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

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 selectionmixed with a recursive utility functional optimization problem is discussed to illustrate the theoretical results.