A class of quadratic reflected BSDEs with singular coefficients

Abstract

In this paper, we study the existence and uniqueness of the solution to a reflected backward stochastic differential equation (RBSDE) with the generator $g(t,y,z)=G_f^F(t,y,z)+f(y)|z|^2$, where $f(y)$ is a locally integrable function defined on an open interval $D$, and $G_f^F(t,y,z)$ is induced by $f$ and a Lipschitz continuous function $F$. Both the solution $Y_t$ and the obstacle $L_t$ of this RBSDE take values in $D$. As applications, we provide a probabilistic interpretation of an obstacle problem for a quadratic PDE with a singular term, whose solution takes values in $D$, and study an optimal stopping problem for the payoff of American options under general utilities.