμ,a)μs(da)]dt+𝔼0T[Ulyμ(t,ytμ,ztμ,a)μs(da)·y~t]dt+𝔼0T[Ulzμ(t,ytμ,ztμ,a)μt(da)·z~t]dt.

By applying Itô’s formula to (Qtμy~t), we get (50)𝔼[Q0μy~0]=𝔼[QTμy~T]-𝔼[0TUlyμ(t,a)μt(da)·y~t]dt-𝔼[0TUlzμ(t,a)μt(da)·z~t]dt+𝔼0TQtμ[Ufμ(t,a)qt(da)-Ufμ(t,a)μt(da)]dt+𝔼0TRtμ[Ugμ(t,a)qt(da)-Ugμ(t,a)μt(da)]dt.

Then, for every q, (49) becomes (51)0𝔼0T[(t,ytμ,ztμ,qt,Qtμ,Rtμ)-(t,ytμ,ztμ,μt,Qtμ,Rtμ)]dt.

Now, let μ and F be an arbitrary element of the σ-algebra t, and set ϕ=qt1F+μt1Ω-F. It is obvious that ϕ is an admissible relaxed control.

Applying the above inequality with ϕ, we get (52)0𝔼[1F{(t,ytμ,ztμ,qt,Qtμ,Rtμ)-(t,ytμ,ztμ,μt,Qtμ,Rtμ)}],Ft.

Which implies that (53)0𝔼[(t,ytμ,ztμ,qt,Qtμ,Rtμ)-(t,ytμ,ztμ,μt,Qtμ,Rtμ)t].

The quantity inside the conditional expectation is t-measurable, and thus the result follows immediately.

4. Sufficient Optimality Conditions for Relaxed Controls

In this section, we study when necessary optimality conditions (48) become sufficient. For any q, we denote by (yq,zq) the solution of (14) controlled by q.

Theorem 9 (sufficient optimality conditions for relaxed controls).

Assume that the functions Ψ and (y,z)(t,y,z,q,Q,R) are convex, and for any q, yTq=0 is an m-dimensional t-measurable random variable such that 𝔼|ξ|2<. Then, μ is an optimal solution of the relaxed control problem {(14), (15), (16)}, if it satisfies (48).

Proof.

Let μ2 be an arbitrary element of (candidate to be optimal). For any μ1, we have (54)𝒥(μ1)-𝒥(μ2)=𝔼[h(y0μ1)-h(y0μ2)]+𝔼0T[Ul(t,ytμ1,ztμ1,a)μ1t(da)-Ul(t,ytμ2,ztμ2,a)μ2t(da)]dt.

Since Ψ are convex, then (55)Ψ(y0μ1)-Ψ(y0μ2)Ψy(y0μ2)(y0μ1-y0μ2).

Thus, (56)𝒥(μ1)-𝒥(μ2)𝔼[Ψy(y0μ2)(y0μ1-y0μ2)]+𝔼0T[Ulq(t,a)μ1t(da)-Ulμ(t,a)μ2t(da)]dt.

We remark from (47) that Q0μ=Ψy(y0μ). Then, we have (57)𝒥(μ1)-𝒥(μ2)𝔼[Ψy(y0μ2)(y0μ1-y0μ2)]+𝔼0T[Ulq(t,a)μ1t(da)-Ulμ(t,a)μ2t(da)]dt.

Thus, (58)𝒥(μ1)-𝒥(μ2)𝔼[Q0μ2(y0μ1-y0μ2)]+𝔼0T[Ulq(t,a)μ1t(da)-Ulμ(t,a)μ2t(da)]dt.

By applying Itô’s formula to Qtμ(ytq-ytμ), we obtain (59)𝔼[Q0μ2(y0μ1-y0μ2)]=𝔼[QTμ2(yTμ1-yTμ2)]-𝔼0Ty(t,ytμ2,ztμ2,μ,Qtμ2,Rtμ2)×(ytμ1-ytμ2)dt-𝔼0Tz(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)×(ztμ1-ztμ2)dt+𝔼0TQtμ2[Ufq(t,a)μ1t(da)-Ufμ(t,a)μ2t(da)]dt+𝔼0TRtμ2[Ugq(t,a)μ1t(da)-Ugμ(t,a)μ2t(da)]dt.

Then, (60)𝒥(μ1)-𝒥(μ2)𝔼0T[(t,ytμ2,ztμ2,μ1t,Qtμ2,Rtμ2)-(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)]dt-𝔼0Ty(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)×(ytμ1-ytμ2)dt-𝔼0Tz(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)×(ztμ1-ztμ2)dt.

Since is convex in (y,z) and linear in μ, then by using the Clarke generalized gradient of evaluated at (yt,zt,μ) and the necessary optimality conditions, that (61)(t,ytμ1,ztμ1,μ1t,Qtμ1,Rtμ1)-(t,ytμ2,ztμ2,μ2t,ptμ2,Qtμ2,Rtμ2)y(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)(ytμ1-ytμ2)+z(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)(ztμ1-ztμ2) or equivalently (62)0(t,ytμ1,ztμ1,μ1t,Qtμ1,Rtμ1)-(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)-y(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)(ytμ1-ytμ2)-z(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)(ztμ1-ztμ2).

Then, from (60), we get (63)𝒥(μ1)-𝒥(μ2)0.

The theorem is proved.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is partially supported by The Algerian PNR Project no. 8/u07/857.

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