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This paper deals with the first-order and second-order necessary optimality conditions concerning the components for discrete-time stochastic systems under weakened convexity assumptions. By means of a new discrete-time backward stochastic equation, we establish a first-order necessary optimality condition concerning every component in the form of a global stochastic maximum principle. Besides, we also obtain an optimality condition concerning one component of vector control in the form of a global stochastic maximum principle and concerning the other in the form of linearised stochastic maximum principle. Moreover, by introducing a new discrete-time backward stochastic matrix equation, we derive diverse second-order necessary optimality conditions of singular and quasi-singular controls concerning the components.

First-order and second-order necessary optimality conditions concerning components for discrete-time stochastic systems