[Paper Review] A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations

Anselm Hudde, Martin Hutzenthaler+1 more|arXiv (Cornell University)|Mar 20, 2019

Stochastic processes and financial applications42 references12 citations

TL;DR

This paper establishes a new stochastic Gronwall-Lyapunov inequality for multi-dimensional Itô processes under a one-sided affine-linear growth condition, enabling improved moment estimates, strong local Lipschitz continuity, strong completeness, and perturbation bounds for SDEs. The key contribution is a sharper L^p moment bound for p ≥ 2 that avoids suboptimal use of Young’s inequality, leading to tighter and more general results than prior approaches.

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.

Motivation & Objective

• To address the lack of sharp L^p moment estimates for p ≥ 2 in Itô processes under affine-linear growth conditions.
• To overcome the suboptimality of prior methods that rely on Young’s inequality in moment derivations.
• To establish stronger theoretical guarantees for SDEs, including strong local Lipschitz continuity in the initial value and strong completeness.
• To derive improved perturbation estimates for numerical approximations and stochastic models.

Proposed method

• Derives a novel stochastic Gronwall-Lyapunov inequality via Itô’s formula and a refined Gronwall-Bellman-Opial inequality, avoiding Young’s inequality.
• Applies the inequality to the p-th moment of the norm of Itô processes under a one-sided affine-linear growth condition.
• Uses an exponential integrating factor to derive an almost sure identity for the p-th moment process.
• Establishes marginal and uniform moment bounds for the supremum of the process, with explicit dependence on initial data and noise coefficients.
• Applies the main inequality to derive improved estimates for SDEs, including exponential moments, strong completeness, and perturbation bounds.
• Validates the method through applications to SDEs with non-global monotonicity, demonstrating tighter bounds than classical approaches.

Experimental results

Research questions

• RQ1 Can a sharper L^p moment bound be derived for multi-dimensional Itô processes with p ≥ 2 under affine-linear growth, avoiding suboptimal use of Young’s inequality?
• RQ2 How can the stochastic Gronwall-Lyapunov inequality improve the analysis of strong local Lipschitz continuity in the initial value of SDE solutions?
• RQ3 What improvements does the new inequality bring to exponential moment estimates and strong completeness of SDEs?
• RQ4 Can the inequality yield new, tighter perturbation estimates for numerical schemes like tamed Euler methods?
• RQ5 To what extent does the new bound extend beyond deterministic coefficients to random, adapted coefficients in SDEs?

Key findings

• The paper establishes a new stochastic Gronwall-Lyapunov inequality that yields tighter L^p moment bounds for p ≥ 2 than previous methods, avoiding suboptimal estimates from Young’s inequality.
• For SDEs satisfying a one-sided affine-linear growth condition, the new bound improves exponential moment estimates and provides stronger control over solution paths.
• The inequality enables a significant improvement in the analysis of strong local Lipschitz continuity in the initial value, even without global monotonicity.
• The method yields new, uniform perturbation estimates (e.g., in Corollary 3.11) that are tighter and more general than existing results, especially for tamed Euler schemes.
• The framework allows for the derivation of twice continuously differentiable solutions to SDEs without requiring global monotonicity of coefficients.
• The theoretical results are validated through applications to SDEs with super-linearly growing coefficients and SPDEs, where classical methods fail.