On the value of a general stochastic differential game with ergodic payoff

In this paper we study a type of zero-sum stochastic differential games with ergodic payoff in which the diffusion system does not need to be non-degenerate. We first show the existence of a viscosity solution of the associated ergodic Hamilton-Jacobi-Bellman-Isaacs equation under the dissipativity condition. With the help of this viscosity solution, we then give the estimates for the upper and lower ergodic value functions by constructing a series of non-degenerate approximating processes and using the sup- and inf-convolution techniques. Finally, we obtain the existence of a value under the Isaacs condition and its representation formulae. In addition, we apply our results to study a type of pollution accumulation problems with the long-run average social welfare.