Volume 36, Issue 2
A First-Order Numerical Scheme for Forward-Backward Stochastic Differential Equations in Bounded Domains

J. Comp. Math., 36 (2018), pp. 237-258.

Published online: 2018-04

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• 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 $∆t$.

• Keywords

Forward-backward stochastic differential equations, Exit time, Dirichlet boundary conditions, Implicit Euler scheme.

60H35, 60H10, 65C20, 65C30

jieyang@sdu.edu.cn (Jie Yang)

zhangg@ornl.gov (Guannan Zhang)

wdzhao@sdu.edu.cn (Weidong Zhao)

• BibTex
• RIS
• TXT
@Article{JCM-36-237, 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 = {

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 $∆t$.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1612-m2016-0582}, url = {http://global-sci.org/intro/article_detail/jcm/12257.html} }
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 -

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 $∆t$.

Jie Yang, Guannan Zhang & Weidong Zhao. (2020). A First-Order Numerical Scheme for Forward-Backward Stochastic Differential Equations in Bounded Domains. Journal of Computational Mathematics. 36 (2). 237-258. doi:10.4208/jcm.1612-m2016-0582
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