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.

In 1994, Pardoux and Peng [

They proved that if

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

The control variable

The criteria to be minimized, over the set

A control

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 [

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:

The expected cost to be minimized, the relaxed model, is defined from

A relaxed control

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

By using the fact that the coefficients

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 [

The paper is organized as follows. In Section

Along this paper, we denote by

We denote by

Let

Note that the collection

Let

For any

We denote similarly by

Let

An admissible strict control is an

We denote by

For any

The expected cost is defined from

The control problem is to minimize the functional

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:

They and all their derivatives with respect to

We assume moreover that there exist constants

Under the above assumptions, for every

The idea for the relaxed strict control problem defined above is to embed the set

A relaxed control

We denote by

Every relaxed control

For any

The expected cost to be minimized, the relaxed model, is defined from

A relaxed control

Existence of an optimal solution for the problem {(

In this section, we study the problem {(

Since the set of relaxed controls

Denote by

From the optimality of

To this end, we need the following classical Lemmas.

Under the assumption (

Let us prove (

Applying Itô’s formula to

From the Young formula, for every

Since

Let

Then,

Choose

From the above inequality, we derive the following two inequalities:

By using (

Let

Then, we have

For simplicity, we put

(i) Proof of (

For simplicity, we put

Since

From the above inequality, we deduce the following two inequalities:

Since

From (

By using (

Let

Let

Then,

For simplicity, we put

Starting from the variational inequality (

The Hamiltonian

Let

such that for every

Since

By applying Itô’s formula to

Then, for every

Now, let

Applying the above inequality with

Which implies that

The quantity inside the conditional expectation is

In this section, we study when necessary optimality conditions (

Assume that the functions

Let

Since

Thus,

We remark from (

Thus,

By applying Itô’s formula to

Then,

Since

Then, from (

The theorem is proved.

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

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