Quadratic Reflected BSDEs with Unbounded Obstacles

TL;DR

This paper studies quadratic reflected backward stochastic differential equations with unbounded obstacles and terminal conditions, establishing fundamental properties and applications to obstacle problems for semi-linear PDEs with quadratic gradient terms.

Contribution

It provides existence, comparison, and stability results for RBSDEs with unbounded obstacles and quadratic growth, and applies these to obstacle problems for semi-linear PDEs.

Findings

Existence of solutions for quadratic RBSDEs with unbounded obstacles.

Comparison and stability results for these RBSDEs.

Application to obstacle problems in semi-linear PDEs with quadratic gradient terms.

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.