Mean-Field Maximum Principle for Optimal Control of Forward–Backward Stochastic Systems with Jumps and its Application to Mean-Variance Portfolio Problem

Abstract

We study mean-field type optimal stochastic control problem for systems governed by mean-field controlled forward–backward stochastic differential equations with jump processes, in which the coefficients depend on the marginal law of the state process through its expected value. The control variable is allowed to enter both diffusion and jump coefficients. Moreover, the cost functional is also of mean-field type. Necessary conditions for optimal control for these systems in the form of maximum principle are established by means of convex perturbation techniques. As an application, time-inconsistent mean-variance portfolio selection mixed with a recursive utility functional optimization problem is discussed to illustrate the theoretical results.

Keywords

Mean-field forward–backward stochastic differential equation with jumps / Optimal stochastic control / Mean-field maximum principle / Mean-variance portfolio selection with recursive utility functional / Time-inconsistent control problem