Stochastic maximum principle for discrete time mean‐field optimal control problems

Abstract

This article studies optimal control of a discrete‐time stochastic differential equation of mean‐field type with coefficients dependent on function of the law and state of the process. A new version of the maximum principle for discrete‐time mean‐field type stochastic optimal control problems is established, using new discrete‐time mean‐field backward stochastic equations. The cost functional is also of mean‐field type. The study derives necessary first‐order and sufficient optimality conditions using adjoint equations that take the form of discrete‐time backward stochastic differential equations with a mean‐field component. An optimization problem for production and consumption choice is used as an example.