Value existence for zero-sum ergodic stochastic differential games
Juan Li, Wenqiang Li, Yanwei Li, Huaizhong Zhao
TL;DR
This work tackles two-player zero-sum stochastic differential games with ergodic payoff under potentially degenerate diffusion. It develops a vanishing discount approach to establish existence of viscosity solutions to ergodic HJBI equations, and introduces non-degenerate approximations together with sup-/inf-convolution techniques to bound the ergodic value functions. Under the Isaacs condition, it proves that the upper and lower ergodic values converge to a common constant and furnishes representation formulas and a dynamic programming principle for the ergodic problem. An application to pollution accumulation with long-run average welfare demonstrates the practical relevance and effectiveness of the theoretical framework.
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.