@Article{JCM-36-2,
author = {Yang, Jie and Zhang, Guannan and Zhao, Weidong},
title = {A First-Order Numerical Scheme for Forward-Backward Stochastic Differential Equations in Bounded Domains},
journal = {Journal of Computational Mathematics},
year = {2018},
volume = {36},
number = {2},
pages = {237--258},
abstract = {<p style="text-align: justify;">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&nbsp;$\mathcal{O}((∆t)^ε$exp($−1/(∆t) ^{2ε})$), if the distance between the start point $X_0$ and the boundary is
at least on the order of&nbsp;$\mathcal{O}((∆t)^{\frac{1}{2}−ε})$ for any fixed $ε &gt; 0$. 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 $∆t$.</p>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.1612-m2016-0582},
url = {https://global-sci.com/article/84464/a-first-order-numerical-scheme-for-forward-backward-stochastic-differential-equations-in-bounded-domains}
}