Value existence for zero-sum ergodic stochastic differential games

TL;DR

This paper establishes the existence of a value and viscosity solutions for zero-sum ergodic stochastic differential games with possibly degenerate diffusion, providing theoretical foundations and applications in social welfare.

Contribution

It proves the existence of a game value and viscosity solutions under ergodic and dissipativity conditions, extending results to degenerate diffusions.

Findings

Existence of viscosity solutions for ergodic HJB-Isaacs equations.

Construction of non-degenerate approximations for degenerate diffusions.

Representation formulas for the game value.

Abstract

In this paper we investigate two-player zero-sum stochastic differential games with an ergodic payoff, in which the diffusion coefficient does not need to be non-degenerate. We first establish the existence of a viscosity solution to the associated ergodic Hamilton-Jacobi-Bellman-Isaacs equation under a dissipativity condition. With the help of this viscosity solution, we then derive estimates for the upper and the lower ergodic value functions by constructing a series of non-degenerate approximating processes combined with the sup- and inf-convolution techniques. Finally, we prove the existence of a value for the game under the Isaacs condition and provide its representation formulae. As an application, we study the pollution accumulation problem with a long-run average social welfare to illustrate our theoretical results.