Second-order necessary optimality conditions for discrete-time stochastic systems

This paper deals with the second-order necessary optimality conditions for discrete-time stochastic optimal control problems under weakened convexity assumptions. Using a special variation of the control, and by virtue of a new discrete-time backward stochastic equation, we establish a more general and constructive first-order necessary optimality condition in the form of a global stochastic maximum principle. Moreover, by introducing a new discrete-time backward stochastic matrix equation, the second-order multipoint necessary optimality conditions of singular controls are derived, which covers and improves the classical second-order necessary optimality conditions of discrete-time stochastic systems.