$\varepsilon$-Nash equilibrium in stochastic differential games with mean-field interaction and controlled jumps

TL;DR

This paper studies a complex stochastic differential game with jumps and mean-field interactions, establishing the existence of approximate Nash equilibria for large player populations and extending classical mean-field game results.

Contribution

It extends classical mean-field game theory to include controlled jumps and provides convergence rates for approximate Nash equilibria in this setting.

Findings

Constructs approximate Nash equilibria for large n-player games with jumps.

Proves convergence of the n-player game to the mean-field limit.

Extends classical results to games with controlled jumps.

Abstract

We consider a symmetric $n$-player nonzero-sum stochastic differential game with controlled jumps and mean-field type interaction among the players. Each player minimizes some expected cost by affecting the drift as well as the jump part of their own private state process. We consider the corresponding limiting mean-field game and, under the assumption that the latter admits a regular Markovian solution, we prove that an approximate Nash equilibrium for the $n$-player game can be constructed for $n$ large enough, and provide the rate of convergence. This extends to a class of games with controlled jumps classical results in mean-field game literature. This paper complements our previous work, where in particular the existence of a mean-field game solution was investigated.