Stochastic Gronwall's inequality in random time horizon and its application to BSDE

TL;DR

This paper extends Gronwall's inequality to stochastic settings with random time horizons and applies it to establish comparison results for BSDEs with random terminal times.

Contribution

It introduces a stochastic Gronwall's inequality applicable to unbounded random time horizons and uses it to prove a comparison theorem for BSDEs with stochastic monotonicity.

Findings

Established a stochastic Gronwall's inequality for unbounded random time horizons.

Proved a comparison theorem for BSDEs with random terminal time.

Demonstrated the inequality's application in stochastic analysis.

Abstract

In this paper, we introduce and prove a stochastic Gronwall's inequality in (unbounded) random time horizon. As an application, we prove a comparison theorem for backward stochastic differential equation (BSDE for short) with random terminal time under stochastic monotonicity condition.