Stochastic Maximum Principle for Controlled Backward Delayed System via Advanced Stochastic Differential Equation

The main contributions of this paper are three old. First, our primary concern is to investigate a class of stochastic recursive delayed control problems that naturally arise with strong backgrounds but have not been well studied yet. For illustration, some concrete examples are provided here. Second, it is interesting that a new class of time-advanced stochastic differential equations (ASDEs) is introduced as the adjoint process via duality relation. To our knowledge, such equations have never been discussed in literature, although they have considerable research values. An existence and uniqueness result for ASDEs is presented. Third, to illustrate our theoretical results, some dynamic optimization problems are discussed based on our stochastic maximum principles. It is interesting that the optimal controls are derived explicitly by solving the associated time-advanced ordinary differential equation (AODE), the counterpart of the ASDE in its deterministic setup.