TY - JOUR
T1 - A First-Order Numerical Scheme for Forward-Backward Stochastic Differential Equations in Bounded Domains
AU - Yang , Jie
AU - Zhang , Guannan
AU - Zhao , Weidong
JO - Journal of Computational Mathematics
VL - 2
SP - 237
EP - 258
PY - 2018
DA - 2018/04
SN - 36
DO - http://doi.org/10.4208/jcm.1612-m2016-0582
UR - https://global-sci.org/intro/article_detail/jcm/12257.html
KW - Forward-backward stochastic differential equations, Exit time, Dirichlet boundary conditions, Implicit Euler scheme.
AB - <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>