Stochastic maximum principle for discrete time mean-field optimal
In this paper, we study the optimal control of a discrete-time
stochastic differential equation (SDE) of mean-field type, where the
coefficients can depend on both a function of the law and the state of
the process. We establish a new version of the maximum principle for
discrete-time mean-field type stochastic optimal control problems.
Moreover, the cost functional is also of the mean-field type. This
maximum principle differs from the classical principle one since we
introduce new discrete-time mean-field backward (matrix) stochastic
equations. Based on the discrete-time mean-field backward stochastic
equations where the adjoint equations turn out to be discrete backward
SDEs with mean field, we obtain necessary first-order and sufficient
optimality conditions for the stochastic discrete mean-field optimal
control problem. To verify, we apply the result to production and
consumption choice optimization problem.