A global maximum principle for optimal control of general mean-field forward-backward stochastic systems with jumps

TL;DR

This paper establishes a necessary maximum principle for optimal control in complex mean-field forward-backward stochastic systems with jumps, accommodating non-convex control sets and control-independent jump coefficients.

Contribution

It introduces new adjoint equations and estimates to derive a second-order expansion of the cost functional for such systems.

Findings

Derived a stochastic maximum principle for mean-field systems with jumps.

Introduced two new adjoint equations and generic estimates for their solutions.

Analyzed the second-order expansion of the cost functional.

Abstract

In this paper we prove a necessary condition of the optimal control problem for a class of general mean-field forward-backward stochastic systems with jumps in the case where the diffusion coefficients depend on control, the control set does not need to be convex, the coefficients of jump terms are independent of control as well as the coefficients of mean-field backward stochastic differential equations depend on the joint law of $(X(t),Y(t))$. Two new adjoint equations are brought in as well as several new generic estimates of their solutions are investigated for analysing the higher terms, especially, those involving the expectation which come from the derivatives of the coefficients with respect to the measure. Utilizing these subtle estimates, the second-order expansion of the cost functional, which is the key point to analyse the necessary condition, is obtained, and whereafter the stochastic maximum principle.