Discrete Gronwall inequalities for demimartingales

TL;DR

This paper develops discrete stochastic Gronwall inequalities for demimartingales, extending previous martingale results, and applies these to bound errors in the backward Euler-Maruyama numerical scheme.

Contribution

It introduces new discrete inequalities for demimartingales, generalizing existing martingale inequalities, with an application to numerical scheme error estimation.

Findings

Derived discrete Gronwall inequalities for demimartingales

Extended previous martingale inequalities to a broader class

Provided an a priori error bound for the backward Euler-Maruyama scheme

Abstract

The aim of this work is to obtain discrete versions of stochastic Gronwall inequalities involving demimartingale sequences. The results generalize the respective theorems for martingales provided by Kruse and Scheutzow (2018) and Hendy et al. (2022). Moreover, we present an application which provides an upper bound for the a priori estimate of the backward Euler-Maruyama numerical scheme.