A Global Maximum Principle for Controlled Conditional Mean-field FBSDEs with Regime Switching

TL;DR

This paper establishes a comprehensive stochastic maximum principle for controlled conditional mean-field FBSDEs with regime switching, extending previous work to recursive utility and including applications like linear-quadratic problems and state constraints.

Contribution

It develops a general maximum principle for non-convex control domains and driver dependencies, advancing the theory for controlled mean-field FBSDEs with regime switching.

Findings

Derived high-dimensional adjoint BSDEs for first- and second-order conditions.

Revealed relations among Taylor expansion terms using adjoint equations.

Applied the maximum principle to linear-quadratic and constrained problems.

Abstract

This paper is devoted to a global stochastic maximum principle for conditional mean-field forward-backward stochastic differential equations (FBSDEs, for short) with regime switching. The control domain is unnecessarily convex and the driver of backward stochastic differential equations (BSDEs, for short) could depend on $Z$. Different from the case of non-recursive utility, the first-order and second-order adjoint equations are both high-dimensional linear BSDEs. Based on the adjoint equations, we reveal the relations among the terms of the first- and second-order Taylor's expansions. A general maximum principle is proved, which develops the work of Nguyen, Yin, and Nguyen [22] to recursive utility. As applications, the linear-quadratic problem is considered and a problem with state constraint is studied.