Well-posedness of quadratic RBSDEs and BSDEs with one-sided growth restrictions

TL;DR

This paper studies the existence and uniqueness of solutions for quadratic RBSDEs and BSDEs with one-sided growth restrictions, introducing new methods and comparison theorems.

Contribution

It provides new existence and uniqueness results for quadratic RBSDEs and BSDEs under one-sided growth conditions, using novel techniques and comparison principles.

Findings

Established existence of solutions for bounded and unbounded cases.

Provided a method for uniqueness when generators are convex or Lipschitz in certain variables.

Developed general comparison theorems for these equations.

Abstract

In this paper, we investigate the well-posedness of bounded and unbounded solutions for reflected backward stochastic differential equations (RBSDEs) and backward stochastic differential equations (BSDEs). The generators of these equations satisfy a one-sided growth restriction on the variable $y$ and have a general quadratic growth in the variable $z$. The solutions $Y_t$ (and the obstacles for RBSDEs) take values in either $\mathbf{R}$ or $(0, \infty)$. We obtain the existence of solutions primarily by using the methods from Essaky and Hassani (2011) and Bahlali et al. (2017). For the uniqueness of solutions, we provide a method applicable when the generators are convex in $(y,z)$ or are (locally) Lipschitz in $y$ and convex in $z$. Our method relies on the $\theta$-difference technique introduced by Briand and Hu (2008), and some novel comparison arguments based on RBSDEs. We also establish some general comparison theorems for such RBSDEs and BSDEs.