Explicit $\theta$-Schemes for Solving Anticipated Backward Stochastic Differential Equations

TL;DR

This paper introduces explicit -schemes for solving anticipated backward stochastic differential equations, transforming the delay process to achieve high-order convergence and demonstrating their stability and accuracy through theoretical analysis and numerical tests.

Contribution

The paper proposes a novel class of explicit -schemes that handle anticipated BSDEs with delays, achieving high-order convergence and stability.

Findings

Schemes are stable and have high-order convergence.

Numerical tests confirm high accuracy.

Error estimates are rigorously proved.

Abstract

In this paper, a class of stable explicit $\theta$-schemes are proposed for solving anticipated backward stochastic differential equations (anticipated BSDEs) which generator not only contains the present values of the solutions but also the future. We subtly transform the delay process of the generator into the current measurable process, resulting in high-order convergence rate. We also analyze the stability of our numerical schemes and strictly prove the error estimates. Various numerical tests powerful demonstrate high accuracy of the proposed numerical schemes.

Tables

Table 1. Table 1 : Errors and convergence rates of Example 4.1 .

Table 2. Table 2: Errors and convergence rates of Example 4.2 .

Equations

$$\left\{\begin{array}[c]{ll}-dY_{t}=\mathbf{f}\left(t,Y_{t},Z_{t},Y_{t+\delta(t)},Z_{t+\zeta(t)}\right)dt-Z_{t}dW_{t},&t\in\left[0,T\right];\\ Y_{t}=\xi_{t},&t\in\left[T,T+S\right];\\ Z_{t}=\eta_{t},&t\in\left[T,T+S\right],\end{array}\right.$$

$$\begin{array}[c]{ll}t+\delta\left(t\right)\leq T+S;\text{ \ }\text{ \ }&t+\zeta(t)\leq T+S,\end{array}$$

$$\begin{array}[c]{l}\int_{t}^{T}g\left(s+\newline \delta\left(s\right)\right)ds\leq L_{0}\int_{t}^{T+S}g\left(s\right)ds;\\ \int_{t}^{T}g\left(s+\zeta\left(s\right)\right)ds\leq L_{0}\int_{t}^{T+S}g\left(s\right)ds.\end{array}$$

$$\left\{\begin{array}[c]{ll}-dY_{t}=\mathbb{E}\left[f\mathbf{(}t,Y_{t},Z_{t},Y_{t+\delta(t)},Z_{t+\delta(t)})|\mathcal{F}_{t}\right]dt-Z_{t}dW_{t},&t\in\left[0,T\right];\\ Y_{t}=\xi_{t},&t\in\left[T,T+S\right];\\ Z_{t}=\eta_{t},&t\in\left[T,T+S\right],\end{array}\right.$$

$$Y_t=u\left(t,W_t\right),Z_t={\nabla }_{x}u\left(t,W_t\right),\forall t\in \left(0,T\right],$$

$$\begin{array}[c]{l}\frac{\partial}{\partial t}\left(t,x\right)+\frac{1}{2}\sum\limits_{i=1}^{d}\frac{\partial^{2}}{\partial x_{i}^{2}}u\left(t,x\right)+\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}}f(t,u\left(t,x\right),\nabla_{x}u\left(t,x\right),\\ \text{ \ \ \ \ \ \ \ \ \ \ \ \ }u(t+\delta\left(t\right),x+\sqrt{\delta\left(t\right)}y),\nabla_{x}u(t+\delta\left(t\right),x+\sqrt{\delta\left(t\right)}y))e^{-\frac{\text{ \ }y^{2}}{2}}dy=0,\end{array}$$

$$0=t_0<t_1<\cdots <t_M=T+S,$$

$$Y_{t_n}=Y_{t_{n+1}}+{\int }_{t_n}^{t_{n+1}}\mathbb{E}[f(s,Y_s,Z_s,Y_{s+\delta (s)},Z_{s+\delta (s)})|{\mathcal{F}}_s]ds-{\int }_{t_n}^{t_{n+1}}{Z}_{s}d{W}_s$$

$$Y_{t_n}={\mathbb{E}}_{t_n}^x[Y_{t_{n+1}}]+{\int }_{t_n}^{t_{n+1}}{\mathbb{E}}_{t_n}^x[f(s,Y_s,Z_s,Y_{s+\delta (s)},Z_{s+\delta (s)})]ds.$$

$${\int }_{t_n}^{t_{n+1}}{\mathbb{E}}_{t_n}^x[f_s]ds={\theta }_1\Delta t{\mathbb{E}}_{t_n}^x[f_{t_n}]+\left(1-{\theta }_1\right)\Delta t{\mathbb{E}}_{t_n}^x[f_{t_{n+1}}]+R_{y_1}^{n,\delta },$$

$$R_{y_1}^{n,\delta }={\int }_{t_n}^{t_{n+1}}\{{\mathbb{E}}_{t_n}^x[f_s]-{\theta }_1{\mathbb{E}}_{t_n}^x[f_{t_n}]+(1-{\theta }_1){\mathbb{E}}_{t_n}^x[f_{t_{n+1}}]\}ds.$$

$$\begin{array}[c]{c}\theta_{1}\Delta t\mathbb{E}_{t_{n}}^{x}\left[f_{t_{n}}\right]=\theta_{1}\Delta t\mathbb{E}_{t_{n}}^{x}\left[k_{n}f_{t_{n}^{-}}+\left(1-k_{n}\right)f_{t_{n}^{+}}\right]+R_{y_{2,1}}^{n,\delta},\\ \text{ }(1-\theta_{1})\Delta t\mathbb{E}_{t_{n}}^{x}\left[f_{t_{n+1}}\right]=(1-\theta_{1})\Delta t\mathbb{E}_{t_{n}}^{x}\left[k_{n+1}f_{t_{n+1}^{-}}+\left(1-k_{n+1}\right)f_{t_{n+1}^{+}}\right]+R_{y_{2,2}}^{n,\delta},\end{array}$$

$$\begin{array}[c]{l}\text{ \ }f_{t_{n}^{-}}=f(t_{n},Y_{t_{n}},Z_{t_{n}},Y_{t_{n+\delta_{n}^{-}}},Z_{t_{n+\delta_{n}^{-}}}),\\ \text{ \ }f_{t_{n}^{+}}=f(t_{n},Y_{t_{n}},Z_{t_{n}},Y_{t_{n+\delta_{n}^{+}}},Z_{t_{n+\delta_{n}^{+}}}),\\ \text{ \ }f_{t_{n+1}^{+}}=f(t_{n+1},Y_{t_{n+1}},Z_{t_{n+1}},Y_{t_{n+1+\delta_{n+1}^{-}}},Z_{t_{n+1+\delta_{n+1}^{-}}}),\\ \text{ \ }f_{t_{n+1}^{+}}=f(t_{n+1},Y_{t_{n+1}},Z_{t_{n+1}},Y_{t_{n+1+\delta_{n+1}^{+}}},Z_{t_{n+1+\delta_{n+1}^{+}}}),\end{array}$$

$$\begin{array}[c]{ll}\delta_{n}^{-}=[\frac{\delta\left(t_{n}\right)}{\Delta t}],&\delta_{n}^{+}=[\frac{\delta\left(t_{n}\right)}{\Delta t}]+1,\\ k_{n}=\frac{\delta_{n}^{+}\Delta t-\delta\left(t_{n}\right)}{\Delta t},&k_{n+1}=\frac{\delta_{n+1}^{+}\Delta t-\delta\left(t_{n+1}\right)}{\Delta t},\end{array}$$

$$\begin{array}[c]{l}Y_{t_{n}}=\mathbb{E}_{t_{n}}^{x}[Y_{t_{n+1}}]+\theta_{1}k_{n}\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n}^{-}}]+\theta_{1}\left(1-k_{n}\right)\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n}^{+}}]+R_{y_{1}}^{n,\delta}\\ \text{ \ \ \ \ \ \ }+(1-\theta_{1})k_{n+1}\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{-}}]+(1-\theta_{1})\left(1-k_{n+1}\right)\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{+}}]+R_{y_{2}}^{n,\delta},\end{array}$$

$$Y_{t_n}={\mathbb{E}}_{t_n}^x[Y_{t_{n+1}}]+\Delta t{\mathbb{E}}_{t_n}^x[f_{t_{n+1}^{-}}]+{\tilde{R}}_y^{n,\delta },$$

$$\begin{array}[c]{l}Y_{t_{n}}=\mathbb{E}_{t_{n}}^{x}[Y_{t_{n+1}}]+\theta_{1}k_{n}\Delta t\mathbb{E}_{t_{n}}^{x}[\bar{f}_{t_{n}^{-}}]+\theta_{1}(1-k_{n})\Delta t\mathbb{E}_{t_{n}}^{x}[\bar{f}_{t_{n}^{+}}]\\ \text{ \ \ \ \ \ \ }+(1-\theta_{1})k_{n+1}\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{-}}]+(1-\theta_{1})\left(1-k_{n+1}\right)\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{+}}]+R_{y}^{n,\delta},\end{array}$$

$$\begin{array}[c]{l}\bar{f}_{t_{n}^{-}}=f(t_{n},\mathbb{E}_{t_{n}}^{x}[Y_{t_{n+1}}]+\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{-}}],Z_{t_{n}},Y_{t_{n+\delta_{n}^{-}}},Z_{t_{n+\delta_{n}^{-}}}),\\ \bar{f}_{t_{n}^{+}}=f(t_{n},\mathbb{E}_{t_{n}}^{x}[Y_{t_{n+1}}]+\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{-}}],Z_{t_{n}},Y_{t_{n+\delta_{n}^{+}}},Z_{t_{n+\delta_{n}^{+}}}),\end{array}$$

$$R_{y_3}^{n,\delta }={\theta }_1{k}_n\Delta t{\mathbb{E}}_{t_n}^x[{\bar{f}}_{t_n^{-}}-f_{t_n^{-}}]+{\theta }_1\left(1-k_n\right)\Delta t{\mathbb{E}}_{t_n}^x[{\bar{f}}_{t_n^{+}}-f_{t_n^{+}}].$$

$$0={\mathbb{E}}_{t_n}^x[Y_{t_{n+1}}\Delta W_{t_n,t_{n+1}}^{⊺}]+{\int }_{t_n}^{t_{n+1}}{\mathbb{E}}_{t_n}^x[f_s\Delta W_{t_n,s}^{⊺}]ds-{\int }_{t_n}^{t_{n+1}}{\mathbb{E}}_{t_n}^x[Z_s]ds.$$

$$\begin{array}[c]{l}\int_{t_{n}}^{t_{n+1}}\mathbb{E}_{t_{n}}^{x}[f_{s}\Delta W_{t_{n},s}^{\intercal}]ds=(1-\theta_{2})k_{n+1}\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{-}}\Delta W_{t_{n+1}}^{\intercal}]+R_{z_{1}}^{n,\delta}\\ \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }+(1-\theta_{2})\left(1-k_{n+1}\right)\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{+}}\Delta W_{t_{n+1}}^{\intercal}]+R_{z_{2}}^{n,\delta},\end{array}$$

$$-{\int }_{t_n}^{t_{n+1}}{\mathbb{E}}_{t_n}^x[Z_s]ds=-{\theta }_3\Delta t{Z}_{t_n}-\left(1-{\theta }_3\right)\Delta t{\mathbb{E}}_{t_n}^x[Z_{t_{n+1}}]+R_{z_3}^{n,\delta },$$

$$R_{z_1}^{n,\delta }$$

$$R_{z_2}^{n,\delta }$$

$$\begin{array}[c]{l}R_{z_{3}}^{n,\delta}=-\int_{t_{n}}^{t_{n+1}}\left\{\mathbb{E}_{t_{n}}^{x}[Z_{s}]-\theta_{3}Z_{t_{n}}-(1-\theta_{3})\mathbb{E}_{t_{n}}^{x}[Z_{t_{n+1}}]\right\}ds.\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\end{array}$$

$$\begin{array}[c]{l}\theta_{3}\Delta tZ_{t_{n}}=\mathbb{E}_{t_{n}}^{x}[Y_{t_{n+1}}\Delta W_{t_{n+1}}^{\intercal}]+(1-\theta_{2})\left(1-k_{n+1}\right)\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{+}}\Delta W_{t_{n+1}}^{\intercal}]\\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ }+(1-\theta_{2})k_{n+1}\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{-}}\Delta W_{t_{n+1}}^{\intercal}]-(1-\theta_{3})\Delta t\mathbb{E}_{t_{n}}^{x}[Z_{t_{n+1}}]+R_{z}^{n,\delta},\end{array}$$

$${\mathbb{E}}_{t_n}^x[Y_{t_{n+1}}\Delta W_{t_{n+1}}^{⊺}]=\Delta t{\mathbb{E}}_{t_n}^x[\nabla Y_{t_{n+1}}]=\Delta t{\mathbb{E}}_{t_n}^x[Z_{t_{n+1}}].$$

$$\begin{array}[c]{l}\theta_{3}\Delta tZ_{t_{n}}=\theta_{3}\mathbb{E}_{t_{n}}^{x}[Y_{t_{n+1}}\Delta W_{t_{n+1}}^{\intercal}]+(1-\theta_{2})k_{n+1}\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{-}}\Delta W_{t_{n+1}}^{\intercal}]\\ \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ }+(1-\theta_{2})\left(1-k_{n+1}\right)\Delta t\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{+}}\Delta W_{t_{n+1}}^{\intercal}]+R_{z}^{n,\delta}.\end{array}$$

$${\theta }_3\Delta t{Z}^n$$

$$+(1-{\theta }_2)\left(1-k_{n+1}\right)\Delta t{\mathbb{E}}_{t_n}^x[f_{+}^{n+1}\Delta W_{t_{n+1}}^{⊺}],$$

$$\left\{\begin{array}[c]{l}\bar{Y}^{n}=\mathbb{E}_{t_{n}}^{x}[Y^{n+1}]+\Delta t\mathbb{E}_{t_{n}}^{x}[f_{-}^{n+1}],\\ Y^{n}=\mathbb{E}_{t_{n}}^{x}[Y^{n+1}]+\theta_{1}k_{n}\Delta t\mathbb{E}_{t_{n}}^{x}\left[\bar{f}_{-}^{n}\right]+\theta_{1}(1-k_{n})\Delta t\mathbb{E}_{t_{n}}^{x}\left[\bar{f}_{+}^{n}\right]\\ \text{ \ \ \ \ \ \ }+(1-\theta_{1})k_{n+1}\Delta t\mathbb{E}_{t_{n}}^{x}[f_{-}^{n+1}]+(1-\theta_{1})(1-k_{n+1})\Delta t\mathbb{E}_{t_{n}}^{x}[f_{+}^{n+1}],\end{array}\right.$$

Full text

Explicit $\theta$-Schemes for Solving Anticipated Backward Stochastic

Differential Equations

Mingshang Hu Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, PR China. humingshang@sdu.edu.cn. Research supported by NSF (No. 11671231) and Young Scholars Program of Shandong University (No. 2016WLJH10).   

Lianzi Jiang Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, PR China. jianglianzi95@163.com.

Abstract. In this paper, a class of stable explicit $\theta$-schemes are proposed for solving anticipated backward stochastic differential equations (anticipated BSDEs) which generator not only contains the present values of the solutions but also the future. We subtly transform the delay process of the generator into the current measurable process, resulting in high-order convergence rate. We also analyze the stability of our numerical schemes and strictly prove the error estimates. Various numerical tests powerful demonstrate high accuracy of the proposed numerical schemes.

Keywords. anticipated BSDEs, $\theta$-Schemes, error estimates, second order, delay process

AMS subject classifications. 60H35, 65C20, 60H10

1 Introduction

We consider the anticipated backward stochastic differential equations (anticipated BSDEs) on a complete filtered probability space $\left(\Omega,\mathcal{F},\mathbb{F},P\right)$, which generator contains not only the present values of $\left(Y_{\cdot},Z_{\cdot}\right)$, but also the future. The general form of the anticipated BSDEs is as follow:

[TABLE]

where $\mathbb{F=(}\mathcal{F}_{t}\mathcal{)}_{t\geq 0}$ is the natural filtration generated by the standard $d$-dimensional Brownian motion $W=$ $(W_{t})_{t\geq 0}$, $\delta(\cdot)$ and $\zeta(\cdot)$ are two $\mathbb{R}^{+}$-valued continuous functions defined on $[0,T]$, satisfying that

1. (i)

there exists a constant $S>0$ such that, for all $t\in\left[0,T\right],$

[TABLE] 2. (ii)

there exists a constant $L_{0}>0$ such that, for all $t\in\left[0,T\right]$ and for all nonnegative and integrable $g(\cdot)$,

[TABLE]

The generator $\mathbf{f(}t,\omega,y,z,\xi,\eta):\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{m\times d}\times L^{2}(\mathcal{F}_{r};\mathbb{R}^{m})\times L^{2}(\mathcal{F}_{r^{\prime}};\mathbb{R}^{m\times d})\rightarrow L^{2}(\mathcal{F}_{t};\mathbb{R}^{m})$ is $\mathcal{F}_{t}$-measurable, for all $t\in\left[0,T\right]$, $r,r^{\prime}\in[t,T+S]$. $\xi_{t}$ and $\eta_{t}$, $t\in[T,T+S]$ are given terminal conditions. The pair of $(Y_{\cdot},Z_{\cdot})$ is called $L^{2}$-solution of the anticipated BSDEs if it is $\mathcal{F}_{t}$-adapted and square integrable processes.

In 1990, Pardoux and Peng [18] first established the framework for the nonlinear BSDEs and rigorous proved the existence and uniqueness results for the solution, their extraordinary works are continued in [11, 19, 20, 21]. In 2009, Peng and Yang [22] introduced a new type of BSDE, namely anticipated BSDEs $\left(\ref{1.1}\right)$. They strictly proved the existence and uniqueness of solutions and extended the duality between stochastic differential delay equations (SDDEs) and anticipated BSDEs to stochastic control problems. Since then, a large amount of related research is booming, [2, 3, 6, 17] etc..

Since analytical solutions of the backward stochastic differential equations are often difficult to obtain, the numerical approximation is indispensable. Many works have be done on the numerical computing of BSDEs, including one step schemes, such as [1, 4, 8, 9, 10, 13, 15, 16, 23, 24, 26, 27] and multi-step schemes [5, 25, 28, 29], but little attention has been paid to the anticipated BSDEs.

In this paper, we just discuss the special case of the anticipated BSDEs in the following form:

[TABLE]

where the generator $\mathbf{f}=\mathbb{E}[f(t,y_{1},z_{1},y_{2},z_{2})|\mathcal{F}_{t}]$, $f:[0,T]\times\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{m\times d}\times\mathbb{R}^{m}\times\mathbb{R}^{m\times d}\rightarrow\mathbb{R}^{m}$ is an $\mathbb{R}^{m}$-valued vector function, $\delta\left(t\right)$ is a $\mathbb{R}^{+}$-valued monotonically increasing continuous function. Moreover, we can easily verify that $\delta\left(t\right)$ satisfies (i) and (ii), the anticipated BSDEs $\left(\ref{1.2}\right)$ satisfy the conditions of existence and uniqueness Theorem in [22], thus have a pair of unique $L^{2}$-solution $\left(Y_{t},Z_{t}\right)$. Assume that under some regularity conditions about $\xi_{t}=\varphi\left(t,W_{t}\right)$, $f$ and $\delta\left(t\right)$, the unique solution $\left(Y_{t},Z_{t}\right)$ of $\left(\ref{1.2}\right)$ can be represented as (see as [11, 12, 14, 19])

[TABLE]

where $u\left(t,x\right)$ is the smooth solution of the following anticipated parabolic partial integral differential equation

[TABLE]

with the terminal condition $u\left(s,x\right)=\varphi\left(s,x\right)$, $s\in[T,T+S]$, and $\nabla_{x}u$ is the gradient of $u$ with respect to the spacial variable $x=\left(x_{1},x_{2},\ldots,x_{d}\right)$. Furthermore, if $f\in C_{b}^{1+k,3+2k,3+2k,3+2k,3+2k}$, $\varphi\in C_{b}^{1+k,4+2k}$ and $\sqrt{\delta(t)}\in C_{b}^{1+k}$, then we have $u\in$ $C_{b}^{1+k,3+2k}$, $k=0,1,2,....$

In this paper, we devote to proposing a class of stable explicit $\theta$-schemes for solving anticipated BSDEs. The central idea is to use the properties of the conditional mathematical expectation to subtly transform the delay process in the generator into the current measurable process, which obtain a high-order local truncation error. We also design effective numerical methods to compute the conditional mathematical expectation with delay process and get good results in our numerical tests. To the best of our knowledge, this is the first attempt to come up with a high order numerical scheme for solving anticipated BSDEs.

For simple representations, we first introduce some notations that will be used extensively throughout the paper:

• •

$\left|\cdot\right|:$ the standard Euclidean norm in $\mathbb{R}$, $\mathbb{R}^{m}$ and $\mathbb{R}^{m\times d}$.

• •

$L^{2}\left(\mathcal{F}_{T};\mathbb{R}^{m}\right):\mathbb{R}^{m}$-valued $\mathcal{F}_{T}$-measurable random variables such that $\mathbb{E}[|\xi|^{2}]<\infty$.

• •

$\mathbb{E}\left[X\right]:$ the conditional mathematical expectation of $X$.

• •

$\mathbb{E}_{t_{n}}^{x}[X]:$ the conditional mathematical expectation of $X$, under the $\sigma$-field $\sigma\{W_{t}=x\}$, i.e. $\mathbb{E}\left[X|W_{t_{n}}=x\right]$.

• •

$\Delta W_{t,s}:W_{s}-W_{t}$, $0\leq t\leq s,$ a standard Brownian motion with mean zero and variance $s-t$.

• •

$C_{b}^{l,k,k,k,k}:$ continuously differentiable functions $f:\left[0,T\right]\times\mathbb{R}^{m}\times\mathbb{R}^{m\times d}\times\mathbb{R}^{m}\times\mathbb{R}^{m\times d}\rightarrow\mathbb{R}$ with uniformly bounded partial derivatives $\partial_{t}^{l_{1}}f$ and $\partial_{y_{1}}^{k_{1}}\partial_{z_{1}}^{k_{2}}\partial_{y_{2}}^{k_{3}}\partial_{z_{2}}^{k_{4}}\\ f$ for $\frac{1}{2}\leq l_{1}\leq l$ and $1\leq\sum_{i=1}^{4}k_{i}\leq 4$, $k_{i}\geq 0$.

• •

$C_{b}^{l,k}:$ continuously differentiable functions $\phi:\left[0,T\right]\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ with uniformly bounded partial derivatives $\partial_{t}^{l_{1}}\phi$ and $\partial_{x}^{k_{1}}\phi$ for $\frac{1}{2}\leq l_{1}\leq l$ and $1\leq k_{1}\leq k$.

• •

$C_{b}^{l}:$ continuously differentiable functions $\psi:\left[0,T\right]\rightarrow\mathbb{R}^{+}$ with uniformly bounded partial derivatives $\partial_{t}^{l_{1}}\psi$ for $1\leq l_{1}\leq l$.

The paper is organized as follow. In section 2, we propose a class of explicit $\theta$-schemes for anticipated BSDEs by choosing different parameters $\theta_{i}(i=1,2,3)$. In section 3, according to the properties of the conditional mathematical expectation, we subtly transform the delay process in the generator into the current process, which obtain a high-order local truncation error. We also analyze the stability of our numerical schemes and strictly prove the error estimates. In section 4, various numerical tests are given to demonstrate high efficiency and accuracy of our schemes. Finally, we come to the conclusions in section 5.

2 Explicit schemes for Anticipated BSDEs

2.1 Reference equations

For the time interval $\left[0,T+S\right]$, we introduce the following uniform partition:

[TABLE]

with $\Delta t=\frac{T+S}{M}$. For simplicity, we consider $S$ is a rational number, by choosing $M$ appropriately, we let $S=K\Delta t$ and $T=N\Delta t$, where $K$ and $N$ are positive integers.

Let $\left(Y_{t,}Z_{t}\right)$ be the solution of $\left(\ref{1.2}\right)$, we consider the $\left(\ref{1.2}\right)$ on the time interval $\left[t_{n},t_{n+1}\right]$, we can obtain easily

[TABLE]

for $n=0,1,\ldots,N-1$. Taking the conditional mathematical expectation $\mathbb{E}_{t_{n}}^{x}\left[\cdot\right]$ on both sides of $\left(\ref{3.1}\right)$, using the formula $\mathbb{E}_{t_{n}}^{x}[\mathbb{E}\left[\cdot|\mathcal{F}_{s}\right]]=\mathbb{E}_{t_{n}}^{x}[\cdot]$, for $s\geq t_{n},$ we get

[TABLE]

For notational simplicity, we denote $f_{s}=f(s,Y_{s},Z_{s},Y_{s+\delta(s)},Z_{s+\delta(s)})$. It is notable that the integrands $\mathbb{E}_{t_{n}}^{x}[f_{s}]$ in $\left(\ref{3.2}\right)$ is a deterministic function, then we can use some numerical integration methods to approximate it. Here we use the $\theta$-scheme to approximate the integral:

[TABLE]

where $\theta_{1}\in\left[0,1\right]$ and

[TABLE]

Since $t_{n}+\delta(t_{n})$ and $t_{n+1}+\delta(t_{n+1})$ may not be on the grid points, we approximate it with the value of the adjacent grid points $t_{n+\delta_{n}^{-}}$, $t_{n+\delta_{n}^{+}}$, $t_{n+1+\delta_{n+1}^{-}}$and $t_{n+1+\delta_{n+1}^{+}}$, then

[TABLE]

where

[TABLE]

and

[TABLE]

the scale coefficients $k_{n}\in[0,1]$, $n=0,1,\ldots,N-1$. Combining $\left(\ref{3.3}\right)$ and $\left(\ref{3.5}\right),$ we deduce that

[TABLE]

where interpolation error $R_{y_{2}}^{n,\delta}=R_{y_{2,1}}^{n,\delta}+R_{y_{2,2}}^{n,\delta}$. In order to propose an explicit numerical scheme to solve $\left\{Y_{t_{n}}\right\}$, we approximate the $Y_{t_{n}}$ in $f_{t_{n}^{-}}$ and  $f_{t_{n}^{+}}$ by the left rectangle formula

[TABLE]

where $\tilde{R}_{y}^{n}=\int_{t_{n}}^{t_{n+1}}\mathbb{E}_{t_{n}}^{x}\left[f_{s}\right]-\mathbb{E}_{t_{n}}^{x}[f_{t_{n+1}^{-}}]ds$. According to $\left(\ref{3.6}\right)$ and $\left(\ref{3.7}\right)$, we obtain the following reference equation:

[TABLE]

where

[TABLE]

and $R_{y}^{n,\delta}=R_{y_{1}}^{n,\delta}+R_{y_{2}}^{n,\delta}+R_{y_{3}}^{n,\delta}$ with

[TABLE]

Now, multiplying $\left(\ref{3.1}\right)$ by $\Delta W_{t_{n},t_{n+1}}^{\intercal}$ and taking the conditional mathematical expectation $\mathbb{E}_{t_{n}}^{x}\left[\cdot\right]$ on both sides of the derived equation, by the Itô isometry formula, we obtain

[TABLE]

To simplify the presentation, we use $\Delta W_{t_{n+1}}^{\intercal}$to denote $\Delta W_{t_{n},t_{n+1}}^{\intercal}$. Similar to the above process, we rewrite the two integral terms on the right side of $\left(\ref{3.9}\right)$ in the following:

[TABLE]

[TABLE]

where $\theta_{2}\in\left[0,1\right]$, $\theta_{3}\in(0,1]$

[TABLE]

and

[TABLE]

According to $\left(\ref{3.9}\right),\left(\ref{3.10}\right)$ and $\left(\ref{3.11}\right)$, we obtain

[TABLE]

where $R_{z}^{n,\delta}=R_{z_{1}}^{n,\delta}+R_{z_{2}}^{n,\delta}+R_{z_{3}}^{n,\delta}$. By $\left(\ref{1.3}\right)$ and the definition of $\mathbb{E}_{t_{n}}^{x}\left[\cdot\right]$, using the formula of integration by part, it is easy to prove that:

[TABLE]

Substituting $\left(\ref{3.13}\right)$ into $\left(\ref{3.12}\right)$, we get the following reference equation:

[TABLE]

2.2 The explicit $\theta$-schemes

Let $\left(Y^{n},Z^{n}\right)$ $(n=N-1,\ldots,0)$ denote the numerical approximation to the solution $\left(Y_{t_{n}},Z_{t_{n}}\right)$ of the anticipated BSDEs $\left(\ref{1.2}\right)$ at time level $t=t_{n}(n=N-1,\ldots,0)$. To maintain the consistency of representation, we use $\left(Y^{N+i},Z^{N+i}\right)$ to denote the given final values $\left(Y_{t_{n}},Z_{t_{n}}\right)$ of the anticipated BSDEs $\left(\ref{1.2}\right)$ at time partition $t=t_{N+i}(i=0,\ldots,K)$. From the two reference equations $\left(\ref{3.8ref}\right)$ and $\left(\ref{3.14ref}\right)$, we propose the following explicit $\theta$-schemes for solving the anticipated BSDEs $\left(\ref{1.2}\right)$.

The explicit $\theta$-**schemes **Given the random variables $Y^{N+i}$ and $Z^{N+i}$, $i=0,\ldots,K$, solve random variables $Y^{n}$ and $Z^{n}$, $n=N-1,\ldots,0$, from

[TABLE]

[TABLE]

with the deterministic parameters $\theta_{i}\in\left[0,1\right](i=1,2)$, $\theta_{3}\in(0,1]$, the scale coefficients $k_{n}\in\left[0,1\right](n=0,1,\ldots,N-1)$ defined in $\left(\ref{Kn}\right)$ and

[TABLE]

Remark 2.1

It can be seen that by choosing different parameters $\theta_{i}\left(i=1,2\right)\in[0,1]$, $\theta_{3}\in(0,1]$, we could get a class of different numerical schemes. When choosing $\theta_{i}=\frac{1}{2}(i=1,2,3)$, the explicit $\theta$-scheme is a second order scheme for solving anticipated BSDEs, which will be confirmed in later numerical experiments.

Remark 2.2

When we calculate the $Y^{n}$ and $Z^{n}$ at each time level, the future values are also involved. By reasonable selecting $M$, the future values can be on the time grid points to ensure the performed of the numerical iterations. In addition, when $Y^{n}$ and $Z^{n}$ are near the time of 0, the point of $\delta_{n}^{-}=0$ may appear, we take $\bar{f}_{-}^{n}=f(t_{n},\bar{Y}^{n},Z^{n},\bar{Y}^{n},Z^{n})$ at this time.

3 Error estimates

In this section, we will give the error estimates of the explicit $\theta$-schemes $\left(\ref{scheme3.2.1}\right)$ and $\left(\ref{scheme3.2.2}\right)$. Before given the main theorem, we first introduce some useful lemmas which play an important role in the later proof.

3.1 Stability analysis

Lemma 3.1** (Discrete Gronwall Lemma)**

Suppose that $N$ and $K$ are two nonnegative integers and $\Delta t$ is any positive number. Let $\left\{A_{n}\right\}$, $n=N-1,N-2,\ldots,0$, satisfy

[TABLE]

where $\alpha$ and $\beta$ are two positive constants. Let $M_{A}=\max\limits_{N\leq j\leq N+K}\left|A_{j}\right|$ and $\hat{T}=T+S=\left(N+K\right)\Delta t$, then for $n=N-1,N-2,\ldots,0$,

[TABLE]

Lemma 3.2

Let $N$ and $K$ are two nonnegative integers, $\Delta t$ is any positive number. Assume $\left\{A_{n}\right\},\left\{B_{n}\right\}$ and $\left\{R^{n}\right\},n=0,1,\ldots,N+K$, are nonnegative sequences, satisfy the inequality

[TABLE]

for $n=0,1,\ldots,N-1$, where $\mathcal{I}_{n}$ $\subseteq$ $\left\{1,2,\ldots,K+1\right\}$ is an indicator set with $\left|\mathcal{I}_{n}\right|\leq I_{0}<\infty$, $C_{1},C_{2}$ and $C_{3}$ are positive constants. Let

[TABLE]

If there exist a constant $C_{4}$ $>0$, such that $C_{1}-I_{0}C_{3}\geq C_{4}$, Then for $n=N-1,N-2,\ldots,0$, it holds that

[TABLE]

where $C$ is a constant depending on $C_{1},C_{2},C_{3},\hat{T}$ and $I_{0}$.

Proof. This proof is similar to the Lemma 4.2 in [29]. See the appendix for details.

Remark 3.3

The Lemma 3.2 is a generalization of the Lemma 4.2 in [29] . When $\Delta t\rightarrow 0$, then $K\rightarrow\infty$ and $\mathcal{I}_{n}$ $\subseteq$ $\left\{1,2,\ldots,K+1\right\}$ can include the infinity number indicator, resulting in the right term of the $\left(\ref{4.2}\right)$ changing from $\Delta t(M_{A}+M_{B})$ to $(M_{A}+M_{B})$.

Without loss of generality, in the following of this section we only consider the case of one-dimensional anticipated BSDEs $(i.e.,m=d=1)$. Conclusions can be generalized to $d$-dimensional anticipated BSDEs where $d>1$. Before the main theorem, we first introduce the following notation:

[TABLE]

and

[TABLE]

Theorem 3.4

Let $\left(Y_{t},Z_{t}\right)$, $t\in\left[0,T+S\right]$, be the solution of the anticipated BSDEs $\left(\ref{1.2}\right)$ and $\left(Y^{n},Z^{n}\right)(n=N+K,\ldots,0)$ defined in the explicit $\theta$-schemes 2.2, $M_{ey}=\max\limits_{N\leq i\leq N+K}\mathbb{E}[|e_{y}^{i}|^{2}],\;M_{ez}=\max\limits_{N\leq i\leq N+K}\mathbb{E}[|e_{z}^{i}|^{2}],\;\hat{T}=\left(N+K\right)\Delta t=T+S$. Assume that $f$ satisfies the regularity conditions with Lipschitz constant $L$. Then for sufficiently small time step $\Delta t$, we have

[TABLE]

for $n=N-1,N-2,\ldots,0$. Here $C$ is a constant depending on $L$ and $\hat{T}$, $R_{y}^{n,\delta}\mathbb{\ }$and $R_{z}^{n,\delta}$are defined in $\left(\ref{3.8ref}\right)$ and $\left(\ref{3.14ref}\right)$, respectively.

Proof.* We complete the proof of the theorem in three steps.*

Step1:The estimate of $e_{y}^{n}$.

Since $\delta(t)$ is a monotonically increasing function, we denote the first n_{0}\which satisfies $\delta_{n}^{-}=1$. While $n\,\geq n_{0}$, then $\delta_{n}^{-}\geq 1$, for each integer $n>0$, Subtract $\left(\ref{3.8ref}\right)$ from $\left(\ref{scheme3.2.2}\right)$, we can obtain:

[TABLE]

Then

[TABLE]

By the definition of $e_{\bar{f}_{-}}^{n},$ $e_{\bar{f}_{+}}^{n},$ $e_{f_{-}}^{n+1}$and $e_{f_{+}}^{n+1}$, using the Lipschitz continuity property of $f$, we have

[TABLE]

and

[TABLE]

Similarly

[TABLE]

and

[TABLE]

We may rewrite

[TABLE]

where $L$ is the Lipschitz constant of $f\left(t,y_{1},z_{1},y_{2},z_{2}\right)$ with respect to $y_{1}$, $z_{1}$, $y_{2}$ and $z_{2}$. Taking square on both side of $\left(\ref{4.15}\right)$, using the inequalities $(a+b)^{2}\leq a^{2}+b^{2}+(1+\gamma\Delta t)a^{2}+(1+\frac{1}{\gamma\Delta t})b^{2}$ and $(\sum_{i=1}^{m}a_{i})^{2}\leq m\sum_{i=1}^{m}a_{i}^{2}$, we get

[TABLE]

Step2:The estimate of $e_{z}^{n}$.

In the same way above, subtracting $\left(\ref{3.14ref}\right)$ from $\left(\ref{scheme3.2.1}\right)$, we have

[TABLE]

Moreover,

[TABLE]

Applying Hölder inequality, we deduce that

[TABLE]

and

[TABLE]

Then we have the estimate

[TABLE]

Dividing both side of $\left(\ref{4.20}\right)$ by $2\theta_{3}^{2}\Delta t$ we have

[TABLE]

Step3: The proof of $\left(\ref{theorem4.1}\right)$.

United $\left(\ref{4.16}\right)$ and $\left(\ref{4.21}\right)$, it is direct to obtain

[TABLE]

which implies

[TABLE]

where $D_{1}=24L^{2}(1+L)^{2}+\frac{12L^{2}}{\theta_{3}^{2}}$, $D_{2}=24L^{2}(1+L)^{2}$. From $\left(\ref{4.23}\right)$ we have

[TABLE]

where $C_{1}^{\ast}=\frac{1}{2}-\frac{4L^{2}}{\gamma}-4L^{2}\Delta t$, $\mathcal{I}_{n}^{\ast}=\left\{1,\delta_{n}^{-},\delta_{n}^{+},\delta_{n+1}^{-}+1,\delta_{n+1}^{+}+1\right\}$. While $n\,<n_{0}$, $\delta_{n}^{-}=0$, by Remark 2.2, similar to the process above, we have

[TABLE]

where $C_{1}^{\ast\ast}=\frac{1}{2}-\frac{8L^{2}}{\gamma}-8L^{2}\Delta t$, $\mathcal{I}_{n}^{\ast\ast}=\left\{1,\delta_{n}^{+},\delta_{n+1}^{-}+1,\delta_{n+1}^{+}+1\right\}$. When $\Delta t$ is sufficiently small, by letting $\gamma$ be big enough, we can choose constants $C_{1},C_{2},C_{3},C_{4}$, which satisfy

[TABLE]

Taking mathematical expectation on both sides of $\left(\ref{4.25}\right)$ and $\left(\ref{4.24}\right)$, rewriting the inequality, we have

[TABLE]

*where $\mathcal{I}_{n}=\left\{1,\delta_{n}^{-},\delta_{n}^{+},\delta_{n+1}^{-}+1,\delta_{n+1}^{+}+1\right\}\backslash\left\{0\right\}$ and R^{n}=(1+\frac{1}{\gamma\Delta t})4$$\mathbb{E}[|R_{y}^{n,\delta}|^{2}]+\frac{3}{\theta_{3}^{2}\Delta t}\mathbb{E}[|R_{z}^{n,\delta}|^{2}]. Applying Lemma 3.2 to $\left(\ref{4.26}\right)$, the conclusion $\left(\ref{theorem4.1}\right)\;$can be drawn.   *

Remark 3.5

Theorem 3.4 implies that our explicit $\theta$-schemes are stable.

3.2 Error estimates for anticipated BSDEs

In this section, we discuss the error estimates of the explicit $\theta$-schemes for anticipate BSDEs. Under some regularity conditions on $f$, $\varphi$ and $\delta(t)$, we first give the estimates of the local truncation errors $R_{y}^{n,\delta}$ and $R_{z}^{n,\delta}$ defined in $\left(\ref{3.8ref}\right)$ and $\left(\ref{3.14ref}\right)$, then based on the Theorem 3.4, we get the error estimates of the explicit $\theta$-schemes 2.2.

Lemma 3.6

Let $R_{y}^{n,\delta}$ and $R_{z}^{n,\delta}$ be the local truncation errors derived from $\left(\ref{3.8ref}\right)$ and $\left(\ref{3.14ref}\right)$, respectively. Assume $\sqrt{\delta(t)}$ is smooth enough, for sufficiently small $\Delta t$, it holds that:

For $\theta_{i}\left(i=1,2\right)\in[0,1]$, $\theta_{3}\in(0,1]$, if $\varphi\in C_{b}^{\frac{1}{2},2}$, $f\in C_{b}^{\frac{1}{2},1,1,1,1}$, we have

[TABLE]

and if $\varphi\in C_{b}^{1,4}$, $f\in C_{b}^{1,3,3,3,3}$, we have

[TABLE] 2. 2.

In particular, for $\theta_{i}=\frac{1}{2}\left(i=1,2,3\right)$, if $\varphi\in C_{b}^{2,6}$, $f\in C_{b}^{2,5,5,5,5}$, we have

[TABLE]

where $C>0$ is a constant just depending on $T$, the upper bounds of derivatives of $\varphi$, $\delta(t)$ and $f$.

Proof.* 1.According to the definition of $R_{y}^{n,\delta}$ and $R_{z}^{n,\delta},$ using the continuity of the $\varphi$ and $f,$ $\left(\ref{truncation cond1}\right)$ can be proved easily. 2.Now we prove $\left(\ref{truncation cond2}\right)$ and $\left(\ref{truncation cond3}\right)$. We first approximate the truncation errors $R_{y_{1}}^{n,\delta}$, $R_{z_{1}}^{n,\delta}$ and $R_{z_{3}}^{n,\delta}$. By $\left(\ref{1.3}\right)$, we can define $f(t,Y_{t},Z_{t},Y_{t+\delta(t)},Z_{t+\delta(t)})=f(t,u(t,W_{t}),\nabla_{x}u(t,W_{t}),u(t+\delta(t),W_{t+\delta(t)}),\nabla_{x}u(t+\delta(t),W_{t+\delta(t)}))=F(t,W_{t},W_{t+\delta(t)})$. If $f\in C_{b}^{1+k,3+2k,3+2k,3+2k,3+2k}$, $\varphi\in C_{b}^{1+k,4+2k}$ and $\sqrt{\delta(t)}\in C_{b}^{1+k}$, then $F\in C_{b}^{1+k,2+2k,2+2k}$, $k=0,1,\ldots$. We rewrite $R_{y_{1}}^{n,\delta}$ as follow*

[TABLE]

By the properties of conditional mathematical expectation we can see

[TABLE]

Similarly, we obtain

[TABLE]

and

[TABLE]

Now we define a new function

[TABLE]

Since $F\in C_{b}^{1+k,2+2k,2+2k}$ and $\int_{-\infty}^{+\infty}e^{-\frac{\;\;z^{2}}{2}}dz<\infty$, we can draw the conclusion that $H\in$ $C_{b}^{1+k,2+2k,2+2k}$. Due to the properties of Brownian motion, It’s worth noting that

[TABLE]

then

[TABLE]

Therefore, $\left(\ref{4.29}\right)$ changes to be

[TABLE]

We denote $H\left(s,W_{s},W_{s}\right)$ by $H_{s}$, by It$\hat{o}$* formula, we know*

[TABLE]

where $L^{0}=\frac{\partial}{\partial t}+\frac{1}{2}(\frac{\partial}{\partial x}+\frac{\partial}{\partial y})^{2}$, $L^{1}=\frac{\partial}{\partial x}+\frac{\partial}{\partial y}$. If $\varphi\in C_{b}^{1,4}$ and $f\in C_{b}^{1,3,3,3,3}$, then $H\in C_{b}^{1,2,2}$, we can deduce

[TABLE]

Furthermore, if $\theta_{1}=\frac{1}{2}$, $\varphi\in C_{b}^{2,6}$ and $f\in C_{b}^{2,5,5,5,5}$, then $H\in C_{b}^{2,4,4}$, continue to apply It$\hat{o}$* formula to the $L^{0}H_{r}$ in equation $\left(\ref{4.36}\right)$, we can deduce*

[TABLE]

then $\left|R_{y}^{n,\delta}\right|\leq C(\Delta t)^{3}$. In addition, by It$\hat{o}^{\prime}$s isometry formula, we have

[TABLE]

where

[TABLE]

Owing to $\mathbb{E}_{t_{n}}^{x}[|\Delta W_{t_{n+1}}|]\leq(\Delta t)^{\frac{1}{2}},$ while $H\in C_{b}^{1,2,2}$, we have $\left|R_{1}\right|\leq C(\Delta t)^{\frac{5}{2}}$ and $\left|R_{2}\right|\leq C(\Delta t)^{2},$ then $\left|R_{z_{1}}^{n,\delta}\right|\leq C(\Delta t)^{2}$. What’s more, while $H\in C_{b}^{2,4,4}$, $\theta_{2}=\frac{1}{2}$, using It$\hat{o}$* formula again*

[TABLE]

and

[TABLE]

thus $|R_{z_{1}}^{n,\delta}|\leq C(\Delta t)^{3}$. Then it is straightforward to show by a computation similar to $R_{y_{1}}^{n,\delta}$ and $R_{z_{1}}^{n,\delta}$

[TABLE]

where $L^{\ast}=\frac{\partial}{\partial t}+\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}$. We also have the estimates

[TABLE]

Next, we estimate the non-grid point interpolation errors $R_{y_{2}}^{n,\delta}$ and $R_{z_{2}}^{n,\delta}$. Similar to $R_{y_{1}}^{n,\delta},$ we rewrite $R_{y_{2}}^{n,\delta}$ and $R_{z_{2}}^{n,\delta}$ as follow

[TABLE]

and

[TABLE]

where we denote $F_{r}=F(t_{n},W_{t_{n}},W_{r})$ and $\hat{F}_{r}=F(t_{n+1},W_{t_{n+1}},W_{r})$. if $F\in C_{b}^{1,2,2}$, we apply It$\hat{o}$* formula to the third variable of $F_{r}$ and $\hat{F}_{r}$, we obtain*

[TABLE]

and

[TABLE]

If $F\in C_{b}^{2,4,4}$, continue to apply It$\hat{o}$* formula to $F_{r}^{(2)}$ and $\hat{F}_{r}^{(2)}$, using the fact $\delta(t_{n})-\delta_{n}^{-}-\left(1-k_{n}\right)\Delta t=\delta(t_{n+1})-\delta_{n+1}^{-}-\left(1-k_{n+1}\right)\Delta t=0$, we can deduce*

[TABLE]

and

[TABLE]

*We can also verify that $|R_{y_{3}}^{n,\delta}|\leq C\left(\Delta t\right)^{3}$. To sum up, we get the estimates of $|R_{y}^{n,\delta}|$ and $|R_{z}^{n,\delta}|$. The proof is complete.  * **

Based on the above discussion, now let us show the error estimates of $Y_{t_{n}}-Y^{n}$ and $Z_{t_{n}}-Z^{n}$.

Theorem 3.7

Let $\left(Y_{t},Z_{t}\right),t\in\left[0,T+S\right]$, be the solution of the anticipated BSDEs $\left(\ref{1.2}\right)$ and $\left(Y^{n},Z^{n}\right)(n=N+K,\ldots,0)$ defined in the explicit scheme 2.2. Suppose that $f$ satisfies the regularity conditions with Lipschitz constant $L$, $\sqrt{\delta(t)}$ is smooth enough, $\hat{T}=\left(N+K\right)\Delta t=T+S$. Then for sufficiently small time step $\Delta t$, we have the following estimates.

For $\theta_{i}\left(i=1,2\right)\in[0,1]$, $\theta_{3}\in(0,1]$, if $\varphi\in C_{b}^{\frac{1}{2},2}$, $f\in C_{b}^{\frac{1}{2},1,1,1,1}$, $M_{ey}=O\left(\Delta t\right)$ and $M_{ez}=O\left(\Delta t\right)$, we have

[TABLE]

and if $\varphi\in C_{b}^{1,4}$, $f\in C_{b}^{1,3,3,3,3}$, $M_{ey}=O\left(\Delta t^{2}\right)$ and $M_{ez}=O\left(\Delta t^{2}\right)$, we have

[TABLE] 2. 2.

In particular, for $\theta_{i}=\frac{1}{2}\left(i=1,2,3\right)$, if $\varphi\in C_{b}^{2,6}$, $f\in C_{b}^{2,5,5,5,5}$, $M_{ey}=O\left(\Delta t^{4}\right)$ and $M_{ez}=O\left(\Delta t^{4}\right)$, we have

[TABLE]

Here $C$ is a constant depending on $L$, $\hat{T}$, $\theta_{i}\left(i=1,2,3\right)$, and the upper bounds of derivatives of $\varphi$, $f$.

Proof.* By Lemma 3.6, for sufficiently time step $\Delta t$, it holds that*

For $\theta_{i}\left(i=1,2\right)\in[0,1]$, $\theta_{3}\in(0,1]$, if $\varphi\in C_{b}^{\frac{1}{2},2}$ and $f\in C_{b}^{\frac{1}{2},1,1,1,1}$, we have

[TABLE]

and if $\varphi\in C_{b}^{1,4}$ and$\ f\in C_{b}^{1,3,3,3,3}$, we have

[TABLE] 2. 2.

In particular, for $\theta_{i}=\frac{1}{2}\left(i=1,2,3\right)$, if $\varphi\in C_{b}^{2,6}$ and $f\in C_{b}^{2,5,5,5,5}$, we have

[TABLE]

*Using Theorem 3.4, the conclusion can be directly obtained.   *

Remark 3.8

The Theorem 3.7 above implies that our explicit $\theta$-schemes could have high order as well as high accuracy for solving anticipated BSDEs. When $f$, $\varphi$ and $\delta(t)$ are smooth enough the convergence order of schemes 2.2 for solving $Y$ and $Z$ is 1. In particular, if the parameters $\theta_{1}=\theta_{2}=\theta_{3}=\frac{1}{2}$, the convergence rate can reach second order.

4 Numerical experiments

In this section, we will give some numerical experiments to illustrate the high accuracy of our explicit $\theta$-schemes for solving the anticipated BSDEs $\left(\ref{1.2}\right)$.

We first give the numerical approximation methods for the conditional mathematical expectation which includes the delay process. To simplify the process, we just give the approximation of $\mathbb{E}_{t_{n}}^{x}[f\left(t_{n+1},Y^{n+1},Y^{n+m+1}\right)]$ $(m=\delta_{n+1}^{-}$, $\delta_{n+1}^{+})$ with the generator $f$ only depends on $Y$

[TABLE]

where $\left\{w_{i}\right\}_{i=1}^{G_{1}}$ and $\left\{w_{j}\right\}_{j=1}^{G_{2}}$ are the weights of Gauss-Hermite quadrature formula, $\left\{p_{i}\right\}_{i=1}^{G_{1}}$ and $\left\{q_{j}\right\}_{j=1}^{G_{2}}$ are the roots of the Hermite polynomials of degree $G_{1}$ and $G_{2}$, respectively. The conditional mathematical expectations $\mathbb{E}_{t_{n}}^{x}[Y^{n+1}]$, $\mathbb{E}_{t_{n}}^{x}\left[\bar{f}_{-}^{n}\right]$, $\mathbb{E}_{t_{n}}^{x}\left[\bar{f}_{+}^{n}\right]$, $\mathbb{E}_{t_{n}}^{x}[Y^{n+1}\Delta W_{t_{n+1}}^{\intercal}]$, $\mathbb{E}_{t_{n}}^{x}[f_{-}^{n+1}\Delta W_{t_{n+1}}^{\intercal}]$ and $\mathbb{E}_{t_{n}}^{x}[f_{+}^{n+1}\Delta W_{t_{n+1}}^{\intercal}]$ can be obtained similarly, see more details in [27, 28].

Now let us take a few notes on our numerical experiments. We take a uniform partition of time and space, $\Delta t$ and $\Delta x$ represent time and space steps, respectively. In our tests, we use Gauss-Hermite quadrature rule to approximate the conditional mathematical expectations and apply cubic spline interpolation to compute spatial non-grid points. In order to balance the time and space error, we set $\Delta t$ and $\Delta x$ satisfy the equality $(\Delta t)^{p+1}=(\Delta x)^{q}$, where $q=4$ is the order of spatial interpolation error and $p$ is the convergence order of the numerical scheme. In the following tables, CR stands for convergence rate with respect to time step $\Delta t$, $|Y_{0}-Y^{0}|$ and $|Z_{0}-Z^{0}|$ denote the absolute errors between the exact and numerical solutions for $Y_{t}$ and $Z_{t}$ at $(t_{0},x_{0})$, respectively.

Example 4.1

In this example, we consider a linear anticipated BSDEs

[TABLE]

with the terminal condition

[TABLE]

The exact solution is $(Y_{t},Z_{t})=(\exp(t)\sin(t+W_{t}),\exp(t)\cos(t+W_{t}))$. We let the Brownian motion start at the time-space point $(0,0)$ and $T=1$, then $S=\frac{1}{4}$, the exact solution $\left(Y_{t},Z_{t}\right)$ at initial time $t_{0}=0$ is $\left(Y_{0},Z_{0}\right)=\left(0,1\right)$. In table 1, we have listed the errors $\left|Y_{0}-Y^{0}\right|$ and $\left|Z_{0}-Z^{0}\right|$ and their convergence rates of our explicit $\theta$-schemes with different time partitions.

Example 4.2

In this example, we test the following nonlinear anticipated BSDEs with $\delta(t)=\frac{1}{4}t^{2}$.

[TABLE]

The exact solution is

[TABLE]

The Brownian motion starts at the time-space point $(0,0),$ $T=1$ and $S=\frac{1}{4}$, the exact solution $\left(Y_{t},Z_{t}\right)$ at initial time $t_{0}=0$ is $\left(Y_{0},Z_{0}\right)=\left(0,1\right)$. In table 2, we show the errors $\left|Y_{0}-Y^{0}\right|$ and $\left|Z_{0}-Z^{0}\right|$ and their convergence rates of our explicit $\theta$-schemes with different time partitions.

Remark 4.3

The numerical results above show that our numerical schemes defined in $\left(\ref{scheme}\right)$ work very well for solving anticipated BSDEs whether the generator is linear or nonlinear.

Remark 4.4

The convergence rate of the scheme depends on the choice of parameters. If $\theta_{i}\left(i=1,2,3\right)$ is not equal to $\frac{1}{2}$, the convergence rate of the schemes is first order, if $\theta_{1}=\theta_{2}=\theta_{3}=\frac{1}{2}$, the convergence rate is second order, which is in good agreement with our theoretical analysis.

5 Conclusions

In this paper, we propose a class of stable explicit $\theta$-schemes for solving anticipated BSDEs. By choosing different $\theta_{i}$ $\left(i=1,2,3\right)$, we obtain many different numerical schemes. We also analyze the stability of our numerical schemes and strictly prove the error estimates. When $\theta_{1}=\theta_{2}=\theta_{3}=\frac{1}{2}$, the scheme is a second order numerical scheme, which is very effective for solving anticipated BSDEs. The numerical tests in section 5 powerful back up the theoretical results. This seems to be the first time to design high order numerical schemes for anticipated BSDEs.

Appendix A Appendix

A.1 The proof of Lemma 3.2

Proof. For $\left(\ref{4.1}\right)$, replacing $n$ by n+s\with $0\leq s\leq N-n-1$ and multiplying $\left(1+C_{2}\Delta t\right)^{s}$ on the both side of the inequality, we rewrite it as follow

[TABLE]

Adding up the both side of $\left(\ref{4.3}\right)$ about $s$ from [math] to $N-n-1$, we get

[TABLE]

Observe that fixed $i\in\mathcal{I}_{n+s}$, we have

[TABLE]

then

[TABLE]

Inserting $\left(\ref{4.5}\right)$ into $\left(\ref{4.4}\right)$, we deduce that:

[TABLE]

Which leads to

[TABLE]

Let

[TABLE]

From $\left(\ref{4.7}\right)$, we have

[TABLE]

Using Lemma 3.1 to $\left(\ref{4.8}\right)$ we obtain

[TABLE]

which is

[TABLE]

Let $C_{5}=$ $e^{\alpha\hat{T}}\max\left(e^{C_{2}\hat{T}},I_{0}C_{3},1,\alpha\right)$, Inserting $\left(\ref{4.9}\right)$ into the right side of $\left(\ref{4.7}\right)$, since $C_{1}-I_{0}C_{3}\geq C_{4}>0$, we get

[TABLE]

where $C_{6}=2e^{C_{2}\hat{T}}\max\left\{1,I_{0}C_{3}\hat{T},\alpha C_{5}\hat{T},\alpha C_{5}\hat{T}^{2}\right\}$. We can complete the proof by using $C=\frac{C_{6}}{\min\left\{1,C_{4}\right\}}$ in $\left(\ref{4.10}\right)$.