On mean-field control problems for backward doubly stochastic systems

¹ Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, Shandong 266237, PR China
² School of Mathematics, Shandong University, Jinan, Shandong 250100, PR China

^* Corresponding author: wangmeng22@mail.sdu.edu.cn

Received: 21 June 2022
Accepted: 5 February 2024

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.