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.