Concave and other generalizations of stochastic Gronwall inequalities

TL;DR

This paper extends stochastic Gronwall inequalities to include concave and other nonlinear forms, aiding in proving existence and uniqueness of solutions for complex path-dependent SDEs driven by Lévy processes.

Contribution

It introduces new nonlinear generalizations of stochastic Gronwall inequalities, specifically focusing on concave and other forms, expanding their applicability in stochastic differential equations.

Findings

Developed concave and other nonlinear stochastic Gronwall inequalities.

Applied these inequalities to establish existence and uniqueness of solutions for path-dependent SDEs.

Enhanced tools for analyzing SDEs with non-Lipschitz conditions.

Abstract

We provide nonlinear generalizations of a class of stochastic Gronwall inequalities that have 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. More precisely, we study generalizations of the Bihari-LaSalle type. Whilst in a closely connected article by the author convex generalizations are studied, we investigate here concave and other generalizations. These types of estimates are useful to obtain existence and uniqueness of global solutions of path-dependent SDEs driven by L\'evy processes under one-sided non-Lipschitz monotonicity and coercivity assumptions.