Quadratic Reflected BSDEs with Unbounded Obstacles

Abstract

In this paper, we analyze a real-valued reflected backward stochastic differential equation (RBSDE) with an unbounded obstacle and an unbounded terminal condition when its generator $f$ has quadratic growth in the $z$-variable. In particular, we obtain existence, comparison, and stability results, and consider the optimal stopping for quadratic $g$-evaluations. As an application of our results we analyze the obstacle problem for semi-linear parabolic PDEs in which the non-linearity appears as the square of the gradient. Finally, we prove a comparison theorem for these obstacle problems when the generator is convex or concave in the $z$-variable.

Key Words: Quadratic reflected backward stochastic differential equations, convex/concave generator, $þ$-difference method, Legenre-Fenchel duality, optimal stopping problems for quadratic $g$-evaluations, stability, obstacle problems for semi-linear parabolic PDEs, viscosity solutions