Linear-Quadratic Stochastic Differential Games on Directed Chain Networks

Yichen Feng, Jean-Pierre Fouque, Tomoyuki Ichiba

Department of Statistics and Applied Probability, South Hall, University of California, Santa Barbara, CA 93106, USA.

Abstract: We study linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque and Ichiba. We solve explicitly for Nash equilibria with a finite number of players and we study more general finite-player games with a mixture of both directed chain interaction and mean field interaction. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. The limit is characterized by Catalan functions and the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain, with or without the presence of mean field interaction.

Keywords: Linear-quadratic stochastic games, directed chain network, Nash equilibrium, Catalan functions, Catalan Markov chain, mean field games.

Pages: 25 – 67 | Full PDF Paper