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

Abstract: In this paper, we study the existence, uniqueness and comparison theorems for the bounded solutions and unbounded solutions of reflected backward stochastic differential equations (RBSDEs) and backward stochastic differential equations (BSDEs), whose generators have a one-sided growth restriction on $y$ and a general quadratic growth in $z$, and whose solutions $Y_t$ (and obstacles) take values in $(a,\infty)$ with $a\geq-\infty$. The existence of solutions was obtained mainly by using the methods from Essaky and Hassani (2011) and Bahlali et al. (2017), and some well-posedness results of ODEs. We provide a method for the uniqueness of solutions of such RBSDEs and BSDEs, whose generators satisfy a $\theta$-domination condition. This condition contains the generators which are convex in $(y,z)$ or are (locally) Lipschitz in $y$ and convex in $z$, and some generators which are not convex. Our method relies on the $\theta$-difference technique borrowed from Briand and Hu (2008), and some comparison arguments based on RBSDEs. We also provide a method to study the comparison theorems for such RBSDEs and BSDEs.