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.

Keywords

Backward stochastic differential equation / Comparison theorem / Quadratic growth / Viscosity solution / American option

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