(untitled) (Journal of Computational Mathematics, 2019)
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 O((∆t)^εexp(−1/(∆t) ^2ε)), if the distance between the start point X_0 and the boundary is at least on the order of O((∆t)^12−ε) for any fixed ε > 0. Hence, in spatial discretization, we set the mesh size ∆x ∼ O((∆t)^12−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.