# A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations

## TL;DR

This paper extends the stochastic Gronwall inequality to provide bounds for higher moments of Itô processes, leading to improved results in SDE theory such as Lipschitz continuity, moment estimates, and stability under perturbations.

## Contribution

It introduces upper bounds for p-th moments of Itô processes with p≥2, complementing existing results for p∈(0,1), and applies these to enhance SDE analysis.

## Key findings

- Improved moment bounds for Itô processes with p≥2.
- Enhanced strong local Lipschitz continuity results for SDE solutions.
- Better perturbation and stability estimates for SDEs.

## Abstract

There are numerous applications of the classical (deterministic) Gronwall inequality. Recently, Michael Scheutzow discovered a stochastic Gronwall inequality which provides upper bounds for $p$-th moments, $p\in(0,1)$, of the supremum of nonnegative scalar continuous processes which satisfy a linear integral inequality. In this article we complement this with upper bounds for $p$-th moments, $p\in[2,\infty)$, of the supremum of general It\^o processes which satisfy a suitable one-sided affine-linear growth condition. As example applications, we improve known results on strong local Lipschitz continuity in the starting point of solutions of stochastic differential equations (SDEs), on (exponential) moment estimates for SDEs, on strong completeness of SDEs, and on perturbation estimates for SDEs.