Title: 

Linear-Quadratic Stochastic Differential Games On Directed Networks

 

 

 

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 in a previous work. We solve explicitly for Nash equilibria in the case of a finite chain with various boundary conditions and in the case of an infinite chain. The latter case is characterized by Catalan functions and the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain. Under equilibrium the variance of a state converges in the infinite time limit. Our analysis is extended to a mixed game with directed chain and mean field interactions. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. It is also extended to the game on a deterministic tree structure. Our ongoing research concerns games with interactions on a random directed chain and a random tree structure. This is joint work with Jean-Pierre Fouque and Tomoyuki Ichiba.