On Zhou's maximum principle for near-optimal control of mean-field forward-backward stochastic systems with jumps and its applications

This paper is concerned with stochastic maximum principle for near-optimal control of nonlinear controlled mean-field forward-backward stochastic systems driven by Brownian motions and random Poisson martingale measure (FBSDEJs in short) where the coefficients depend on the state of the solution process as well as on its marginal law through its expected value. Necessary conditions of near-optimality are derived where the control domain is non-convex. Under some additional hypotheses, we prove that the near-maximum condition on the Hamiltonian function in integral form is a sufficient condition for "-optimality. Our result is derived by using the spike variation method, Ekeland's variational principle and some estimates of the state and adjoint processes, along with Clarke's generalised gradient for non-smooth data. This paper extends the results obtained by Zhou (1998) to a class of mean-field stochastic control problems involving mean-field FBSDEJs. As an application, mean-variance portfolio selection mixed with a recursive utility functional optimisation problem is discussed to illustrate our theoretical results.