Optimal Control Problems Governed by MFSDEs with multi-defaults

TL;DR

This paper addresses optimal control problems involving mean-field stochastic differential equations with multiple defaults, providing theoretical solutions, maximum principles, and explicit examples for recursive utility optimization.

Contribution

It introduces a method to decompose complex MMFSDE control problems into simpler subproblems and establishes existence, uniqueness, and solution techniques for these systems.

Findings

Derived maximum principles for subproblems

Proved existence and uniqueness of solutions

Obtained explicit solutions for recursive utility control

Abstract

In this paper, we solve an optimal control problem governed by a system of mean-field stochastic differential equations with multiple defaults (MMFSDEs). We transform the global optimal control problem into several optimal control subproblems governed by a system of mean-field stochastic differential equations with single default (SMFSDEs) and derive both the sufficient and necessary maximum principles for these subproblems. We also give the existence and uniqueness of solutions to the MMFSDEs and the mean-field backward stochastic differential equations with multiple defaults (MMFBSDEs), respectively. Finally, as an example, our results are applied to obtain the explicit solution for an optimal control problem whose cost function is considered as a recursive utility process with multiple defaults.