A Stochastic Gronwall Lemma

TL;DR

This paper establishes a stochastic version of Gronwall's inequality, providing bounds on the moments of solutions to certain stochastic integral inequalities using martingale techniques.

Contribution

It introduces a stochastic Gronwall lemma with explicit bounds and offers a simplified proof of a key martingale inequality with near-optimal constants.

Findings

Bound on the p-th moment of the supremum of Z in terms of H

Explicit numerical bounds for martingale inequality constants

Simplified proof technique for martingale inequalities

Abstract

We prove a stochastic Gronwall lemma of the following type: if $Z$ is an adapted nonnegative continuous process which satisfies a linear integral inequality with an added continuous local martingale $M$ and a process $H$ on the right hand side, then for any $p \in (0,1)$ the $p$-th moment of the supremum of $Z$ is bounded by a constant $\kappa_p$ (which does not depend on $M$) times the $p$-th moment of the supremum of $H$. Our main tool is a martingale inequality which is due to D. Burkholder. We provide an alternative simple proof of the martingale inequality which provides an explicit numerical value for the constant $c_p$ appearing in the inequality which is at most four times as large as the optimal constant.