Backward Stochastic Differential Equations Coupled with Value Function and Related Optimal Control Problems

We get a new type of controlled backward stochastic differential equations (BSDEs), namely, the BSDEs, coupled with value function. We prove the existence and the uniqueness theorem as well as a comparison theorem for such BSDEs coupled with value function by using the approximation method. We get the related dynamic programming principle (DPP) with the help of the stochastic backward semigroup which was introduced by Peng in 1997. By making use of a new, more direct approach, we prove that our nonlocal Hamilton‐Jacobi‐Bellman (HJB) equation has a unique viscosity solution in the space of continuous functions of at most polynomial growth. These results generalize the corresponding conclusions given by Buckdahn et al. (2009) in the case without control.