Stochastic maximum principle for delayed doubly stochastic control systems and their applications

ABSTRACT

In this paper, we investigate the optimal control problems for delayed doubly stochastic control systems. We first discuss the existence and uniqueness of the delayed doubly stochastic differential equation by martingale representation theorem and contraction mapping principle. As a necessary condition of the optimal control, we deduce a stochastic maximum principle under some assumption. At the same time, a sufficient condition of optimality is obtained by using the duality method. At the end of the paper, we apply our stochastic maximum principle to a class of linear quadratic optimal control problem and obtain the explicit expression of the optimal control.

KEYWORDS:

• Stochastic maximum principle
• doubly stochastic differential equation
• delay system
• optimal control

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was partially supported by China Automobile Industry Innovation and Development Joint Fund [U1564213] and NSFC – National Natural Science Foundation of China [grant number 11371169].