Maximum Principle for Mean Field Type Control Problems with General Volatility Functions

TL;DR

This paper develops a maximum principle for mean field control problems with complex volatility dependencies, using a Hilbert space approach to derive necessary and sufficient conditions.

Contribution

It introduces a novel Hilbert space embedding method to analyze mean field control problems with state, measure, and control-dependent volatility functions.

Findings

Derived necessary and sufficient optimality conditions.

Established a system of forward-backward stochastic differential equations.

Provided a new analytical framework for complex volatility functions.

Abstract

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward-backward stochastic differential equations.