On mean-field stochastic maximum principle for near-optimal controls for Poisson jump diffusion with applications

Abstract

In this paper, we study mean-field type stochastic control problems for systems described by mean-field stochastic differential equations with jump processes, in which the coefficients contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. We derive necessary as well as sufficient conditions of near-optimality for our model, using Ekeland’s variational principle, spike variation method and some estimates of the state and adjoint processes. Under certain concavity conditions with non-negative derivatives, we prove that the near-maximum condition on the Hamiltonian function in integral form is a sufficient condition for near-optimality. Our result differs from the classical one in the sense that here the adjoint equation has a mean-field type, while the second-order adjoint equation remains the same as in the classical case. As an application, our results are applied to a mean-variance portfolio selection where explicit expression of the near-optimal portfolio selection strategy is obtained in the state feedback form involving both state process and its marginal distribution, via the solutions of Riccati ordinary differential equations.