Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

TL;DR

This paper proves the existence of smooth solutions to the Hamilton-Jacobi-Bellman-Isaacs system in nonzero-sum stochastic differential games, establishing the regularity of Nash payoffs in unbounded domains.

Contribution

It demonstrates the existence of smooth solutions to the HJBI system for nonzero-sum stochastic differential games in unbounded domains, covering both continuous and discontinuous generators.

Findings

Existence of smooth solutions to HJBI system in unbounded domains.

Smooth Nash payoffs derived from the HJBI solutions.

Applicable to both continuous and discontinuous generators.

Abstract

In this paper we deal with the problem of existence of a smooth solution of the Hamilton-Jacobi-Bellman-Isaacs (HJBI for short) system of equations associated with nonzero-sum stochastic differential games. We consider the problem in unbounded domains either in the case of continuous generators or for discontinuous ones. In each case we show the existence of a smooth solution of the system. As a consequence, we show that the game has smooth Nash payoffs which are given by means of the solution of the HJBI system and the stochastic process which governs the dynamic of the controlled system.