Bounded Variation Singular Stochastic Control and Dynkin Game

We consider a bounded variation singular stochastic control problem with value V in a general situation with control of a diffusion and nonlinear cost functional defined as solution to a backward stochastic differential equation (BSDE). Associated with this is a Dynkin game with value u. We establish the well-known relation frac (partial)/(partial)x V=u for this general situation. A saddle point for the Dynkin game is given by the pair of first action times of an optimal control. The methods are from stochastic analysis and include a priori estimates, pathwise construction, and comparison theorems for forward stochastic differential equations (FSDE) and BSDE.