Forward and Backward Mean-Field Stochastic Partial Differential Equation and Optimal Control

TL;DR

This paper investigates solutions and optimal control strategies for forward and backward mean-field stochastic PDEs, establishing existence, uniqueness, and maximum principles, with applications to linear-quadratic control problems and controlled stochastic PDEs.

Contribution

It provides new theoretical results on existence, uniqueness, and optimality conditions for mean-field stochastic PDEs, including Pontryagin's maximum principles and practical applications.

Findings

Proved continuous dependence theorems for mean-field stochastic PDEs

Established existence and uniqueness of solutions

Derived necessary and sufficient optimality conditions

Abstract

This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. We first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then we establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin's maximum principles. To illustrate the theoretical results, we apply stochastic maximum principles to study an example, an infinite-dimensional linear-quadratic control problem of mean-field type. Further an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type are studied.