Stochastic Recursive Optimal Control Problem with Mixed Delay under Viscosity Solution's Framework

This paper is concerned with the stochastic recursive optimal control problem with mixed delay. The connection between Pontryagin's maximum principle and Bellman's dynamic programming principle is discussed. Without containing any derivatives of the value function, relations among the adjoint processes and the value function are investigated by employing the notions of super- and sub-jets introduced in defining the viscosity solutions. Stochastic verification theorem is also given to verify whether a given admissible control is really optimal.