A first-order numerical scheme for forward-backward stochastic differential equations in bounded domains^*

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 (Yt, Zt) with respect to Xt to avoid direct approximation of the involved random exit time. Especially, in the one-dimensional case, we prove that the probability of Xt exiting the domain within ∆t is on the order of O((∆t)^ε exp(−1/(∆t)^2ε)), if the distance between the start point X0 and the boundary is at least on the order of O((∆t) ^{21 −ε}) for any fixed ε > 0. Hence, in spatial discretization, we set the mesh size ∆x ∼ O((∆t) ^{21 −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.