Sharp convex generalizations of stochastic Gronwall inequalities

TL;DR

This paper extends stochastic Gronwall inequalities to convex Bihari-LaSalle type, providing sharp constants and enabling new criteria for exponential moment finiteness in path-dependent SDEs, with applications to reaction-diffusion systems.

Contribution

It introduces convex generalizations of stochastic Gronwall inequalities with sharp constants, extending their applicability to path-dependent SDEs and related stochastic analysis problems.

Findings

Sharp convex stochastic Gronwall inequalities established.

New criteria for exponential moments of path-dependent SDEs derived.

Applications demonstrated in reaction-diffusion systems with noise.

Abstract

We provide generalizations of a class of stochastic Gronwall inequalities that has been studied by von Renesse and Scheutzow (2010), Scheutzow (2013), Xie and Zhang (2020) and Mehri and Scheutzow (2021). This class of stochastic Gronwall inequalities is a useful tool for SDEs. Our focus are convex generalizations of the Bihari-LaSalle type. The constants we obtain are sharp. In particular, we provide new sharp constants for the stochastic Gronwall inequalities. The proofs are connected to a domination inequality by Lenglart (1977), an inequality by Pratelli (1976) and a characterization of Lenglart's concept of domination via the Snell envelope. The inequalities we study appear for example in connection with exponential moments of solutions to path-dependent SDEs: For non-path-dependent SDEs, criteria for the finiteness of exponential moments are known. To be able to extend these proofs to the path-dependent case, a convex generalization of a stochastic Gronwall inequality seems necessary. Using the results of this paper, we obtain a criterion for the finiteness of exponential moments which is similar to that known for non-path-dependent SDEs. Stochastic Gronwall inequalities can also be applied to study other types of SDEs than path-dependent SDEs: An estimate of this paper is applied by Agresti and Veraar (2023) to prove global well-posedness for reaction-diffusion systems with transport noise.