Near-Relaxed Control Problem of Fully Coupled Forward–Backward Doubly System

Abstract

In this paper, we are concerned with an optimal control problem where the system is driven by a fully coupled forward–backward doubly stochastic differential equation. We study the relaxed model for which an optimal solution exists. This is an extension of initial control problem, where admissible controls are measure valued processes. We establish necessary as well as sufficient optimality conditions to the relaxed one.

Keywords

Fully coupled forward–backward doubly stochastic differential equation / Relaxed control / Maximum principle / Adjoint equation / Variational principle