Bounded variation singular stochastic control and associated Dynkin game

We consider an optional control problem for a one dimensional Itô diffusion and a stochastic game of optimal stopping associated with it. Their value functions satisfy .... and an optimal control defines a sadddle point for the game. This extends earlier results to the case of bounded variation control and general nonadditive cost functionals in the form of a controlled FBSDE. Our approach uses probabilistic methods such as comparison theorems, and a pathwise construction of policies