Nash equilibrium payoffs for stochastic differential games with jumps and coupled nonlinear cost functionals

TL;DR

This paper studies Nash equilibrium payoffs in two-player stochastic differential games with jumps, using coupled backward stochastic differential equations and PDEs, extending previous work with new dynamic programming principles.

Contribution

It introduces a dynamic programming principle for stopping times in jump-diffusion games and characterizes Nash equilibria via viscosity solutions of coupled PDEs.

Findings

Existence and characterization of Nash equilibrium payoffs.

Lower and upper value functions are viscosity solutions of coupled PDEs.

Generalization of previous results to jump-diffusion settings.

Abstract

In this paper we investigate Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games whose cost functionals are defined by a system of coupled backward stochastic differential equations. We obtain an existence theorem and a characterization theorem for Nash equilibrium payoffs. For this end the problem is described equivalently by a stochastic differential game with jumps. But, however, unlike the work by Buckdahn, Hu and Li [9], here the important tool of a dynamic programming principle for stopping times has to be developed. Moreover, we prove that the lower and upper value functions are the viscosity solutions of the associated coupled systems of PDEs of Isaacs type, respectively. Our results generalize those by Buckdahn, Cardaliaguet and Rainer [7] and by Lin [16].