On the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains

TL;DR

This paper proves the dynamic programming principle for nondegenerate stochastic differential games within domains, accommodating arbitrary stopping times and minimal regularity assumptions, advancing the theoretical foundation of stochastic game analysis.

Contribution

It extends the dynamic programming principle to more general stopping times and weaker regularity conditions in stochastic differential games.

Findings

Established the DPP for stochastic differential games with measurable coefficients

Allowed arbitrary and randomized stopping times in the framework

Provided a foundation for future removal of regularity assumptions

Abstract

We prove the dynamic programming principe for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. The zeroth-order "coefficient" and the "free" term are only assumed to be measurable. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. The main assumption, which will be removed in a subsequent article, is that there exists a sufficiently regular solution of the Isaacs equation.