# On Optimal Control Problem for Backward Stochastic Doubly Systems

Authors: Adel Chala
Journal: ISRN Applied Mathematics
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

In 1994, Pardoux and Peng [1] 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)dB←t-ZtdWt,YT=ξ.They proved that iff 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. [2] 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 [3], 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 [4] 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 [5] 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 variableν=(νt), called strict control, is an ℱt-adapted process with values in some sets U of ℝk. We denote by 𝒰 the class of all strict controls.The criteria to be minimized, over the set𝒰, 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 controlu∈𝒰 is called optimal if it satisfies
(4)J(u)=inf⁡ν∈𝒰J(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 [6], and it was developed by Chala [7, 8]. We introduce then a bigger new class ℛ 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 fromℛ into ℝ by
(6)𝒥(q)=𝔼[Ψ(y0q)+∫0T‍∫U‍l(t,ytq,ztq,a)qt(da)dt].A relaxed controlμ is called optimal if it solves
(7)𝒥(μ)=inf⁡q∈ℛ𝒥(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 byμ an optimal relaxed control, and q is an arbitrary element of ℛ, 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 coefficientsf, 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 [9] and Bahlali [6], 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 [10] and Han, Peng, and Wu [11].The paper is organized as follows. In Section2, 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 byC some positive constants and for simplicity, we need the following matrix notations. We denote by ℳn×d(ℝ) the space of n×d real matrix and ℳn×n(ℝ) the linear space of vectors M=(M1,...,Md) where Mi∈ℳn×n(ℝ). For any M,N∈ℳn×nd(ℝ), L,S∈ℳn×d(ℝ), Q∈ℳn×d(ℝ), α,β∈ℝn, and γ∈ℝd, we use the following notations:αβ=∑i=1n‍αiβi is the product scalar in ℝn;  LS=∑i=1d‍LiSi, where Li and Si are the ith column of L and S;  ML=∑i=1d‍MiLi∈ℝn;  M(αγ)=∑i=1d‍(Miα)γi∈ℝn;  MN=∑i=1d‍(MiNi)∈ℳn×n(ℝ);  MQN=∑i=1d‍MiQNi∈ℳn×n(ℝ);  MQγ=∑i=1d‍MiQγγi∈ℳn×n(ℝ).We denote byL* the transpose of the matrix L and M*=(M1*,...,Md*).


## 2. Formulation of the Problem

Let(Ω,ℱ,(ℱt(B,W))t≥0,ℙ) be a probability space, on which a d-dimensional Brownian motions W=(Wt:0≤t≤T) and B=(Bt:0≤t≤T) are defined. For each t∈[0,T], we define ℱt(B,W)≜ℱt,TB∨ℱtW, where for any process {pt}, ℱs,tp=σ{p(r)-p(s);s≤r≤t}∨𝒩, ℱtp=ℱ0,tp. Note that the collection {ℱt,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 σ1∨σ2 denotes the σ-fields generated by σ1∪σ2.Note that the collection{(ℱt(B,W))t,t∈[0,T]} is neither increasing nor decreasing, and it does not constitute a filtration.LetT be a positive real number and U a nonempty subset of ℝk.


### 2.1. The Strict Control Problem

For anyn∈ℕ, let ℳ2(0,T;ℝn) denote the set of n-dimensional jointly measurable random processes {φt,t∈[0,T]} which satisfy
(8)(i):𝔼[∫0T‍|φt|2dt]<∞,(ii):φtis(ℱt(B,W))measurable,for  a.e.t∈[0,T].We denote similarly by𝒮2([0,T];ℝn) the set of continuous n-dimensional random processes which satisfy
(9)(i):𝔼[sup⁡0≤t≤T|φt|2]<∞,(ii):φtis(ℱt(B,W))measurable,foranyt∈[0,T].LetT be a strictly positive real number and U is a nonempty subset of ℝk.Definition 1.
(
y
t
,
z
t
) is said to be a solution of (1), if and only if, (yt,zt)∈ℳ2(ℝn×d)×𝒮2(ℝn×m) and it satisfies (1).Definition 2.
An admissible strict control is anℱt-adapted process ν=(νt) with values in U such that 𝔼[sup⁡t∈[0,T]|νt|2]<∞.
We denote by𝒰 the set of all admissible controls.For anyν∈𝒰, 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]×ℝm×ℳm×d(ℝ)×U→ℝm,  g:[0,T]×ℝm×ℳm×d(ℝ)×U→ℝm×k, and ξ is an n-dimensional ℱ0-measurable random variable such that 𝔼|ξ|2<∞.The expected cost is defined from𝒰 into ℝ by
(11)J(ν)=𝔼[Ψ(y0ν)+∫0T‍l(t,ytν,ztν,νt)dt],
where Ψ:ℝm→ℝ,  l:[0,T]×ℝm×ℳm×d(ℝ)×U→ℝ.The control problem is to minimize the functionalJ over 𝒰, if u∈𝒰 is an optimal solution; that is,
(12)J(u)=inf⁡ν∈𝒰J(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 constantsC>0 and 0≤ζ<1 such that for any (w,t)∈Ω×[0,T],  (Y1,Z1);  (Y2,Z2)∈ℝm×ℳm×d(ℝ),
(Hypo02)(i):|f(t,Y1,Z1)-f(t,Y2,Z2)|2≤C(|Y1-Y2|2+∥Z1-Z2∥2),(ii):∥g(t,Y1,Z1)-g(t,Y2,Z2)∥2≤C|Y1-Y2|2+ζ(∥Z1-Z2∥2).Under the above assumptions, for everyν∈𝒰, (10) has a unique strong solution and the functional cost J is well defined from 𝒰 into ℝ.


### 2.2. The Relaxed Model

The idea for the relaxed strict control problem defined above is to embed the setU 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 ℙ(U)-valued process q, where ℙ(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 ℙ(U)-valued process, progressively measurable with respect to (ℱt)t and such that for each t,  1]0,t[qt is ℱt-measurable.
We denote byℛ the set of all admissible relaxed controls.Remark 4.
Every relaxed controlq 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 ℙ(U). The set U is embedded into the set ℛ of relaxed process by the mapping f:ν∈𝒰↦fυ(dt,da)=δυt(da)dt∈ℛ, where δυ is the atomic measure concentrated at a single point υ.For anyq∈ℛ, 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 fromℛ into ℝ by
(15)𝒥(q)=𝔼[Ψ(y0q)+∫0T‍∫U‍l(t,ytq,ztq,a)qt(da)dt].A relaxed controlρ is called optimal if it solves
(16)𝒥(μ)=inf⁡q∈ℛ𝒥(q).Existence of an optimal solution for the problem {(14), (15), (16)} has been solved.


## 2.1. The Strict Control Problem

For anyn∈ℕ, let ℳ2(0,T;ℝn) denote the set of n-dimensional jointly measurable random processes {φt,t∈[0,T]} which satisfy
(8)(i):𝔼[∫0T‍|φt|2dt]<∞,(ii):φtis(ℱt(B,W))measurable,for  a.e.t∈[0,T].We denote similarly by𝒮2([0,T];ℝn) the set of continuous n-dimensional random processes which satisfy
(9)(i):𝔼[sup⁡0≤t≤T|φt|2]<∞,(ii):φtis(ℱt(B,W))measurable,foranyt∈[0,T].LetT be a strictly positive real number and U is a nonempty subset of ℝk.Definition 1.
(
y
t
,
z
t
) is said to be a solution of (1), if and only if, (yt,zt)∈ℳ2(ℝn×d)×𝒮2(ℝn×m) and it satisfies (1).Definition 2.
An admissible strict control is anℱt-adapted process ν=(νt) with values in U such that 𝔼[sup⁡t∈[0,T]|νt|2]<∞.
We denote by𝒰 the set of all admissible controls.For anyν∈𝒰, 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]×ℝm×ℳm×d(ℝ)×U→ℝm,  g:[0,T]×ℝm×ℳm×d(ℝ)×U→ℝm×k, and ξ is an n-dimensional ℱ0-measurable random variable such that 𝔼|ξ|2<∞.The expected cost is defined from𝒰 into ℝ by
(11)J(ν)=𝔼[Ψ(y0ν)+∫0T‍l(t,ytν,ztν,νt)dt],
where Ψ:ℝm→ℝ,  l:[0,T]×ℝm×ℳm×d(ℝ)×U→ℝ.The control problem is to minimize the functionalJ over 𝒰, if u∈𝒰 is an optimal solution; that is,
(12)J(u)=inf⁡ν∈𝒰J(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 constantsC>0 and 0≤ζ<1 such that for any (w,t)∈Ω×[0,T],  (Y1,Z1);  (Y2,Z2)∈ℝm×ℳm×d(ℝ),
(Hypo02)(i):|f(t,Y1,Z1)-f(t,Y2,Z2)|2≤C(|Y1-Y2|2+∥Z1-Z2∥2),(ii):∥g(t,Y1,Z1)-g(t,Y2,Z2)∥2≤C|Y1-Y2|2+ζ(∥Z1-Z2∥2).Under the above assumptions, for everyν∈𝒰, (10) has a unique strong solution and the functional cost J is well defined from 𝒰 into ℝ.


## 2.2. The Relaxed Model

The idea for the relaxed strict control problem defined above is to embed the setU 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 ℙ(U)-valued process q, where ℙ(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 ℙ(U)-valued process, progressively measurable with respect to (ℱt)t and such that for each t,  1]0,t[qt is ℱt-measurable.
We denote byℛ the set of all admissible relaxed controls.Remark 4.
Every relaxed controlq 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 ℙ(U). The set U is embedded into the set ℛ of relaxed process by the mapping f:ν∈𝒰↦fυ(dt,da)=δυt(da)dt∈ℛ, where δυ is the atomic measure concentrated at a single point υ.For anyq∈ℛ, 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 fromℛ into ℝ by
(15)𝒥(q)=𝔼[Ψ(y0q)+∫0T‍∫U‍l(t,ytq,ztq,a)qt(da)dt].A relaxed controlρ is called optimal if it solves
(16)𝒥(μ)=inf⁡q∈ℛ𝒥(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 controlsℛ 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 ℛ.Denote by(ytθ,ztθ) the solution of (14) associated with μθ.From the optimality ofμ, the variational inequality will be derived from the fact that
(18)0≤𝒥(μθ)-𝒥(μ).To this end, we need the following classical Lemmas.Lemma 5.
Under the assumption (13) and (Hypo02), we have
(19)lim⁡θ→0[sup⁡0≤t≤T𝔼|ytθ-ytμ|2]=0,(20)lim⁡θ→0𝔼[∫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 s≤T, we have
(21)𝔼|ytθ-ytμ|2+𝔼∫tT‍|zsθ-zsμ|2ds=2𝔼∫tT|(ysθ-ysμ)[∫Uf(s,ysθ,zsθ,a)μsθ(da)-∫Uf(s,ysμ,zsμ,a)μs(da)]|ds+𝔼∫tT|[∫Ug(s,ysθ,zsθ,a)μsθ(da)-∫Ug(s,ysμ,zsμ,a)μs(da)]|ds.
From the Young formula, for everyε>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+𝔼∫tT‍|zsθ-zsμ|2ds≤(1+α)𝔼∫tT‍|(ysθ-ysμ)|2ds+θ2C(1+α)𝔼∫tT|∫Uf(s,ysθ,zsθ,a)qs(da)-∫Uf(s,ysθ,zsθ,a)μs(da)|2ds+θ2C𝔼∫tT|∫Ug(s,ysθ,zsθ,a)qs(da)-∫Ug(s,ysθ,zsθ,a)μs(da)|2ds+C(1+α)𝔼∫tT|∫Uf(s,ysθ,zsθ,a)μs(da)-∫Uf(s,ysμ,zsθ,a)μs(da)|2ds+C(1+α)𝔼∫tT|∫Uf(s,ysμ,zsθ,a)μs(da)-∫Uf(s,ysμ,zsμ,a)μs(da)|2ds+C𝔼∫tT|∫Ug(s,ysθ,zsθ,a)μs(da)-∫Ug(s,ysμ,zsθ,a)μs(da)|2ds+C𝔼∫tT|∫Ug(s,ysμ,zsμ,a)μs(da)-∫Ug(s,ysμ,zsμ,a)μs(da)|2ds.
Sinceg and f are uniformly Lipschitz with respect to y and z and from (Hypo02), where 0≤ζ<1, then
(23)𝔼|ytθ-ytμ|2+(1-C(1+α)-ζ)𝔼∫tT|zsθ-zsμ|2ds≤(C(1+α)+(1+α)+C)𝔼∫tT|(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,C≤1-ζ2,2c+ζ-11-ζ,C>1-ζ2.
Then,1-C/(1+α)-ζ≥(1-ζ)/2>0.
Choosec1=1-C/(1+α)-ζ,  c2=C/(1+α)+(1+α)+C, then we have
(26)𝔼|ytθ-ytμ|2+c1𝔼∫tT‍|zsθ-zsμ|2ds≤c2𝔼∫tT‍|(ysθ-ysμ)|2ds+Mθ.
From the above inequality, we derive the following two inequalities:(27)𝔼|ytθ-ytμ|≤c2𝔼∫tT‍|(ysθ-ysμ)|2ds+Mθ,(28)𝔼∫tT‍|zsθ-zsμ|2ds≤c2c1𝔼∫tT‍|(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.
Lety~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)lim⁡θ→0[sup⁡0≤t≤T𝔼|ytθ-ytμθ-y~t|2]=0,(31)lim⁡θ→0𝔼[∫0T‍|ztθ-ztμθ-z~t|2dt]=0.Proof.
For simplicity, we putYtθ=(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+𝔼∫tT‍|Zsθ|2ds≤1ε𝔼∫tT‍|Ysθ|2ds+ε𝔼∫tT|[∫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+ε𝔼∫tT‍|[∫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 putΛtθ(a)=(t,ytμ+αθ(Ytθ+y~t),ztμ+αθ(Ztθ+z~t),a), we have the following inequality:
(33)𝔼|Ytθ|2+𝔼∫tT‍|Zsθ|2ds≤1ε𝔼∫tT‍|Ysθ|2ds+ε𝔼∫tT‍|FsyYsθ+FszZsθ-ηsθ|2ds+ε𝔼∫tT‍|GsyYsθ+GszZsθ-ηs′θ|2ds,
where
(34)Fty=-∫01‍∫U‍fy(Λtθ(a))μt(da)dα,Gty=∫01‍∫U‍gy(Λtθ(a))μt(da)dα,Ftz=-∫01‍∫U‍fz(Λtθ(a))μt(da)dα,Gtz=∫01‍∫U‍gz(Λtθ(a))μt(da)dα,
and ηtθ, ηt′θ are given by
(35)ηtθ=∫tT∫U[fy(Λsθ(a))(ysθ-ysμ)+fz(Λsθ(a))(zsθ-zsμ)]qs(da)ds-∫tT∫U[fy(Λsθ(a))(ysθ-ysμ)+fz(Λsθ(a))(zsθ-zsμ)]μs(da)ds,ηt′θ=∫tT∫U[gy(Λsθ(a))(ysθ-ysμ)+gz(Λsθ(a))(zsθ-zsμ)]qs(da)ds-∫tT∫U[gy(Λsθ(a))(ysθ-ysμ)+gz(Λsθ(a))(zsθ-zsμ)]μs(da)ds.
SinceFty, Ftz, Gty, and Gtz are bounded, then
(36)𝔼|Ytθ|2+𝔼∫tT‍|Zsθ|2ds≤1ε𝔼∫tT‍|Ysθ|2ds+εC𝔼∫tT‍|Ysθ|2ds+εC𝔼∫Γ‍|Zsθ|2ds+Πtθ,
where Πtθ=εC𝔼∫tT|ηsθ|2ds+εC𝔼∫tT|ηs′θ|2ds. Choose ε=1/2C, then we have
(37)𝔼|Ytθ|2+12𝔼∫tT‍|Zsθ|2ds≤(2C+12)𝔼∫tT‍|Ysθ|2ds+Πtθ.
From the above inequality, we deduce the following two inequalities:(38)𝔼|Ytθ|2≤(2C+12)𝔼∫tT‍|Ysθ|2ds+Πtθ,(39)𝔼∫tT‍|Zsθ|2ds≤2(2C+12)𝔼∫tT‍|Ysθ|2ds+2Πtθ.
Sincefy and fz and gy and gz are continuous and bounded, then from (19) and (20), we have
(40)lim⁡θ→0(𝔼∫tT‍|ηsθ|2ds+𝔼∫tT‍|ηs′θ|2ds)=0.
From (40), we deduce that
(41)lim⁡θ→0Πtθ=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.
Letμ be an optimal relaxed control minimizing the cost 𝒥 over ℛ and (ytμ,ztμ) the associated optimal trajectory. Then, for any q∈ℛ, we have
(42)0≤𝔼[Ψy(y0μ)·y~0]+𝔼∫0T‍[∫Ul(t,ytμ,ztμ,a)qt(da)-∫U‍l(t,ytμ,ztμ,a)μs(da)]dt+𝔼∫0T‍[∫U‍ly(t,ytμ,ztμ,a)μs(da)·y~t]dt+𝔼∫0T‍[∫U‍lz(t,ytμ,ztμ,a)μt(da)·z~t]dt.Proof.
Letμ be an optimal relaxed control minimizing the cost 𝒥 over ℛ, then from (18), and by using the definition of μtθ, we have
(43)0≤𝔼[Ψ(y0θ)-Ψ(y0μ)]+θ𝔼∫0T‍[∫U‍l(t,ytθ,ztθ,a)qt(da)-∫U‍l(t,ytθ,ztθ,a)μt(da)]dt+𝔼∫0T‍∫U‍[l(t,ytθ,ztθ,a)-l(t,ytμ,ztμ,a)l(t,ytθ,ztθ,a)]μt(da)dt.
Then,(44)0≤𝔼∫01‍[Ψy(y0μ+αθ(y~0+Y0θ))y~0]dα+𝔼∫0T‍∫01‍∫U‍[ly(Λ¯tθ(a))·y~t+lz(Λ¯tθ(a))·z~t]μt(da)dαdt+𝔼∫0T‍[∫U‍l(t,ytμ,ztμ,a)qt(da)-∫U‍l(t,ytμ,ztμ,a)μt(da)]dt+βtθ,
where βtθ is given by
(45)βtθ=𝔼∫01‍[Ψy(y0μ+αθ(y~0+Y0θ))·Y0θ]dα+𝔼∫0T‍∫01‍∫U‍[ly(Λ¯tθ(a))·Ytθ+lz(Λ¯tθ(a))·Ztθ]μt(da)dt+𝔼∫0T‍∫01‍∫U‍[ly(Λ¯tθ(a))(ytθ-ytμ)+lz(Λ¯tθ(a))(ztθ-ztμ)]qt(da)dt-𝔼∫0T‍∫01‍∫U‍[ly(Λ¯tθ(a))(ytθ-ytμ)+lz(Λ¯tθ(a))(ztθ-ztμ)]μt(da)dt.
For simplicity, we putΛ¯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 lim⁡θ→0βtθ=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 Hamiltonianℋ is defined from [0,T]×ℝm×ℳn×m(ℝ)×ℙ(U)×ℳn×d(ℝ) into ℝ by
(46)ℋ(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).
Letμ be an optimal relaxed control minimizing the functional 𝒥 over ℛ and (ytμ,ztμ) the solution of (14) associated with μ. Then, there exists unique adapted process (Qμ)∈Lℱ2([0,T];ℝm) 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 everyqt∈ℙ(U),
(48)ℋ(t,ytμ,ztμ,qt,Qtμ,Rtμ)≥ℋ(t,ytμ,ztμ,μt,Qtμ,Rtμ),∀μt∈ℙ(U),ae,  as.Proof.
SinceQ0μ=Ψy(y0μ), then (42) becomes
(49)0≤𝔼[Q0μy~0]+𝔼∫0T‍[∫Ulμ(t,ytμ,ztμ,a)qt(da)-∫U‍lμ(t,ytμ,ztμ,a)μs(da)]dt+𝔼∫0T‍[∫U‍lyμ(t,ytμ,ztμ,a)μs(da)·y~t]dt+𝔼∫0T‍[∫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[∫0T‍∫U‍lyμ(t,a)μt(da)·y~t]dt-𝔼[∫0T‍∫U‍lzμ(t,a)μt(da)·z~t]dt+𝔼∫0T‍Qtμ[∫U‍fμ(t,a)qt(da)-∫U‍fμ(t,a)μt(da)]dt+𝔼∫0T‍Rtμ[∫U‍gμ(t,a)qt(da)-∫U‍gμ(t,a)μt(da)]dt.
Then, for everyq∈ℛ, (49) becomes
(51)0≤𝔼∫0T‍[ℋ(t,ytμ,ztμ,qt,Qtμ,Rtμ)-ℋ(t,ytμ,ztμ,μt,Qtμ,Rtμ)]dt.
Now, letμ∈ℛ and F be an arbitrary element of the σ-algebra ℱt, and set ϕ=qt1F+μt1Ω-F. It is obvious that ϕ is an admissible relaxed control.
Applying the above inequality withϕ, we get
(52)0≤𝔼[1F{ℋ(t,ytμ,ztμ,qt,Qtμ,Rtμ)-ℋ(t,ytμ,ztμ,μt,Qtμ,Rtμ)}],∀F∈ℱt.
Which implies that(53)0≤𝔼[ℋ(t,ytμ,ztμ,qt,Qtμ,Rtμ)-ℋ(t,ytμ,ztμ,μt,Qtμ,Rtμ)ℱt].
The quantity inside the conditional expectation isℱt-measurable, and thus the result follows immediately.


## 3.1. Preliminary Results

Since the set of relaxed controlsℛ 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 ℛ.Denote by(ytθ,ztθ) the solution of (14) associated with μθ.From the optimality ofμ, the variational inequality will be derived from the fact that
(18)0≤𝒥(μθ)-𝒥(μ).To this end, we need the following classical Lemmas.Lemma 5.
Under the assumption (13) and (Hypo02), we have
(19)lim⁡θ→0[sup⁡0≤t≤T𝔼|ytθ-ytμ|2]=0,(20)lim⁡θ→0𝔼[∫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 s≤T, we have
(21)𝔼|ytθ-ytμ|2+𝔼∫tT‍|zsθ-zsμ|2ds=2𝔼∫tT|(ysθ-ysμ)[∫Uf(s,ysθ,zsθ,a)μsθ(da)-∫Uf(s,ysμ,zsμ,a)μs(da)]|ds+𝔼∫tT|[∫Ug(s,ysθ,zsθ,a)μsθ(da)-∫Ug(s,ysμ,zsμ,a)μs(da)]|ds.
From the Young formula, for everyε>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+𝔼∫tT‍|zsθ-zsμ|2ds≤(1+α)𝔼∫tT‍|(ysθ-ysμ)|2ds+θ2C(1+α)𝔼∫tT|∫Uf(s,ysθ,zsθ,a)qs(da)-∫Uf(s,ysθ,zsθ,a)μs(da)|2ds+θ2C𝔼∫tT|∫Ug(s,ysθ,zsθ,a)qs(da)-∫Ug(s,ysθ,zsθ,a)μs(da)|2ds+C(1+α)𝔼∫tT|∫Uf(s,ysθ,zsθ,a)μs(da)-∫Uf(s,ysμ,zsθ,a)μs(da)|2ds+C(1+α)𝔼∫tT|∫Uf(s,ysμ,zsθ,a)μs(da)-∫Uf(s,ysμ,zsμ,a)μs(da)|2ds+C𝔼∫tT|∫Ug(s,ysθ,zsθ,a)μs(da)-∫Ug(s,ysμ,zsθ,a)μs(da)|2ds+C𝔼∫tT|∫Ug(s,ysμ,zsμ,a)μs(da)-∫Ug(s,ysμ,zsμ,a)μs(da)|2ds.
Sinceg and f are uniformly Lipschitz with respect to y and z and from (Hypo02), where 0≤ζ<1, then
(23)𝔼|ytθ-ytμ|2+(1-C(1+α)-ζ)𝔼∫tT|zsθ-zsμ|2ds≤(C(1+α)+(1+α)+C)𝔼∫tT|(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,C≤1-ζ2,2c+ζ-11-ζ,C>1-ζ2.
Then,1-C/(1+α)-ζ≥(1-ζ)/2>0.
Choosec1=1-C/(1+α)-ζ,  c2=C/(1+α)+(1+α)+C, then we have
(26)𝔼|ytθ-ytμ|2+c1𝔼∫tT‍|zsθ-zsμ|2ds≤c2𝔼∫tT‍|(ysθ-ysμ)|2ds+Mθ.
From the above inequality, we derive the following two inequalities:(27)𝔼|ytθ-ytμ|≤c2𝔼∫tT‍|(ysθ-ysμ)|2ds+Mθ,(28)𝔼∫tT‍|zsθ-zsμ|2ds≤c2c1𝔼∫tT‍|(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.
Lety~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)lim⁡θ→0[sup⁡0≤t≤T𝔼|ytθ-ytμθ-y~t|2]=0,(31)lim⁡θ→0𝔼[∫0T‍|ztθ-ztμθ-z~t|2dt]=0.Proof.
For simplicity, we putYtθ=(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+𝔼∫tT‍|Zsθ|2ds≤1ε𝔼∫tT‍|Ysθ|2ds+ε𝔼∫tT|[∫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+ε𝔼∫tT‍|[∫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 putΛtθ(a)=(t,ytμ+αθ(Ytθ+y~t),ztμ+αθ(Ztθ+z~t),a), we have the following inequality:
(33)𝔼|Ytθ|2+𝔼∫tT‍|Zsθ|2ds≤1ε𝔼∫tT‍|Ysθ|2ds+ε𝔼∫tT‍|FsyYsθ+FszZsθ-ηsθ|2ds+ε𝔼∫tT‍|GsyYsθ+GszZsθ-ηs′θ|2ds,
where
(34)Fty=-∫01‍∫U‍fy(Λtθ(a))μt(da)dα,Gty=∫01‍∫U‍gy(Λtθ(a))μt(da)dα,Ftz=-∫01‍∫U‍fz(Λtθ(a))μt(da)dα,Gtz=∫01‍∫U‍gz(Λtθ(a))μt(da)dα,
and ηtθ, ηt′θ are given by
(35)ηtθ=∫tT∫U[fy(Λsθ(a))(ysθ-ysμ)+fz(Λsθ(a))(zsθ-zsμ)]qs(da)ds-∫tT∫U[fy(Λsθ(a))(ysθ-ysμ)+fz(Λsθ(a))(zsθ-zsμ)]μs(da)ds,ηt′θ=∫tT∫U[gy(Λsθ(a))(ysθ-ysμ)+gz(Λsθ(a))(zsθ-zsμ)]qs(da)ds-∫tT∫U[gy(Λsθ(a))(ysθ-ysμ)+gz(Λsθ(a))(zsθ-zsμ)]μs(da)ds.
SinceFty, Ftz, Gty, and Gtz are bounded, then
(36)𝔼|Ytθ|2+𝔼∫tT‍|Zsθ|2ds≤1ε𝔼∫tT‍|Ysθ|2ds+εC𝔼∫tT‍|Ysθ|2ds+εC𝔼∫Γ‍|Zsθ|2ds+Πtθ,
where Πtθ=εC𝔼∫tT|ηsθ|2ds+εC𝔼∫tT|ηs′θ|2ds. Choose ε=1/2C, then we have
(37)𝔼|Ytθ|2+12𝔼∫tT‍|Zsθ|2ds≤(2C+12)𝔼∫tT‍|Ysθ|2ds+Πtθ.
From the above inequality, we deduce the following two inequalities:(38)𝔼|Ytθ|2≤(2C+12)𝔼∫tT‍|Ysθ|2ds+Πtθ,(39)𝔼∫tT‍|Zsθ|2ds≤2(2C+12)𝔼∫tT‍|Ysθ|2ds+2Πtθ.
Sincefy and fz and gy and gz are continuous and bounded, then from (19) and (20), we have
(40)lim⁡θ→0(𝔼∫tT‍|ηsθ|2ds+𝔼∫tT‍|ηs′θ|2ds)=0.
From (40), we deduce that
(41)lim⁡θ→0Πtθ=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.
Letμ be an optimal relaxed control minimizing the cost 𝒥 over ℛ and (ytμ,ztμ) the associated optimal trajectory. Then, for any q∈ℛ, we have
(42)0≤𝔼[Ψy(y0μ)·y~0]+𝔼∫0T‍[∫Ul(t,ytμ,ztμ,a)qt(da)-∫U‍l(t,ytμ,ztμ,a)μs(da)]dt+𝔼∫0T‍[∫U‍ly(t,ytμ,ztμ,a)μs(da)·y~t]dt+𝔼∫0T‍[∫U‍lz(t,ytμ,ztμ,a)μt(da)·z~t]dt.Proof.
Letμ be an optimal relaxed control minimizing the cost 𝒥 over ℛ, then from (18), and by using the definition of μtθ, we have
(43)0≤𝔼[Ψ(y0θ)-Ψ(y0μ)]+θ𝔼∫0T‍[∫U‍l(t,ytθ,ztθ,a)qt(da)-∫U‍l(t,ytθ,ztθ,a)μt(da)]dt+𝔼∫0T‍∫U‍[l(t,ytθ,ztθ,a)-l(t,ytμ,ztμ,a)l(t,ytθ,ztθ,a)]μt(da)dt.
Then,(44)0≤𝔼∫01‍[Ψy(y0μ+αθ(y~0+Y0θ))y~0]dα+𝔼∫0T‍∫01‍∫U‍[ly(Λ¯tθ(a))·y~t+lz(Λ¯tθ(a))·z~t]μt(da)dαdt+𝔼∫0T‍[∫U‍l(t,ytμ,ztμ,a)qt(da)-∫U‍l(t,ytμ,ztμ,a)μt(da)]dt+βtθ,
where βtθ is given by
(45)βtθ=𝔼∫01‍[Ψy(y0μ+αθ(y~0+Y0θ))·Y0θ]dα+𝔼∫0T‍∫01‍∫U‍[ly(Λ¯tθ(a))·Ytθ+lz(Λ¯tθ(a))·Ztθ]μt(da)dt+𝔼∫0T‍∫01‍∫U‍[ly(Λ¯tθ(a))(ytθ-ytμ)+lz(Λ¯tθ(a))(ztθ-ztμ)]qt(da)dt-𝔼∫0T‍∫01‍∫U‍[ly(Λ¯tθ(a))(ytθ-ytμ)+lz(Λ¯tθ(a))(ztθ-ztμ)]μt(da)dt.
For simplicity, we putΛ¯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 lim⁡θ→0βtθ=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 Hamiltonianℋ is defined from [0,T]×ℝm×ℳn×m(ℝ)×ℙ(U)×ℳn×d(ℝ) into ℝ by
(46)ℋ(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).
Letμ be an optimal relaxed control minimizing the functional 𝒥 over ℛ and (ytμ,ztμ) the solution of (14) associated with μ. Then, there exists unique adapted process (Qμ)∈Lℱ2([0,T];ℝm) 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 everyqt∈ℙ(U),
(48)ℋ(t,ytμ,ztμ,qt,Qtμ,Rtμ)≥ℋ(t,ytμ,ztμ,μt,Qtμ,Rtμ),∀μt∈ℙ(U),ae,  as.Proof.
SinceQ0μ=Ψy(y0μ), then (42) becomes
(49)0≤𝔼[Q0μy~0]+𝔼∫0T‍[∫Ulμ(t,ytμ,ztμ,a)qt(da)-∫U‍lμ(t,ytμ,ztμ,a)μs(da)]dt+𝔼∫0T‍[∫U‍lyμ(t,ytμ,ztμ,a)μs(da)·y~t]dt+𝔼∫0T‍[∫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[∫0T‍∫U‍lyμ(t,a)μt(da)·y~t]dt-𝔼[∫0T‍∫U‍lzμ(t,a)μt(da)·z~t]dt+𝔼∫0T‍Qtμ[∫U‍fμ(t,a)qt(da)-∫U‍fμ(t,a)μt(da)]dt+𝔼∫0T‍Rtμ[∫U‍gμ(t,a)qt(da)-∫U‍gμ(t,a)μt(da)]dt.
Then, for everyq∈ℛ, (49) becomes
(51)0≤𝔼∫0T‍[ℋ(t,ytμ,ztμ,qt,Qtμ,Rtμ)-ℋ(t,ytμ,ztμ,μt,Qtμ,Rtμ)]dt.
Now, letμ∈ℛ and F be an arbitrary element of the σ-algebra ℱt, and set ϕ=qt1F+μt1Ω-F. It is obvious that ϕ is an admissible relaxed control.
Applying the above inequality withϕ, we get
(52)0≤𝔼[1F{ℋ(t,ytμ,ztμ,qt,Qtμ,Rtμ)-ℋ(t,ytμ,ztμ,μt,Qtμ,Rtμ)}],∀F∈ℱt.
Which implies that(53)0≤𝔼[ℋ(t,ytμ,ztμ,qt,Qtμ,Rtμ)-ℋ(t,ytμ,ztμ,μt,Qtμ,Rtμ)ℱt].
The quantity inside the conditional expectation isℱt-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∈ℛ, we denote by (yq,zq) the solution of (14) controlled by q.Theorem 9 (sufficient optimality conditions for relaxed controls).
Assume that the functionsΨ and (y,z)↦ℋ(t,y,z,q,Q,R) are convex, and for any q∈ℛ, yTq=0 is an m-dimensional ℱt-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.
Letμ2 be an arbitrary element of ℛ (candidate to be optimal). For any μ1∈ℛ, we have
(54)𝒥(μ1)-𝒥(μ2)=𝔼[h(y0μ1)-h(y0μ2)]+𝔼∫0T‍[∫Ul(t,ytμ1,ztμ1,a)μ1t(da)-∫U‍l(t,ytμ2,ztμ2,a)μ2t(da)]dt.
SinceΨ are convex, then
(55)Ψ(y0μ1)-Ψ(y0μ2)≥Ψy(y0μ2)(y0μ1-y0μ2).
Thus,(56)𝒥(μ1)-𝒥(μ2)≥𝔼[Ψy(y0μ2)(y0μ1-y0μ2)]+𝔼∫0T‍[∫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)]+𝔼∫0T‍[∫Ulq(t,a)μ1t(da)-∫U‍lμ(t,a)μ2t(da)]dt.
Thus,(58)𝒥(μ1)-𝒥(μ2)≥𝔼[Q0μ2(y0μ1-y0μ2)]+𝔼∫0T‍[∫Ulq(t,a)μ1t(da)-∫U‍lμ(t,a)μ2t(da)]dt.
By applying Itô’s formula toQtμ(ytq-ytμ), we obtain
(59)𝔼[Q0μ2(y0μ1-y0μ2)]=𝔼[QTμ2(yTμ1-yTμ2)]-𝔼∫0T‍ℋy(t,ytμ2,ztμ2,μ,Qtμ2,Rtμ2)×(ytμ1-ytμ2)dt-𝔼∫0T‍ℋz(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)×(ztμ1-ztμ2)dt+𝔼∫0T‍Qtμ2[∫U‍fq(t,a)μ1t(da)-∫U‍fμ(t,a)μ2t(da)]dt+𝔼∫0T‍Rtμ2[∫Ugq(t,a)μ1t(da)-∫U‍gμ(t,a)μ2t(da)]dt.
Then,(60)𝒥(μ1)-𝒥(μ2)≥𝔼∫0T‍[ℋ(t,ytμ2,ztμ2,μ1t,Qtμ2,Rtμ2)-ℋ(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)]dt-𝔼∫0T‍ℋy(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)×(ytμ1-ytμ2)dt-𝔼∫0T‍ℋz(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)×(ztμ1-ztμ2)dt.
Sinceℋ is convex in (y,z) and linear in μ, then by using the Clarke generalized gradient of ℋ evaluated at (yt,zt,μ) and the necessary optimality conditions, that
(61)ℋ(t,ytμ1,ztμ1,μ1t,Qtμ1,Rtμ1)-ℋ(t,ytμ2,ztμ2,μ2t,ptμ2,Qtμ2,Rtμ2)≥ℋy(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)(ytμ1-ytμ2)+ℋz(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)(ztμ1-ztμ2)
or equivalently
(62)0≤ℋ(t,ytμ1,ztμ1,μ1t,Qtμ1,Rtμ1)-ℋ(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)-ℋy(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)(ytμ1-ytμ2)-ℋz(t,ytμ2,ztμ2,μ2t,Qtμ2,Rtμ2)(ztμ1-ztμ2).
Then, from (60), we get
(63)𝒥(μ1)-𝒥(μ2)≥0.
The theorem is proved.



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*Source: 903912-2014-03-04.xml*