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

TL;DR

This paper establishes the dynamic programming principle for nondegenerate stochastic differential games in domains, allowing arbitrary stopping times without assuming solvability of the Isaacs equation, and proves the uniqueness of value functions.

Contribution

It extends the dynamic programming principle to more general stopping times and removes the need for Isaacs equation solvability assumptions.

Findings

Dynamic programming principle holds for arbitrary stopping times.

Value functions are uniquely determined by Isaacs equations.

No assumptions on classical or viscosity solutions of Isaacs equations.

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. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. There is no assumption about solvability of the the Isaacs equation in any sense (classical or viscosity). The zeroth-order "coefficient" and the "free" term are only assumed to be measurable in the space variable. We also prove that value functions are uniquely determined by the functions defining the corresponding Isaacs equations and thus stochastic games with the same Isaacs equation have the same value functions.