On Solving Finitely Reflected Backward Stochastic Differential Equations

Abstract

Classical theory gives a closed form representation of the density p(t,x), a solution to a linear parabolic PDE, via the Feynman-Kac Formula of the underlying diffusion process. In the non-linear PDE case there is no closed form representation for p(t,x) and instead one solves a SDE running back in time whose initial (deterministic value) coincides with p(t,x). This method of solving semi-linear parabolic PDEs is an effective alternative to known numerical schemes. Furthermore, the FBSDE approach allows for treatment of non-smooth coefficients in the PDE that cannot be handled by classical deterministic methods. One of the most important extensions of BSDEs is that of adding reflections. Roughly speaking, the solution of a Reflected BSDE (RBSDE) is forced to remain within some region by a so-called reflection process. We prove the existence and uniqueness of FR-FBSDE (Finitely Reflected Forward Backward SDE) along with a Donsker-type computational algorithm for effective approximate solution. Applications to option pricing in finance serve as an illustration of our results.