A First-Order Numerical Scheme for Forward-Backward Stochastic Differential Equations in Bounded Domains

Abstract

We propose a novel numerical scheme for decoupled forward-backward stochastic differential equations (FBSDEs) in bounded domains, which corresponds to a class of nonlinear parabolic partial differential equations with Dirichlet boundary conditions. The key idea is to exploit the regularity of the solution ($Y_t$, $Z_t$) with respect to $X_t$ to avoid direct approximation of the involved random exit time. Especially, in the one-dimensional case, we prove that the probability of $X_t$ exiting the domain within $∆t$ is on the order of $\mathcal{O}((∆t)^ε$exp($−1/(∆t) ^{2ε})$), if the distance between the start point $X_0$ and the boundary is at least on the order of $\mathcal{O}((∆t)^{\frac{1}{2}−ε})$ for any fixed $ε > 0$. Hence, in spatial discretization, we set the mesh size $∆x ∼ \mathcal{O}((∆t)^{\frac{1}{2}−ε})$, so that all the interior grid points are sufficiently far from the boundary, which makes the error caused by the exit time decay sub-exponentially with respect to $∆t$. The accuracy of the approximate solution near the boundary can be guaranteed by means of high-order piecewise polynomial interpolation. Our method is developed using the implicit Euler scheme and cubic polynomial interpolation, which leads to an overall first-order convergence rate with respect to $∆t$.