  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 stochastic optimal
control problems. Moreover, the cost functional is also of the mean-field type.
This maximum principle differs from the classical principle since we introduce
new discrete-time backward (matrix) stochastic equations. Based on the
discrete-time 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
optimal control problem. To verify, we apply the result to production and
consumption choice optimization problem.
