G-BSDEs with time-varying monotonicity condition

TL;DR

This paper addresses backward stochastic differential equations driven by $G$-Brownian motion whose generator has time-varying monotonicity in $y$ and Lipschitz in $z$. The authors employ a Yosida approximation to construct Lipschitz approximants $f_{alpha}$, solve the corresponding $G$-BSDEs, and derive uniform estimates to pass to the limit. They prove existence and uniqueness for the original $G$-BSDE (and for the full form with $g$) by establishing convergence in $M_G^2$ and identifying the limit as a solution. The work extends well-posedness for nonlinear $G$-BSDEs under time varying monotonicity, with potential applications in robust finance and stochastic control under model uncertainty.

Abstract

In this paper, we study backward stochastic differential equations driven by G-Brownian motion where the generator has time-varying monotonicity with respect to y and Lipsitz property with respect to z. Through the Yosida approximation, we have proved the existence and uniqueness of the solutions to these equations.

Theorems & Definitions (15)

• Theorem 2.1: hu2009representationdenis2011function
• Definition 2.2
• Theorem 2.3: hu2014backward
• Theorem 2.4: hu2014backwardhu2024BSDEsong2011some
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 4.1
• Lemma 4.2