We consider a non zero sum stochastic differential game with a maximum n players, where 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 new players 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 special coupling in the zero order terms. We provide an existence result for solutions in appropriate spaces that allow to construct Nash optimal feedback controls. The paper is related to a previous result in a similar setting for two players leading to a parabolic system of Bellman equations . Here, we study the elliptic case (infinite horizon) and present the generalisation to more than two players.
Systems of PDE; L∞ estimates; regularity; stochastic differential games; controlled birth/death processes;
Alain Bensoussan, Jens Frehse et Christine Grün, « Stochastic differential games with a varying number of players », Communications on Pure and Applied Analysis, vol. 13, septembre 2014, p. 1719–1736.
Communications on Pure and Applied Analysis, vol. 13, septembre 2014, p. 1719–1736