Communications in Mathematical Sciences

Volume 13 (2015)

Number 3

Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

On a system of PDEs associated to a game with a varying number of players

Pages: 623 – 639

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n3.a2

Authors

Alain Bensoussan (University of Texas at Dallas, Richardson, Texas, U.S.A.; and City University of Hong Kong, Kowloon Tong, Hong Kong)

Jens Frehse (Institute for Applied Mathematics, University of Bonn, Germany)

Christine Grün (Université de Toulouse, France)

Abstract

We consider a general Bellman type system of parabolic partial differential equations with a special coupling in the zero order terms. We show the existence of solutions in $L^p((0,T); W^{2,p} (\mathcal{O})) \cap W{1,p} ((0,T) \times \mathcal{O})$ by establishing suitable a priori bounds. The system is related to a certain 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. We assume that 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.

Keywords

Bellman systems, regularity for PDEs, Nash points, stochastic differential games

2010 Mathematics Subject Classification

35B45, 35B65, 35J47, 49N70, 91A15, 91A23

Published 3 March 2015