Asymptotic $C^{1,\gamma}$-regularity for value functions to uniformly elliptic dynamic programming principles

TL;DR

This paper establishes an asymptotic $C^{1,eta}$ regularity estimate for value functions associated with uniformly elliptic dynamic programming principles, enabling the derivation of regularity results for the limit PDE.

Contribution

It introduces a novel asymptotic regularity estimate for value functions in elliptic dynamic programming, bridging discrete and continuous PDE regularity.

Findings

Proves an asymptotic $C^{1,eta}$ estimate for value functions.

Enables passage to the limit with discrete gradients.

Derives $C^{1,eta}$ regularity for the limit PDE.

Abstract

In this paper we prove an asymptotic $C^{1,\gamma}$-estimate for value functions of stochastic processes related to uniformly elliptic dynamic programming principles. As an application, this allows us to pass to the limit with a discrete gradient and then to obtain a $C^{1,\gamma}$-result for the corresponding limit PDE.