On regularity properties and approximations of value functions for stochastic differential games in domains

TL;DR

This paper establishes approximation techniques for value functions in stochastic differential games, providing bounds on derivatives and Lipschitz constants, applicable in both domain-restricted and whole-space scenarios.

Contribution

It introduces a method to approximate value functions with bounded second derivatives, enhancing understanding of their regularity in stochastic differential games.

Findings

Value functions can be approximated with bounded second derivatives.

Lipschitz estimates for value functions are established in various settings.

Approximation constants depend on the parameter K.

Abstract

We prove that for any constant $K\geq1$, the value functions for time homogeneous stochastic differential games in the whole space can be approximated up to a constant over $K$ by value functions whose second-order derivatives are bounded by a constant times $K$. On the way of proving this result we prove that the value functions for stochastic differential games in domains and in the whole space admit estimates of their Lipschitz constants in a variety of settings.