Stochastic differential games with a variable number of players

Abstract

In this talk we consider an infinite horizon non zero sum stochastic differential game with a maximum of two players. The players control a diffusion in order to minimise a certain cost functional. During the game it is possible that present players may die or a new player may appear. The death, respectively the birth time of a player is exponentially distributed with intensities that depend on the diffusion and the controls of the players who are alive. We show how the game is related to a system of partial differential equations with a quadratic growth condition on the first order terms and a special coupling in the zero order terms. We provide an existence result for solutions in adequate spaces that allow to construct Nash optimal feedback controls. (With Alain Bensoussan, Jens Frehse)