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

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.

33 pages. Compared to the previous version, Equation (1.5) (i.e., Assumption (4A1-(iii))) has been slightly revised for clarity