A discrete stochastic Gronwall Lemma

TL;DR

This paper develops a discrete stochastic version of the Gronwall Lemma, providing a key tool for analyzing discrete stochastic processes and numerical methods like the backward Euler-Maruyama scheme.

Contribution

It introduces a novel discrete stochastic Gronwall Lemma and applies it to derive an a priori estimate for the backward Euler-Maruyama method.

Findings

Established a discrete stochastic Gronwall Lemma

Provided an a priori estimate for the backward Euler-Maruyama method

Extended deterministic inequalities to stochastic discrete processes

Abstract

We derive a discrete version of the stochastic Gronwall Lemma found in [Scheutzow, IDAQP, 2013]. The proof is based on a corresponding deterministic version of the discrete Gronwall Lemma and an inequality bounding the supremum in terms of the infimum for time discrete martingales. As an application the proof of an a priori estimate for the backward Euler-Maruyama method is included.