Maximum principle for discrete-time stochastic optimal control problem under distribution uncertainty

In this paper, we study a discrete-time stochastic optimal control problem under distribution uncertainty with convex control domain. By weak convergence method and Sion's minimax theorem, we obtain the variational inequality for cost functional under a reference probability $P^{\ast}$. Moreover, under the square integrability condition for noise and control, we establish the discrete-time stochastic maximum principle under $P^{\ast}$. Finally, we introduce a backward algorithm to calculate the reference probability $P^{\ast}$ and the optimal control $u^{\ast}$.