On mean-field control problems for backward doubly stochastic systems

TL;DR

This paper develops a stochastic maximum principle for mean-field backward doubly stochastic control problems, establishing necessary and sufficient conditions for optimality and analyzing related well-posedness issues.

Contribution

It introduces a maximum principle for mean-field backward doubly stochastic systems and proves well-posedness of associated coupled equations, advancing control theory in this complex setting.

Findings

Derived necessary and sufficient optimality conditions

Proved well-posedness of mean-field coupled equations

Illustrated applications with examples

Abstract

This article is concerned with stochastic control problems for backward doubly stochastic differential equations of mean-field type, where the coefficient functions depend on the joint distribution of the state process and the control process. We obtain the stochastic maximum principle which serves as a necessary condition for an optimal control, and we also prove its sufficiency under proper conditions. As a byproduct, we prove the well-posedness for a type of mean-field fully coupled forward-backward doubly stochastic differential equation arising naturally from the control problem, which is of interest in its own right. Some examples are provided to illustrate the applications of our results to control problems in the types of scalar interaction and first order interaction.