Optimality conditions in a stochastic optimal control problem of integro-differential systems

The optimal control problem of stochastic systems described by ordinary linear integro-differential equations is considered. In the case of a linear control object and a linear quality functional, a necessary and sufficient optimality condition of the L.S. Pontryagin maximum principle type is obtained. Further, in the case of a nonlinear, continuously differentiable and convex quality functional, a sufficient optimality condition is established.