On Variant Reflected Backward SDEs, with Applications

We study a new type of reflected backward stochastic differential equations (RBSDEs), where the reflecting process enters the drift in a nonlinear manner. This type of the reflected BSDEs is based on a variance of the Skorohod problem studied recently by Bank and El Karoui (2004), and is hence named the “Variant Reflected BSDEs” (VRBSDE) in this paper. The special nature of the Variant Skorohod problem leads to a hidden forward-backward feature of the BSDE, and as a consequence this type of BSDE cannot be treated in a usual way. We shall prove that in a small-time duration most of the well-posedness, comparison, and stability results are still valid, although some extra conditions on the boundary process are needed. We will also provide some possible applications where the VRBSDE can be potentially useful. These applications show that the VRBSDE could become a novel tool for some problems in finance and optimal stopping problems where no existing methods can be easily applicable.