#### 期刊菜单

Sharp Convergence Rates of a Class of Multistep Formulae for Backward Stochastic Differential Equations
DOI: 10.12677/AAM.2022.115269, PDF, HTML, XML, 下载: 47  浏览: 72  国家自然科学基金支持

Abstract: Since the well-posedness of its solution was established, the backward stochastic differential equa-tion has been applied in many research fields, such as stochastic optimal control, partial differential equations, financial mathematics, risk measurement, nonlinear expectation and so on. This paper studies the convergence rate of a class of multistep formulae for backward stochastic differential equations with the help of remainder of the quadrature rule over the uniform grid. Due to the ap-plication of barycentric Lagrange interpolation with Chebyshev grids, computational accuracy of the multistep formula is greatly improved. Numerical results indicate that the proposed convergence order is sharp.

1. 引言

1973年，Bismut [1] 首次引入了线性倒向随机微分方程的概念，并研究了该类方程解的存在唯一性。基于如下的BSDE，

$\left\{\begin{array}{l}-\text{d}{y}_{t}=f\left(t,{y}_{t},{z}_{t}\right)\text{d}t-{z}_{t}\text{d}{W}_{t}\\ {y}_{T}=\epsilon \end{array}$

Pardoux和Peng [2] 得到了非线性倒向随机微分方程解的存在唯一性。这一重要研究成果奠定了正倒向随机微分方程组的理论基础。从那时起，BSDE得到了许多研究者的广泛研究。在之后的研究中，通过非线性Feynman-Kac公式，Ma，Protter和Yong [3] 提出了研究正倒向随机微分方程组的四步法，该方法证明了在任意时间区间上的解的存在唯一性。1991年，Peng [4] 得到了正倒向随机微分方程(FBSDEs)和偏微分方程之间的直接关系，然后在 [5] 中，他还找到了随机控制问题的最大值原理。

2. 多步半离散格式

$\left\{\Omega ,\mathcal{F},P,{\left\{{\mathcal{F}}_{t}\right\}}_{0\le t\le T}\right\}$ 是一个完整的、滤波的概率空间，在这上面定义了一个标准的d维布朗运动 ${W}_{t}$。如果一个过程 $\left({y}_{t},{z}_{t}\right):\left[0,T\right]×\Omega \to {ℝ}^{m}×{ℝ}^{m×d}$ 它是 $\left\{{\mathcal{F}}_{t}\right\}$ -适应，平方可积而且满足(1)，

${y}_{t}=\xi +{\int }_{t}^{T}f\left(s,{y}_{s},{z}_{s}\right)\text{d}s-{\int }_{t}^{T}{z}_{s}\text{d}{W}_{s},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,T\right),$ (1)

Pardoux和Peng证明了(1)的解的存在唯一性，我们假设初值 ${y}_{t}$ 形式为 $\phi \left({W}_{t}\right)$，然后 的解 $\left({y}_{t},{z}_{t}\right)$ 能够表示为如下形式，

${y}_{t}=u\left(t,{W}_{t}\right){z}_{t}=\nabla u\left(t,{W}_{t}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall t\in \left[0,T\right).$

$\frac{\partial u}{\partial t}+\frac{1}{2}{\sum }_{i=1}^{d}\frac{{\partial }^{2}u}{\partial {x}_{i}^{2}}+f\left(t,u,\nabla u\right)=0.$

${y}_{{t}_{n}}={y}_{{t}_{n+k}}+{\int }_{{t}_{n}}^{{t}_{n+k}}f\left(s,{y}_{s},{z}_{s}\right)\text{d}s-{\int }_{{t}_{n}}^{{t}_{n+k}}{z}_{s}\text{d}{W}_{s},$ (2)

${y}_{{t}_{n}}={\mathbb{E}}_{{t}_{n}}^{x}\left[{y}_{{t}_{n+k}}\right]+{\int }_{{t}_{n}}^{{t}_{n+k}}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left(s,{y}_{s},{z}_{s}\right)\right]\text{d}s.$ (3)

$0={\mathbb{E}}_{{t}_{n}}^{x}\left[{y}_{{t}_{n+l}}\Delta {W}_{{t}_{n+l}}\right]+{\int }_{{t}_{n}}^{{t}_{n+l}}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left(s,{y}_{s},{z}_{s}\right)\Delta {W}_{S}\right]\text{d}s-{\int }_{{t}_{n}}^{{t}_{n+l}}{\mathbb{E}}_{{t}_{n}}^{x}\left[{z}_{s}\right]\text{d}s.$

${\int }_{{t}_{n}}^{{t}_{n+k}}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left(s,{y}_{s},{z}_{s}\right)\right]\text{d}s={\int }_{{t}_{n}}^{{t}_{n+k}}{P}_{{K}_{y}}^{{t}_{n},x}\left(s\right)\text{d}s+{R}_{y}^{n},$

${R}_{y}^{n}={\int }_{{t}_{n}}^{{t}_{n+k}}\left\{{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left(s,{y}_{s},{z}_{s}\right)\right]-{P}_{{K}_{y}}^{{t}_{n},x}\left(s\right)\right\}\text{d}s.$

${\int }_{{t}_{n}}^{{t}_{n+k}}{P}_{{K}_{y}}^{{t}_{n},x}\left(s\right)\text{d}s=k\Delta t{\sum }_{i=0}^{{K}_{y}}{b}_{{K}_{y},i}^{k}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left({t}_{n+i},{y}_{{t}_{n+i}},{z}_{{t}_{n+i}}\right)\right],$ (4)

${b}_{{k}_{y},i}^{k}=\frac{1}{k\Delta t}{\int }_{{t}_{n}}^{{t}_{n+k}}{\prod }_{\begin{array}{l}j=0\\ j\ne i\end{array}}^{{K}_{y}}\left(\frac{s-{t}_{n+j}}{{t}_{n+i}-{t}_{n+j}}\right)\text{d}s,$

${b}_{{k}_{y},i}^{k}=\frac{1}{k}{\int }_{0}^{t}{\prod }_{\begin{array}{l}j=0\\ j\ne i\end{array}}^{{K}_{y}}\left(\frac{s-j}{i-j}\right)\text{d}s,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,{K}_{y}.$ (5)

${y}_{{t}_{n}}={\mathbb{E}}_{{t}_{n}}^{x}\left[{y}_{{t}_{n+k}}\right]+k\Delta t{\sum }_{i=0}^{{K}_{y}}{b}_{{K}_{y},i}^{k}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left({t}_{n+i},{y}_{{t}_{n+i}},{z}_{{t}_{n+i}}\right)\right]+{R}_{y}^{n}.$ (6)

$0={\mathbb{E}}_{{t}_{n}}^{x}\left[{z}_{{t}_{n+l}}\right]+{\sum }_{i=0}^{{K}_{z}}{b}_{{K}_{z},i}^{l}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left({t}_{n+i},{y}_{{t}_{n+i}},{z}_{{t}_{n+i}}\right)\Delta {W}_{{t}_{n+i}}\right]-{\sum }_{i=0}^{{K}_{z}}{b}_{{K}_{z},i}^{l}{\mathbb{E}}_{{t}_{n}}^{x}\left[{z}_{{t}_{n+i}}\right]+{R}_{z}^{n},$ (7)

$\begin{array}{c}{R}_{z}^{n}={\int }_{{t}_{n}}^{{t}_{n+l}}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left(s,{y}_{s},{z}_{s}\right)\Delta {W}_{S}\right]\text{d}s-{\int }_{{t}_{n}}^{{t}_{n+l}}{\mathbb{E}}_{{t}_{n}}^{x}\left[{z}_{s}\right]\text{d}s+{\sum }_{i=0}^{{K}_{z}}{b}_{{K}_{z},i}^{l}{\mathbb{E}}_{{t}_{n}}^{x}\left[{z}_{{t}_{n+i}}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-l\Delta t{\sum }_{i=0}^{{K}_{z}}{b}_{{K}_{z},i}^{l}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left({t}_{n+i},{y}_{{t}_{n+i}},{z}_{{t}_{n+i}}\right)\Delta {W}_{{t}_{n+i}}\right].\end{array}$

${\mathbb{E}}_{{t}_{n}}^{x}\left[{y}_{{t}_{n+l}}\Delta {W}_{{t}_{n+l}}\right]=l\Delta t{\mathbb{E}}_{{t}_{n}}^{x}\left[{z}_{{t}_{n+l}}\right].$ (8)

3. 最优收敛阶证明

${y}_{t}=\phi \left({W}_{t}\right)+{\int }_{t}^{T}f\left(s,{y}_{s}\right)\text{d}s-{\int }_{t}^{T}{z}_{s}\text{d}{W}_{s}$ (9)

${y}_{{t}_{n}}={\mathbb{E}}_{{t}_{n}}^{x}\left[{y}_{{t}_{n+{k}_{y}}}\right]+{k}_{y}\Delta t{\sum }_{i=0}^{{K}_{y}}{b}_{{K}_{y},i}^{k}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left({t}_{n+i},{y}_{{t}_{n+i}}\right)\right]+{R}_{y}^{n},$

$0={\mathbb{E}}_{{t}_{n}}^{x}\left[{z}_{{t}_{n+1}}\right]+{\sum }_{i=0}^{{K}_{z}}{b}_{{K}_{z},i}^{l}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left({t}_{n+i},{y}_{{t}_{n+i}}\right)\Delta {W}_{{t}_{n+i}}\right]-{\sum }_{i=0}^{{K}_{z}}{b}_{{K}_{z},i}^{l}{\mathbb{E}}_{{t}_{n}}^{x}\left[{z}_{{t}_{n+i}}\right]+\frac{1}{\Delta t}{R}_{z}^{n}.$ (10)

${y}^{n}={\mathbb{E}}_{{t}_{n}}^{x}\left[{y}^{n+{K}_{y}}\right]+{K}_{y}\Delta t{\sum }_{i=0}^{{K}_{y}}{b}_{{K}_{y},i}^{{K}_{y}}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left({t}_{n+i},{y}^{n+i}\right)\right],$

$0={\mathbb{E}}_{{t}_{n}}^{x}\left[{z}^{n+1}\right]+{\sum }_{i=0}^{{K}_{z}}{b}_{{K}_{z},i}^{1}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left({t}_{n+i},{y}^{n+i}\right)\Delta {{W}^{\prime }}_{{t}_{n+i}}\right]-{\sum }_{i=0}^{{K}_{z}}{b}_{{K}_{z},i}^{1}{\mathbb{E}}_{{t}_{n}}^{x}\left[{z}^{n+i}\right].$ (11)

$|{R}_{y}^{n}|\le C{\left(\Delta t\right)}^{{k}_{y}+2}$${k}_{y}$ 为奇数时；

$|{R}_{y}^{n}|\le C{\left(\Delta t\right)}^{{k}_{y}+3}$${k}_{y}$ 为偶数时；

$|{R}_{z}^{n}|\le C{\left(\Delta t\right)}^{{k}_{z}+2}$

$\underset{0\le n\le N}{\mathrm{sup}}\mathbb{E}\left[|{y}_{{t}_{n}}-{y}^{n}|\right]\le C{\left(\Delta t\right)}^{{K}_{y}+1}$${k}_{y}$ 为奇数时；

$\underset{0\le n\le N}{\mathrm{sup}}\mathbb{E}\left[|{y}_{{t}_{n}}-{y}^{n}|\right]\le C{\left(\Delta t\right)}^{{K}_{y}+2}$${k}_{y}$ 为偶数时；

${e}_{y}^{n}={\mathbb{E}}_{{t}_{n}}^{x}\left[{e}_{y}^{n+{K}_{y}}\right]+{K}_{y}\Delta t{\sum }_{i=0}^{{K}_{y}}{b}_{{K}_{y},i}^{{K}_{y}}{\mathbb{E}}_{{t}_{n}}^{x}\left[f\left({t}_{n+i},{y}_{{t}_{n+i}}\right)-f\left({t}_{n+i},{y}^{n+i}\right)\right]+{R}_{y}^{n}.$ (12)

$|{e}_{y}^{n}|\le {\mathbb{E}}_{{t}_{n}}^{x}\left[|{e}_{y}^{n+{K}_{y}}|\right]+L{K}_{y}\Delta t{\sum }_{i=0}^{{K}_{y}}{b}_{{K}_{y},i}^{{K}_{y}}{\mathbb{E}}_{{t}_{n}}^{x}\left[|{e}_{y}^{n+i}|\right]+|{R}_{y}^{n}|,$ (13)

$\mathbb{E}\left[|{e}_{y}^{n}|\right]\le \mathbb{E}\left[|{e}_{y}^{n+{K}_{y}}|\right]+L{K}_{y}\Delta t{\sum }_{i=0}^{{K}_{y}}{b}_{{K}_{y},i}^{{K}_{y}}\mathbb{E}\left[|{e}_{y}^{n+i}|\right]+\mathbb{E}\left[|{R}_{y}^{n}|\right].$ (14)

${N}_{k}=\left[\frac{N-n}{{K}_{y}}\right]$。对于满足 $1\le s\le {N}_{k}$ 的整数s，我们也有同样的估计

$\mathbb{E}\left[|{e}_{y}^{n+\left(s-1\right){K}_{y}}|\right]\le \mathbb{E}\left[|{e}_{y}^{n+s{K}_{y}}|\right]+L{K}_{y}\Delta t{\sum }_{i=0}^{{K}_{y}}{b}_{{K}_{y},i}^{{K}_{y}}\mathbb{E}\left[|{e}_{y}^{n+\left(s-1\right){K}_{y}+i}|\right]+\mathbb{E}\left[|{R}_{y}^{n+\left(s-1\right){K}_{y}}|\right].$ (15)

$\mathbb{E}\left[|{e}_{y}^{n}|\right]\le \mathbb{E}\left[|{e}_{y}^{n+{N}_{k}{K}_{y}}|\right]+2L{K}_{y}\Delta t{\sum }_{i=0}^{{N}_{k}{K}_{y}}\mathbb{E}\left[|{e}_{y}^{n+i}|\right]+{\sum }_{i=0}^{{N}_{k}-1}\mathbb{E}\left[|{R}_{y}^{n+i{K}_{y}}|\right],$ (16)

$\left(1-2L{K}_{y}\Delta t\right)\mathbb{E}\left[|{e}_{y}^{n}|\right]\le \mathbb{E}\left[|{e}_{y}^{n+{N}_{k}{K}_{y}}|\right]+2L{K}_{y}\Delta t{\sum }_{i=n+1}^{{N}_{k}{K}_{y}}\mathbb{E}\left[|{e}_{y}^{i}|\right]+{\sum }_{i=0}^{{N}_{k}-1}\mathbb{E}\left[|{R}_{y}^{n+i{K}_{y}}|\right].$ (17)

$\mathbb{E}\left[|{e}_{y}^{n}|\right]\le {\text{e}}^{{N}_{1}T}\left(D{M}_{0}+\Delta t{K}_{y}{N}_{1}{M}_{0}\right)={\text{e}}^{{N}_{1}T}\left(D+\Delta t{K}_{y}{N}_{1}\right){M}_{0}.$

$\mathbb{E}\left[|{R}_{y}^{n+i{K}_{y}}|\right]\le C{\left(\Delta t\right)}^{{k}_{y}+2}$${k}_{y}$ 为奇数时；

$\mathbb{E}\left[|{R}_{y}^{n+i{K}_{y}}|\right]\le C{\left(\Delta t\right)}^{{k}_{y}+3}$${k}_{y}$ 为偶数时，

${\sum }_{i=0}^{{N}_{k}-1}\mathbb{E}\left[|{R}_{y}^{n+i{K}_{y}}|\right]\le N\mathbb{E}\left[|{R}_{y}^{n+i{K}_{y}}|\right]\le CN{\left(\Delta t\right)}^{{K}_{y}+2}=C{\left(\Delta t\right)}^{{K}_{y}+1}$${k}_{y}$ 为奇数时；

${\sum }_{i=0}^{{N}_{k}-1}\mathbb{E}\left[|{R}_{y}^{n+i{K}_{y}}|\right]\le N\mathbb{E}\left[|{R}_{y}^{n+i{K}_{y}}|\right]\le CN{\left(\Delta t\right)}^{{K}_{y}+3}=C{\left(\Delta t\right)}^{{K}_{y}+2}$${k}_{y}$ 为偶数时。

${M}_{0}\le C{\left(\Delta t\right)}^{{K}_{y}+1}$${k}_{y}$ 为奇数时；

${M}_{0}\le C{\left(\Delta t\right)}^{{K}_{y}+2}$${k}_{y}$ 为偶数时。

$\underset{0\le n\le N}{\mathrm{sup}}\mathbb{E}\left[|{z}_{{t}_{n}}-{z}^{n}|\right]\le C{\left(\Delta t\right)}^{\mathrm{min}\left({K}_{y}+1,{K}_{z}\right)},$

4. 数值实验

${h}^{n}=\underset{x\in {ℝ}^{q}}{\mathrm{max}}\underset{{x}_{i}\in {D}_{h}^{n}}{\mathrm{min}}|x-{x}_{i}|=\underset{x\in {ℝ}^{q}}{\mathrm{max}}dist\left(x,{D}_{h}^{n}\right).$

${y}_{i}^{n}={\stackrel{^}{\mathbb{E}}}_{{t}_{n}}^{xi}\left[{\stackrel{^}{y}}^{n+{K}_{y}}\right]+{K}_{y}\Delta t{\sum }_{j=0}^{{K}_{y}}{b}_{{K}_{y},j}^{{K}_{y}}{\stackrel{^}{\mathbb{E}}}_{{t}_{n}}^{xi}\left[f\left({t}_{n+j},{\stackrel{^}{y}}^{n+j},{\stackrel{^}{z}}^{n+j}\right)\right]+{K}_{y}\Delta t{b}_{{K}_{y},0}^{{K}_{y}}f\left({t}_{n},{y}_{i}^{n},{z}_{i}^{n}\right),$

$0={\stackrel{^}{\mathbb{E}}}_{{t}_{n}}^{xi}\left[{\stackrel{^}{z}}^{n+1}\right]+{\sum }_{j=0}^{{K}_{z}}{b}_{{K}_{z},j}^{1}{\stackrel{^}{\mathbb{E}}}_{{t}_{n}}^{xi}\left[f\left({t}_{n+j},{\stackrel{^}{y}}^{n+j},{\stackrel{^}{z}}^{n+j}\right)\Delta {{W}^{\prime }}_{{t}_{n+j}}\right]-{\sum }_{j=0}^{{K}_{z}}{b}_{{K}_{z},j}^{1}{\stackrel{^}{\mathbb{E}}}_{{t}_{n}}^{xi}\left[{\stackrel{^}{z}}^{n+j}\right]-{b}_{{K}_{z},0}^{1}{z}_{i}^{n}.$

${\stackrel{^}{\mathbb{E}}}_{{t}_{n}}^{{x}_{i}}\left[{\stackrel{^}{y}}^{n+k}\right]=\frac{1}{{\pi }^{\frac{d}{2}}}{\sum }_{j=1}^{L}{\omega }_{j}{\stackrel{^}{y}}^{n+k}\left({x}_{i}+\sqrt{2k\Delta t}{a}_{j}\right).$

$\left\{\begin{array}{l}-\text{d}{y}_{t}=\left(-{y}_{t}^{3}+2.5{y}_{t}^{2}-1.5{y}_{t}\right)\text{d}t-{z}_{t}\text{d}{W}_{t}\\ {y}_{T}=\frac{\mathrm{exp}\left({W}_{T}+T\right)}{\mathrm{exp}\left({W}_{T}+T\right)+1}\end{array}$

$\left\{\begin{array}{l}{y}_{t}=\frac{\mathrm{exp}\left({W}_{t}+t\right)}{\mathrm{exp}\left({W}_{t}+t\right)+1}\\ {z}_{t}=\frac{\mathrm{exp}\left({W}_{t}+t\right)}{{\left(\mathrm{exp}\left({W}_{t}+t\right)+1\right)}^{2}}\end{array}$

Table 1. Error and convergence rate of | y 0 − y 0 | in Example 1

Table 2. Error and convergence rate of | z 0 − z 0 | in Example 1

$\left\{\begin{array}{l}-\text{d}{y}_{t}=-{y}_{t}\left(1-{y}_{t}\right)\left(\frac{3}{4}-{y}_{t}\right)\text{d}t-{z}_{t}\text{d}{W}_{t}\\ {y}_{T}=\frac{1}{\mathrm{exp}\left(-{W}_{T}-\frac{T}{4}\right)+1}\end{array}$

$\left\{\begin{array}{l}{y}_{t}=\frac{1}{\mathrm{exp}\left(-{W}_{t}-\frac{t}{4}\right)+1}\\ {z}_{t}=\frac{\mathrm{exp}\left(-{W}_{t}-\frac{t}{4}\right)}{{\left[\mathrm{exp}\left(-{W}_{t}-\frac{t}{4}\right)+1\right]}^{2}}\end{array}$

Table 3. Error and convergence rate of | y 0 − y 0 | in Example 2

Table 4. Error and convergence rate of | z 0 − z 0 | in Example 2

5. 结论

NOTES

*通讯作者Email: 954982638@qq.com

 [1] Bismut, J.M. (1973) Conjugate Convex Functions in Optimal Stochastic Control. Journal of Mathematical Analysis and Applications, 44, 384-404. https://doi.org/10.1016/0022-247X(73)90066-8 [2] Pardoux, E. and Peng, S. (1990) Adapted Solution of a Backward Stochastic Differential Equation. Systems and Control Letters, 14, 55-61. https://doi.org/10.1016/0167-6911(90)90082-6 [3] Ma, J., Protter, P. and Yong, J. (1994) Solving For-ward-Backward Stochastic Differential Equations Explicitly—A Four Step Scheme. Probability Theory and Related Fields, 98, 339-359. https://doi.org/10.1007/BF01192258 [4] Peng, S. (1991) Probabilistic Interpretation for Sys-tems of Quasilinear Parabolic Partial Differential Equations. Stochastics and Stochastic Reports, 37, 61-74. https://doi.org/10.1080/17442509108833727 [5] Peng, S. (1990) A General Stochastic Maximum Principle for Optimal Control Problems. SIAM Journal on Control and Optimization, 28, 966-979. https://doi.org/10.1137/0328054 [6] Chevance, D. (1997) Numerical Methods for Backward Stochastic Differen-tial Equations. In: Rogers, L. and Talay, D., Eds., Numerical Methods in Finance, Cambridge University Press, Cam-bridge, 232-244. https://doi.org/10.1017/CBO9781139173056.013 [7] Bender, C. and Denk, R. (2007) A Forward Scheme for Backward SDEs. Stochastic Processes and Their Applications, 117, 1793-1812. https://doi.org/10.1016/j.spa.2007.03.005 [8] Peng, S. (1999) A Linear Approximation Algorithm Using BSDE. Pacific Economic Review, 4, 285-292. https://doi.org/10.1111/1468-0106.00079 [9] Mémin, J., Peng, S. and Xu, M.Y. (2008) Convergence of Solutions of Discrete Reflected Backward SDEs and Simulations. Acta Mathematicae Applicatae Sinica, English Series, 24, 1-18. https://doi.org/10.1007/s10255-006-6005-6 [10] Zhang, J. (2004) A Numerical Scheme for BSDEs. The Annals of Applied Probability, 14, 459-488. https://doi.org/10.1214/aoap/1075828058 [11] Cvitanic, J. and Zhang, J. (2005) The Steepest Descent Method for Forward-Backward SDEs. Electronic Journal of Probability, 10, 1468-1495. https://doi.org/10.1214/EJP.v10-295 [12] Zhao, W., Chen, L. and Peng, S. (2006) A New Kind of Accurate Nu-merical Method for Backward Stochastic Differential Equations. SIAM Journal on Scientific Computing, 28, 1563-1581. https://doi.org/10.1137/05063341X [13] Zhao, W., Wang, J. and Peng, S. (2009) Error Estimates of the θ-Scheme for Backward Stochastic Differential Equations. Discrete and Continuous Dynamical Systems-B, 12, 905-924. https://doi.org/10.3934/dcdsb.2009.12.905 [14] Zhao, W., Fu, Y. and Zhou, T. (2014) New Kinds of High-Order Multistep Schemes for Coupled Forward Backward Stochastic Differential Equations. SIAM Journal on Scientific Com-puting, 36, A1731-A1751. https://doi.org/10.1137/130941274 [15] Chassagneux, J.F. (2014) Linear Multistep Schemes for BSDEs. SIAM Journal on Numerical Analysis, 52, 2815-2836. https://doi.org/10.1137/120902951 [16] Chassagneux, J.F. and Crisan, D. (2014) Runge-Kutta Schemes for Back-ward Stochastic Differential Equations. The Annals of Applied Probability, 24, 679-720. https://doi.org/10.1214/13-AAP933 [17] Tang, T., Zhao, W. and Zhou, T. (2017) Deferred Correction Methods for Forward Backward Stochastic Differential Equations. Numerical Mathematics: Theory, Methods and Applications, 10, 222-242. https://doi.org/10.4208/nmtma.2017.s02 [18] Zhang, C., Wu, J. and Zhao, W. (2019) One-Step Multi-Derivative Methods for Backward Stochastic Differential Equations. Numerical Mathematics: Theory, Methods and Applications, 12, 1213-1230. https://doi.org/10.4208/nmtma.OA-2018-0122 [19] Zhou, Q. and Sun, Y. (2021) High Order One-step Methods for Backward Stochastic Differential Equations via Itô-Taylor Expansion. Discrete and Continuous Dynamical Sys-tems-B. https://doi.org/10.3934/dcdsb.2021233 [20] Zhao, W., Zhang, G. and Ju, L. (2010) A Stable Multistep Scheme for Solving Backward Stochastic Differential Equations. SIAM Journal on Numerical Analysis, 48, 1369-1394. https://doi.org/10.1137/09076979X [21] Berrut, J.P. and Trefethen, L.N. (2004) Barycentric Lagrange Interpolation. SIAM Review, 46, 501-517. https://doi.org/10.1137/S0036144502417715