Dynkin Games via Dirichlet Forms and Singular Control of One-Dimensional Diffusions

We consider a zero-sum game of optimal stopping in which each of the opponents has the right to stop a one dimensional diffusion process. There are two types of costs. The first is accumulated continuously at the rate H(Xt) where Xt is the current position of the process. In addition there is a cost associated with the stopping of the process. It is given by the function f1(x) for the first player and the function f2(x) for the second player, where x is the position of the process when the stopping option is exercised. We study the solution of the free boundary problem associated with this game via Dirichlet forms on the appropriate functional space. Integrating the value function of the game we get a solution to another free boundary problem which yields the optimal return function for a singular stochastic control problem.