H\"older regularity for stochastic processes with bounded and measurable increments

TL;DR

This paper establishes H"older regularity estimates for expectations of general discrete stochastic processes, linking probabilistic and PDE techniques, and highlighting effects of discretization step size.

Contribution

It extends Krylov-Safonov regularity results to discrete stochastic processes and discretized PDEs with bounded, measurable increments, incorporating Pucci-type inequalities.

Findings

H"older estimates hold for a broad class of discrete processes

Discretization step size influences regularity results

Analytic and probabilistic methods are combined in proofs

Abstract

We obtain an asymptotic H\"older estimate for expectations of a quite general class of discrete stochastic processes. Such expectations can also be described as solutions to a dynamic programming principle or as solutions to discretized PDEs. The result, which is also generalized to functions satisfying Pucci-type inequalities for discrete extremal operators, is a counterpart to the Krylov-Safonov regularity result in PDEs. However, the discrete step size $\varepsilon$ has some crucial effects compared to the PDE setting. The proof combines analytic and probabilistic arguments.