A Stochastic Gronwall Lemma and Well-Posedness of Path-Dependent SDEs Driven by Martingale Noise

TL;DR

This paper establishes existence and uniqueness of solutions for path-dependent stochastic differential equations driven by martingale noise, introducing a novel stochastic Gronwall lemma to handle the challenges posed by cadlag martingales.

Contribution

It presents a new stochastic Gronwall lemma for cadlag martingales and applies it to prove well-posedness of path-dependent SDEs under minimal assumptions.

Findings

Existence and uniqueness of solutions established

New stochastic Gronwall lemma proved for cadlag martingales

Method extends analysis of path-dependent SDEs with martingale noise

Abstract

We show existence and uniqueness of solutions of stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity and coercivity assumptions on the coefficients with a bound in terms of the supremum norm. In this set-up, the usual proof using the ordinary Gronwall lemma together with the Burkholder-Davis-Gundy inequality seems impossible. In order to solve this problem, we prove a new and quite general stochastic Gronwall lemma for cadlag martingales using Lenglart's inequality.