A mean-field maximum principle for optimal control of forward–backward stochastic differential equations with Poisson jump processes

We consider mean-field type stochastic optimal control problems for systems governed by special mean-field forward–backward 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. Necessary conditions of optimal control for these systems in the form of a maximum principle are established by means of spike variation techniques. Our result differs from the classical one in the sense that here the adjoint equation has a mean-field type. The control domain is not assumed to be convex. © 2013, Springer-Verlag Berlin Heidelberg.