High-order Combined Multi-step Scheme for Solving Forward Backward Stochastic Differential Equations

Abstract

In this work, in order to obtain higher-order schemes for solving forward backward stochastic differential equations, we propose a new multi-step scheme by adopting the high-order multi-step method in Zhao et al. (SIAM J. Sci. Comput., 36(4): A1731-A1751, 2014) with the combination technique. Two reference ordinary differential equations containing the conditional expectations and their derivatives are derived from the backward component. These derivatives are approximated by using the finite difference methods with multi-step combinations. The resulting scheme is a semi-discretization in the temporal direction involving conditional expectations, which are solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Our new proposed multi-step scheme allows for higher convergence rate up to ninth order, and are more efficient. Finally, we provide a numerical illustration of the convergence of the proposed method.

1 Introduction

Recently, the forward-backward stochastic differential equation (FBSDE) becomes an important tool for formulating many problems in various areas including physics and financial mathematics. We are interested in the numerical approximation of the general FBSDEs

$$\begin{aligned} \left\{ \begin{array}{l} \,\,\, dX_t = a(t, X_t, Y_t, Z_t)\,dt + b(t, X_t, Y_t, Z_t)\,dW_t, \quad \text{ forward } \text{ component }\\ \,\,\,\,\,\, X_0=x_0,\\ -dY_t = f(t, X_t, Y_t, Z_t)\,dt - Z_t\,dW_t,\text{ backward } \text{ component }\\ \quad Y_T=\xi =g(X_T) \end{array}\right. \end{aligned}$$

(1)

on a filtered complete probability space $(\varOmega , \mathcal {F}, P)$ with the natural filtration $(\mathcal {F}_t)_{0\le t \le T},$ where $a:[0, T] \times \mathbb {R}^n \times \mathbb {R}^m \times \mathbb {R}^{m\times d} \rightarrow \mathbb {R}^n$ and $b:[0, T] \times \mathbb {R}^n \times \mathbb {R}^m \times \mathbb {R}^{m\times d} \rightarrow \mathbb {R}^{n\times d},$ are drift and diffusion coefficients in the forward component, respectively; $W_t=(W^1_t,\ldots , W^d_t)^T$ is a d-dimensional Brownian motion (all Brownian motions are independent with each other); $f(t, X_t, Y_t, Z_t): [0, T] \times \mathbb {R}^n \times \mathbb {R}^m \times \mathbb {R}^{m\times d} \rightarrow \mathbb {R}^m$ is the driver function and $\xi$ is the square-integrable terminal condition. We see that the terminal condition $Y_T$ depends on final value of the forward component. Note that a, b and f are all $\mathcal {F}_t$-adapted, and a triple $(X_t, Y_t, Z_t)$ is called an $L^2$-adapted solution of (1) if it is $\mathcal {F}_t$-adapted, square integrable, and satisfies

$$\begin{aligned} \left\{ \begin{array}{l} X_t = X_0 + \int _0^t a(s, X_s, Y_s, Z_s)\,ds + \int _0^t b(s, X_s, Y_s, Z_s)\,dW_s,\\ Y_t = \xi + \int _t^T f(s, X_s, Y_s, Z_s)\,ds - \int _t^T Z_s\,dW_s. \end{array}\right. \end{aligned}$$

(2)

One obtains decoupled FBSDEs if a and b are independent with $Y_t$ and $Z_t$ in (1), which become backward stochastic differential equations (BSDEs) when $a=0$ and $b=1.$

The existence and uniqueness of solution of the BSDEs assuming the Lipschitz conditions on $f,a,b~\text{ and }~g$ are proven by Pardoux and Peng [25, 26]. The uniqueness of solution is extended under more general assumptions for f in [20], but only in the one-dimensional case. The existence and uniqueness of solution of FBSDEs have been studied in [21, 28].

In recent years, many numerical methods have been proposed for the BSDEs and FBSDEs. We list some of them here: [2,3,4, 8,9,10,11, 14, 15, 18, 19, 22,23,24, 30, 32,33,34, 36, 37, 39,40,43], and many others. In this literature, the high-order methods rely on the high-order approaches for both the forward and backward components, where are clearly difficult and computationally expensive to achieve.

Moreover, Zhao et al. proposed in [38] the high-order multi-step schemes for FBSDEs, which can keep high-order accuracy while using the Euler method to solve the forward component. This is quite interesting since the use of Euler method can dramatically simplify the entire computations. However, the convergence rate is restricted to sixth order, since the stability condition can not be satisfied for a higher order. For this reason, we adopt this method in this work, and we combine some multi-steps to achieve higher rate of convergence. More precisely, following the idea, proposed in [38], two reference ordinary differential equations (ODEs) can be firstly derived for (2), which contain the conditional expectations and their derivatives. To approximate these derivatives, the authors in [38] use the numerical methods derived using Taylor’s expansions for multi-time levels, say $t_i, i=0,1,\ldots , k,$ k is a positive integer. The resulting multi-step scheme is stable only up to that $k=6.$ In order to achieve a better stability for higher-order methods, we propose new finite difference methods (FDMs) by novelly combining Taylor’s expansions for some multi-time levels, e.g., $t_i,$ $t_{i+1}$ $t_{i+2}$ and $t_{i+3},$ $i=0,1,\ldots , k.$ This is to say that we propose new multi-step schemes by using the FDMs with the combination of multi-steps for a better stability. Furthermore, we investigate what is the best combinations. The resulting conditional expectations are solved by using the Gaussian quadrature rules. And thanks to the local property of the generator of diffusion processes, the forward component, $X_t$ can be simply solved by using the Euler method while keeping a high rate of convergence for the numerical solution of FBSDE. Numerical experiments are presented to demonstrate the improvement in the rate of convergence.

In the next section, we start with preliminaries on FBSDEs and derive in Sect. 3 the approximations of derivatives by using the FDM with combined multi-steps. In Sect. 4, we derive the reference ODEs, based on which the semi-discrete higher-order multi-step schemes are introduced for solving decoupled FBSDEs. Section 5 is devoted to the fully discrete higher-order schemes. In Sect. 6, these methods are extended to solve a coupled FBSDE. In Sect. 7, several numerical experiments on the decoupled and coupled FBSDEs including two-dimensional applications are provided to show the higher efficiency and accuracy. Finally, Sect. 8 concludes this work.

2 Preliminaries

As mentioned before, throughout the paper we assume that $(\varOmega , \mathcal {F}, P)$ is a complete, filtered probability space. A standard d-dimensional Brownian motion $W_t$ with a finite terminal time T is defined, and the forward component, $X_t$ generates the filtration $\mathcal {F}_t=\sigma \{X_s, 0\le s \le t\}.$ And the usual hypotheses should be satisfied. We denote the set of all $\mathcal {F}_t$-adapted and square integrable processes in $\mathbb {R}^d$ with $L^2=L^2(0,T;\mathbb {R}^d),$ and list following notations to be used:

• $|\cdot |:$ the Euclidean norm in $\mathbb {R},$ $\mathbb {R}^n$ and $\mathbb {R}^{n\times d};$

• $\mathcal {F}_t^{s,x}:$ $\sigma$-algebra generated by the diffusion process $\{X_r, s\le r\le t, X_s=x\};$

• ${\mathbb {E}}_{t}^{s,x}[\cdot ]:$ conditional expectation under $\mathcal {F}_t^{s,x},$ i.e., ${\mathbb {E}}_{t}^{s,x}[\cdot |\mathcal {F}_t^{s,x}];$

• $C_b^k:$ the set of continuous functions with uniformly bounded derivatives up to order k; 

• $C^{k_1,k_2}:$ the set of functions with continuous partial derivatives $\frac{\partial }{\partial t}$ and $\frac{\partial }{\partial x}$ up to $k_1$ and $k_2,$ respectively;

• $C_L:$ the set of uniformly Lipschitz continuous function with respect to the spatial variables;

• $C^{\frac{1}{2}}_L:$ the subset of $C_L$ such that its element is Hölder-$\frac{1}{2}$ continuous with respect to time, with uniformly bounded Lipschitz and Hölder constants.

Let $X_t$ be a diffusion process

$$\begin{aligned} X_t = x_0 + \int _0^t a(s, X_s)\,ds + \int _0^t b(s, X_s)\,dW_s \end{aligned}$$

(3)

starting at $(t_0, x_0)$ and $t\in [t_0, T],$ which has a unique solution. Note that ${\mathbb {E}}_{s}^{x}[X_t]:={\mathbb {E}}_{s}^{s,x}[X_t]$ is equal to $\mathbb {E}[X_t|X_s=x]$ for all $s \le t$ with the Markov property of the diffusion process. Given a measurable function $g:[0, T] \times \mathbb {R}^n \rightarrow \mathbb {R},$ ${\mathbb {E}}_{s}^{x}[g(t, X_t)]$ is a function of (t, s, x),  whose partial derivative with respect to t reads

$$\begin{aligned} \frac{\partial {\mathbb {E}}_{s}^{x}[g(t, X_t)]}{\partial t}=\lim _{\tau \rightarrow 0^+}\frac{{\mathbb {E}}_{s}^{x}[g(t+\tau , X_{t+\tau })]-{\mathbb {E}}_{s}^{x}[g(t, X_t)]}{\tau } \end{aligned}$$

provided that the limit exists and is finite.

Definition 1

(Generator) The generator $\mathcal {A}_t^x$ of $X_t$ satisfying (3) on a measurable function $g:[0, T] \times \mathbb {R}^n \rightarrow \mathbb {R}$ is defined by

$$\begin{aligned} \mathcal {A}_t^xg(t,x)=\lim _{h \rightarrow 0^+}\frac{{\mathbb {E}}_{t}^{x}[g(t+h, X_{t+h})]-g(t, x)}{h},\quad x\in \mathbb {R}^n. \end{aligned}$$

Theorem 1

Let $X_t$ be the diffusion process defined by (3), then it holds

$$\begin{aligned} \mathcal {A}_t^x f(t, x)=\mathcal {L}_{t, x}f(t, x) \end{aligned}$$

(4)

for $f \in C^{1,2}\left( [0,T]\times \mathbb {R}^n\right)$ with

$$\begin{aligned} \mathcal {L}_{t,x}=\frac{\partial }{\partial t}+\displaystyle \sum _{i}a_i(t,x)\frac{\partial }{\partial x_i}+\frac{1}{2}\displaystyle \sum _{i,j}(bb^{\top })_{i,j}(t,x)\frac{\partial ^2}{\partial x_i \partial x_j}. \end{aligned}$$

The proof can be simply completed by using the Itô’s lemma and the dominated convergence theorem.

Remark 1

From (4) one can straightforwardly deduce that

$$\begin{aligned} \mathcal {A}_t^{X_t} f(t, X_t)=\mathcal {L}_{t, X_t}f(t, X_t), \end{aligned}$$

which is a stochastic process.

By using the Itô’s lemma and Theorem 1 we calculate

$$\begin{aligned} \left. \frac{d\,{\mathbb {E}}_{t_0}^{x_0}[f(t, X_t)]}{d t}\right| _{t=t_0}=\lim _{t \rightarrow t_0^+}\frac{{\mathbb {E}}_{t_0}^{x_0}[f(t, X_{t})]-g(t_0, x_0)}{t-t_0}=\mathcal {L}_{t, x}f(t_0, x_0)=\mathcal {A}_t^{x} f(t_0, x_0), \end{aligned}$$

from which we deduce Theorem 2 as follows.

Theorem 2

Assume that $f \in C^{1,2}\left( [0,T]\times \mathbb {R}^n\right)$ and ${\mathbb {E}}_{t_0}^{x_0}\left[ \left| \mathcal {L}_{t, X_t} f(t, X_t)\right| \right] <\infty ,$ let $t_0<t$ be a fixed time point, and $x_0\in \mathbb {R}^n$ be a fixed space point, it holds that

$$\begin{aligned} \frac{d\,{\mathbb {E}}_{t_0}^{x_0}[f(t, X_t)]}{d t}={\mathbb {E}}_{t_0}^{x_0}\left[ \mathcal {A}_{t}^{ X_t} f(t, X_t)\right] ,\quad t\ge t_0. \end{aligned}$$

Furthermore, one has the following identity

$$\begin{aligned} \left. \frac{d\,{\mathbb {E}}_{t_0}^{x_0}[f(t, X_t)]}{d t}\right| _{t=t_0}=\left. \frac{d\,{\mathbb {E}}_{t_0}^{x_0}[f(t, \tilde{X}_t)]}{d t}\right| _{t=t_0}, \end{aligned}$$

(5)

where $\tilde{X}_t$ is an approximating diffusion process defined by

$$\begin{aligned} \tilde{X}_t = x_0 + \int _0^t \tilde{a}\,ds + \int _0^t \tilde{b}\,dW_s, \end{aligned}$$

$\tilde{a}_t=\tilde{a}(t,\tilde{X}_t;t_0,x_0)$ and $\tilde{b}_t=\tilde{b}(t,\tilde{X}_t;t_0,x_0)$ are smooth functions of $(t,\tilde{X}_t)$ with the parameter $(t_0,x_0)$ which satisfy

$$\begin{aligned} \tilde{a}(t_0,\tilde{X}_{t_0};t_0,x_0)=a(t_0,x_0)~\text{ and } ~\tilde{b}(t_0,\tilde{X}_{t_0};t_0,x_0)=b(t_0,x_0). \end{aligned}$$

It has been noted in [38] that the different approximations of (5) can be obtained by choosing different $\tilde{a}_t$’s and $\tilde{b}_t$’s. One can simply e.g., choose $\tilde{a}(s,\tilde{X}_{s};t_0,x_0)=a(t_0,x_0)$ and $\tilde{b}(s,\tilde{X}_{s};t_0,x_0)=b(t_0,x_0)$ for all $s \in [t_0, t].$

For existence, regularity and representation of solutions of decoupled FBSDEs we refer to [23, 27, 35]. In the following of this section we will present some of those. We denote the forward stochastic differential equation (SDE) starting from (s, x) with $X_t^{s,x}$ and consider the decoupled FBSDEs

$$\begin{aligned} \left\{ \begin{array}{l} X_t^{s,x} = x + \int _s^t a(r, X^{s,x}_r)\,ds + \int _s^t b(r, X^{s,x}_r)\,dW_r,\\ Y^{s,x}_t = g(X^{s,x}_T) + \int _t^T f(r, X^{s,x}_r, Y^{s,x}_r, Z^{s,x}_r)\,dr - \int _t^T Z ^{s,x}_r\,dW_r, \end{array}\right. \end{aligned}$$

(6)

where $t\in [s, T],$ and the superscript $^{s,x}$ will be omitted when the context is clear.

Throughout the paper, we shall often make use of the following standing assumptions:

• 1. The functions $a, b \in C_b^1,$ and assume

  $$\begin{aligned} \sup _{0\le t\le T}\left\{ |a(t,0)|+|b(t,0)|\right\} \le L, \end{aligned}$$

  where the common constant $L>0$ denotes all the Lipschitz constants.

• 2. $n=d$ and we assume that b satisfies

  $$\begin{aligned}b(t,x)b^\top (t,x) \ge \frac{1}{L} I_n,\quad \forall (t,x)\in [0,T]\times \mathbb {R}^n.\end{aligned}$$

• 3. $a, b, f, g\in C_L,$ and assume that

  $$\begin{aligned} \sup _{0\le t\le T}|f(t,0,0,0)|+|g(0)| \le L, \end{aligned}$$

  where L denotes all the Lipschitz constants.

• 4. $a, b, f \in C_L^{\frac{1}{2}}.$

Under the above conditions, it is clear that (6) is well-posed; the resulting integrands by taking conditional expectation on both side of the backward component is continuous with respect to time; the nonlinear Feynman-Kac formula [23, 27] can be given as follows.

Theorem 3

Let $u \in C^{1,2}\left( [0,T]\times \mathbb {R}^n\right)$ be a classical solution to the following PDE

$$\begin{aligned}\mathcal {L}_{t,x}u(t,x)+f(t,x,u(t,x),\nabla u(t,x)b(t,x))=0,\quad u(T,x)=g(x),\end{aligned}$$

then $Y^{s,x}_t = u(t, X_t^{s,x}),$ $Z^{s,x}_t=\nabla _x u(t, X_t^{s,x})b(t,X_t^{s,x}),\, \forall t\in (s,T]$ is the unique solution to (6).

3 Calculation of the Weights in the FDM for Approximating Derivative

In this section we calculate the weights in the FDM for approximating the function derivatives, e.g., $\frac{du(t)}{dt}.$ Let $u(t)\in C_b^{k+1}, k$ is a positive integer, and $t_i=i\varDelta t,$ i.e., $t_0<t_1<\cdots <t_k.$

3.1 Combination of Two Temporal Points

We consider the Taylor’s expansions of $u(t_i)$ and $u(t_{i+1}), i=0,\ldots ,k$

$$\begin{aligned} {\left\{ \begin{array}{ll} u(t_i)=\displaystyle \sum _{j=0}^{k}\frac{(\varDelta t_i)^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}(\varDelta t_i)^{k+1},\\ u(t_{i+1})=\displaystyle \sum _{j=0}^{k}\frac{(\varDelta t_{i+1})^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}(\varDelta t_{i+1})^{k+1}, \end{array}\right. } \end{aligned}$$

from which we can deduce

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \sum _{i=0}^{k}\alpha _{k,i} u(t_i)=\displaystyle \sum _{j=0}^{k}\frac{\displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_i)^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}\left( \displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_i)^{k+1}\right) ,\\ \displaystyle \sum _{i=0}^{k}\alpha _{k,i}u(t_{i+1})=\displaystyle \sum _{j=0}^{k}\frac{\displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_{i+1})^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}\left( \displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_{i+1})^{k+1}\right) , \end{array}\right. } \end{aligned}$$

where $\alpha _{k,i}, i=0,1,\ldots ,k$ are real numbers. Straightforwardly, we obtain

$$\begin{aligned} \begin{aligned} \displaystyle \sum _{i=0}^{k}\alpha _{k,i} (u(t_i)+u(t_{i+1}))=&\displaystyle \sum _{j=0}^{k}\frac{\displaystyle \sum _{i=0}^{k}\alpha _{k,i}\left( (\varDelta t_i)^j+(\varDelta t_{i+1})^j\right) }{j!}\frac{d^ju}{dt^j}(t_0)\\&+\mathcal {O}\left( \displaystyle \sum _{i=0}^{k}\alpha _{k,i}\left( (\varDelta t_i)^{k+1}+(\varDelta t_{i+1})^{k+1}\right) \right) , \end{aligned} \end{aligned}$$

and also

$$\begin{aligned} \frac{du}{dt}(t_0)=\displaystyle \sum _{i=0}^{k}\alpha _{k,i}\left( u(t_i)+u(t_{i+1})\right) + \mathcal {O}\left( \displaystyle \sum _{i=0}^{k}\alpha _{k,i}\left( (\varDelta t_i)^{k+1}+(\varDelta t_{i+1})^{k+1}\right) \right) \end{aligned}$$

(7)

by choosing

$$\begin{aligned} \frac{\displaystyle \sum _{i=0}^{k}\alpha _{k,i}\left( (\varDelta t_i)^j+(\varDelta t_{i+1})^j\right) }{j!}= {\left\{ \begin{array}{ll} 1,\quad j=1,\\ 0, \quad j \ne 1. \end{array}\right. } \end{aligned}$$

(8)

Note that $t_i=i\varDelta t$ and $t_{i+1}=(i+1)\varDelta t,$ the conditions in (8) are equivalent to the following system

$$\begin{aligned} \begin{bmatrix} 2 &{} 2 &{} 2 &{} \dots &{} 2 \\ 1 &{} 3 &{} 5 &{} \dots &{} k+(k+1) \\ 1 &{} 5 &{} 13 &{} \dots &{} k^2+(k+1)^2 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 1 &{} 1^k+2^k &{} 2^k+3^k &{} \dots &{} k^k+(k+1)^k \end{bmatrix} \times \begin{bmatrix} \alpha _{k,0}\varDelta t \\ \alpha _{k,1}\varDelta t \\ \alpha _{k,2}\varDelta t \\ \vdots \\ \alpha _{k,k}\varDelta t \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0\\ \vdots \\ 0 \end{bmatrix}, \end{aligned}$$

which can be solved for $\alpha _{k,i}\varDelta t, i=0,\ldots ,k.$ We refer to the algorithm proposed in [13] for those solutions. We report $\alpha _{k,i}\varDelta t$ for $k=1, 2,\ldots , 7$ in Table 1. The related multi-step schemes proposed in this paper is unstable from $k=8,$ which will be explained below.

Table 1 The values of $\alpha _{k,i}\varDelta t$ for combining two temporal points

The multi-step schemes (combining two temporal points) can be obtained by approximating the reference ODEs (see Sect. 4.1) with (7). Considering the ODE

$$\begin{aligned} \frac{Y(t)}{dt}=f(t,Y(t)),\quad t\in [0,T) \end{aligned}$$

(9)

with the known terminal condition Y(T),  we investigate the stability, see also [38]. Applying (7) to (9) one obtains the multi-step scheme as

$$\begin{aligned} \alpha _{k,0}Y^n + \displaystyle \sum _{j=1}^{k}\left( \alpha _{k,j-1} +\alpha _{k,j}\right) Y^{n+j}+\alpha _{k,k}Y^{n+k+1}=f(t_n, Y^n) \end{aligned}$$

(10)

under the uniform time partition $0=t_0<t_1<\cdots <t_N=T.$ (10) is stable if the roots $\{\lambda _{k,j}\}^k_{j=1}$ of the characteristic equation

$$\begin{aligned} P(\lambda )=\alpha _{k,0} \lambda ^{k+1} + \displaystyle \sum _{j=1}^{k}\left( \alpha _{k,j-1}+\alpha _{k,j}\right) \lambda ^{k+1-j} + \alpha _{k,k}\lambda ^0 \end{aligned}$$

(11)

satisfy the following root conditions [6]

• $|\lambda _{k,j}|\le 1,$

• $P^{'}(\lambda _{k,j})\ne 0$ if $|\lambda _{k,j}|=1$ (simple roots).

By the definition of $\alpha _{k,j}$ in Tabel 1, it can be checked that 1 is the simple root of the latter characteristic function for each k,  except which we list the maximum absolute values of the roots for $k=2,\ldots ,8$ in Table 2. We see that the multi-step scheme (10) is unstable for $k\ge 8.$

Table 2 The maximum absolute root of (11) except the simple roots

However, compared to the multi-step scheme proposed in [38](unstable $\ge 7$), stability for $k=7$ has been achieved, i.e., 1-order higher convergence rate is obtained. Combination of more temporal points can be done similarly, and the resulting multi-step schemes have different instabilities. In our investigation we find that the multi-step scheme by combining four temporal points are stable for $k\le 9,$ which is the best. Thus, we show its detailed derivation in next subsection and apply it for the numerical experiments.

3.2 Combination of Four Temporal Points

Similarly but slightly different to the multi-step scheme in Sect. 3.1, we consider the Taylor’s expansions of $u(t_i),$ $u(t_{i+1}),$ $u(t_{i+2})$ and $u(t_{i+3}), i=0,\ldots ,k,$

$$\begin{aligned} {\left\{ \begin{array}{ll} u(t_i)=\displaystyle \sum _{j=0}^{k}\frac{(\varDelta t_i)^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}(\varDelta t_i)^{k+1},\\ u(t_{i+1})=\displaystyle \sum _{j=0}^{k}\frac{(\varDelta t_{i+1})^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}(\varDelta t_{i+1})^{k+1},\\ u(t_{i+2})=\displaystyle \sum _{j=0}^{k}\frac{(\varDelta t_{i+2})^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}(\varDelta t_{i+2})^{k+1},\\ u(t_{i+3})=\displaystyle \sum _{j=0}^{k}\frac{(\varDelta t_{i+3})^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}(\varDelta t_{i+3})^{k+1}, \end{array}\right. } \end{aligned}$$

from which we deduce

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \sum _{i=0}^{k}\alpha _{k,i} u(t_i)=\displaystyle \sum _{j=0}^{k}\frac{\displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_i)^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}\left( \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_i)^{k+1}\right) ,\\ \displaystyle \sum _{i=0}^{k}\alpha _{k,i}u(t_{i+1})=\displaystyle \sum _{j=0}^{k}\frac{\displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_{i+1})^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}\left( \displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_{i+1})^{k+1}\right) ,\\ \displaystyle \sum _{i=0}^{k}\alpha _{k,i}u(t_{i+2})=\displaystyle \sum _{j=0}^{k}\frac{\displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_{i+2})^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}\left( \displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_{i+2})^{k+1}\right) ,\\ \displaystyle \sum _{i=0}^{k}\alpha _{k,i}u(t_{i+3})=\displaystyle \sum _{j=0}^{k}\frac{\displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_{i+3})^j}{j!}\frac{d^ju}{dt^j}(t_0)+\mathcal {O}\left( \displaystyle \sum _{i=0}^{k}\alpha _{k,i}(\varDelta t_{i+3})^{k+1}\right) , \end{array}\right. } \end{aligned}$$

where $\alpha _{k,i}, i=0,1,\ldots ,k$ are real numbers as well. Straightforwardly, we obtain

$$\begin{aligned} \begin{aligned} \displaystyle&\sum _{i=0}^{k}\alpha _{k,i} (u(t_i)+u(t_{i+1})+u(t_{i+2})+u(t_{i+3}))\\&\quad =\displaystyle \sum _{j=0}^{k}\frac{\displaystyle \sum _{i=0}^{k}\alpha _{k,i}\left( (\varDelta t_i)^j+(\varDelta t_{i+1})^j+(\varDelta t_{i+2})^j+(\varDelta t_{i+3})^j\right) }{j!}\frac{d^ju}{dt^j}(t_0)\\&\qquad +\underbrace{\mathcal {O}\left( \displaystyle \sum _{i=0}^{k}\alpha _{k,i}\left( (\varDelta t_i)^{k+1}+(\varDelta t_{i+1})^{k+1}+(\varDelta t_{i+2})^{k+1}+(\varDelta t_{i+3})^{k+1}\right) \right) }_{:=\epsilon } \end{aligned} \end{aligned}$$

(12)

and also

$$\begin{aligned} \frac{du}{dt}(t_0)=\displaystyle \sum _{i=0}^{k}\alpha _{k,i}\left( u(t_i)+u(t_{i+1})+u(t_{i+2})+u(t_{i+3})\right) +\epsilon \end{aligned}$$

(13)

by choosing

$$\begin{aligned} \frac{\displaystyle \sum _{i=0}^{k}\alpha _{k,i}\left( (\varDelta t_i)^j+(\varDelta t_{i+1})^j+(\varDelta t_{i+2})^j+(\varDelta t_{i+3})^j\right) }{j!}= {\left\{ \begin{array}{ll} 1,\quad j=1,\\ 0, \quad j \ne 1, \end{array}\right. } \end{aligned}$$

which are equivalent to the following system

$$\begin{aligned}&\begin{bmatrix} 4 &{} 4 &{} &{} \dots &{} 4 \\ 6 &{} 10 &{} &{} \dots &{} k+(k+1)+(k+2)+(k+3) \\ 14 &{} 30 &{} &{} \dots &{} k^2+(k+1)^2+(k+2)^2+(k+3)^2\\ \vdots &{} \vdots &{} &{} \vdots &{} \vdots \\ 1^k+2^k+3^k &{} 1^k+2^k+3^k+4^k &{} &{} \dots &{} k^k+(k+1)^k+(k+2)^k+(k+3)^k \end{bmatrix} \\&\quad \times \begin{bmatrix} \alpha _{k,0}\varDelta t \\ \alpha _{k,1}\varDelta t \\ \alpha _{k,2}\varDelta t \\ \vdots \\ \alpha _{k,k}\varDelta t \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0\\ \vdots \\ 0 \end{bmatrix}. \end{aligned}$$

In Table 3 we report solutions of the latter system for $k=1,\ldots , 9.$

Table 3 The values of $\alpha _{k,i}\varDelta t$ for combining four temporal points

Applying (13) to (9) one obtain the multi-step scheme as

$$\begin{aligned} \begin{aligned} \alpha _{k,0}Y^n&+(\alpha _{k,0}+\alpha _{k,1})Y^{n+1}+(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2})Y^{n+2}\\&+\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) Y^{n+j}\\&+(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k})Y^{n+k+1} \\ {}&+(\alpha _{k,k-1}+\alpha _{k,k})Y^{n+k+2}+\alpha _{k,k}Y^{n+k+3}=f(t_n, Y^n), \end{aligned} \end{aligned}$$

(14)

whose characteristic equation reads

$$\begin{aligned} \begin{aligned}&\alpha _{k,0}\lambda ^{k+3} +(\alpha _{k,0}+\alpha _{k,1})\lambda ^{k+2}+(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2})\lambda ^{k+1}\\&\qquad +\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) \lambda ^{k+3-j}\\&\qquad +(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k})\lambda ^{2}+(\alpha _{k,k-1}+\alpha _{k,k})\lambda ^{1}+\alpha _{k,k}\lambda ^{0}=0. \end{aligned} \end{aligned}$$

(15)

By the definition of $\alpha _{k,j}$ in Table 3, it can be checked that 1 is the simple root of the latter characteristic function for each k. The maximum absolute values of the roots for $k=2,\ldots ,10$ expect the simple roots are listed in Table 4, we see that the multi-step scheme (14) is stable for $k\le 9.$

Table 4 The maximum absolute root of (15) except the simple roots

We remark that the stability cannot be guaranteed for $k>9$ by combining more temporal points, e.g., the multi-step scheme constructed by combining five temporal points is stable for $k\le 8.$

4 The Semi-discrete Multi-step Scheme for Decoupled FBSDEs

Following the idea in [38] we derive the semi-discrete scheme for (2) in the decoupled case. We consider the time interval [0, T] with the following partition

$$\begin{aligned} 0=t_0<t_1<t_2<\cdots t_{N_T}=T. \end{aligned}$$

We denote $t_{n+k}-t_n$ by $\varDelta t_{n,k}$ and $W_{t_{n+k}} -W_{t_n}$ by $\varDelta W_{n,k},$ i.e., $\varDelta t_{t_n,t}=t-t_n$ and $\varDelta W_{t_n,t}=W_t-W_{t_n}$ for $t\ge t_n.$

4.1 Two Reference ODEs

Let $(X_t,Y_t,Z_t)$ be the solution of the decoupled FBSDEs (2). By taking conditional expectation $\mathbb {E}_{t_n}^x[\cdot ]$ on both sides of the backward component in (2) one obtains the integral equation

$$\begin{aligned} \mathbb {E}_{t_n}^x\left[ Y_t\right] =\mathbb {E}_{t_n}^x\left[ \xi \right] + \int _t^T\mathbb {E}_{t_n}^x\left[ f(s,X_s,Y_s,Z_s)\right] \,ds,\quad \forall t \in [t_n,T]. \end{aligned}$$

As explained in Sect. 2, the integrand in the latter integral equation is continuous with respect to the time. By taking the derivative with respect to t on both sides one thus obtain the first reference ODE

$$\begin{aligned} \frac{d\,\mathbb {E}_{t_n}^x\left[ Y_t\right] }{dt}=-\mathbb {E}_{t_n}^x\left[ f(t,X_t,Y_t,Z_t)\right] ,\quad \forall t \in [t_n,T]. \end{aligned}$$

(16)

Furthermore, we have

$$\begin{aligned} Y_{t_n}=Y_t+\int _{t_n}^{t}f(s,X_s,Y_s,Z_s)\,ds-\int _{t_n}^{t}Z_s\,dW_s,\quad t\in [t_n,T]. \end{aligned}$$

By multiplying both sides of the latter equation by $\varDelta W_{t_n,t}^{\top }$ and again taking the conditional expectation $\mathbb {E}_{t_n}^x[\cdot ]$ on its both sides we obtain

$$\begin{aligned} 0=\mathbb {E}_{t_n}^x\left[ Y_t\varDelta W_{t_n,t}^{\top }\right] +\int _{t_n}^{t}\mathbb {E}_{t_n}^x\left[ f(s,X_s,Y_s,Z_s)\varDelta W_{t_n,s}^{\top }\right] \,ds-\int _{t_n}^{t}\mathbb {E}_{t_n}^x\left[ Z_s\right] \,ds, \end{aligned}$$

$t\in [t_n,T].$ Similarly, we obtain the second reference ODE

$$\begin{aligned} \frac{d\,\mathbb {E}_{t_n}^x\left[ Y_t\varDelta W_{t_n,t}^{\top }\right] }{dt}=-\mathbb {E}_{t_n}^x\left[ f(t,X_t,Y_t,Z_t)\varDelta W_{t_n,t}^{\top }\right] + \mathbb {E}_{t_n}^x\left[ Z_t\right] ,\quad t \in [t_n,T]. \end{aligned}$$

(17)

by taking the derivative with respect to $t \in [t_n, T].$

Remark 2

The both ODEs (16) and (17) are the derived reference equations for (2). The next step is to derive the numerical schemes by approximating the conditional expectations and the derivatives in (16) and (17).

4.2 The Semi-discrete Scheme

Let $\bar{a}(t,x)$ and $\bar{b}(t,x)$ be smooth functions for $t\in [t_n,T]$ and $x\in \mathbb {R}^n$ satisfying $\bar{a}(t,x)=a(t,x)$ and $\bar{b}(t,x)=b(t,x),$ and thus define the diffusion process

$$\begin{aligned} \bar{X}_{t}^{t_n,x} = x + \int _{t_n}^{t}\bar{a}(s,\bar{X}_{s}^{t_n,x})\,ds+\int _{t_n}^{t}\bar{b}(s,\bar{X}_{s}^{t_n,x})\,dW_s. \end{aligned}$$

(18)

Let $(X_{t}^{t_n,x},Y_{t}^{t_n,x},Z_{t}^{t_n,x})$ be the solution of the decoupled FBSDEs, i.e., $Y_{t}^{t_n,x}$ and $Z_{t}^{t_n,x}$ can be represented by $u(t,X_{t}^{t_n,x})$ and $\nabla _x u(t,X_{t}^{t_n,x}) b(s,X_{s}^{t_n,x}),$ respectively, see Theorem 3.

Therefore, we set $\bar{Y}_t^{t_n,x}=u(t,\bar{X}_{t}^{t_n,x})$ and $\bar{Z}_t^{t_n,x}=\nabla _x u(t,\bar{X}_{t}^{t_n,x}) b(s,\bar{X}_{s}^{t_n,x}),$ By Theorem 2, we have

$$\begin{aligned} \left. \frac{d\,{\mathbb {E}}_{t_n}^{x}[{Y}_{t}^{t_n,x}]}{d t}\right| _{t=t_n}= \left. \frac{d\,{\mathbb {E}}_{t_n}^{x}[{\bar{Y}}_{t}^{t_n,x}]}{d t}\right| _{t=t_n} \end{aligned}$$

and

$$\begin{aligned} \left. \frac{d\,{\mathbb {E}}_{t_n}^{x}[{Y}_{t}^{t_n,x}\varDelta W_{t_n,t}^{\top }]}{d t}\right| _{t=t_n}= \left. \frac{d\,{\mathbb {E}}_{t_n}^{x}[{\bar{Y}}_{t}^{t_n,x}\varDelta W_{t_n,t}^{\top }]}{d t}\right| _{t=t_n}. \end{aligned}$$

Finally, we apply (13) to terms on the right hand side of both the latter equations to obtain

$$\begin{aligned} \begin{aligned}&\left. \frac{d\,{\mathbb {E}}_{t_n}^{x}[{Y}_{t}^{t_n,x}]}{d t}\right| _{t=t_n} \\&\quad = \displaystyle \sum _{i=0}^{k}\alpha _{k,i}{\mathbb {E}}_{t_n}^{x} \left[ {\bar{Y}}_{t_{n+i}}^{t_n,x}+{\bar{Y}}_{t_{n+i+1}}^{t_n,x} +{\bar{Y}}_{t_{n+i+2}}^{t_n,x} +{\bar{Y}}_{t_{n+i+3}}^{t_n,x}\right] +{\bar{R}}_{y,n}^{k}\\&\quad =\alpha _{k,0}{\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n}}^{t_n,x}\right] +(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+1}}^{t_n,x}\right] +(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+2}}^{t_n,x}\right] \\&\qquad +\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) {\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+j}}^{t_n,x}\right] \\&\qquad +(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+1}}^{t_n,x}\right] \\&\qquad +(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+2}}^{t_n,x}\right] +\alpha _{k,k}{\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+3}}^{t_n,x}\right] +{\bar{R}}_{y,n}^{k} \end{aligned} \end{aligned}$$

(19)

and

$$\begin{aligned} \begin{aligned}&\left. \frac{d\,{\mathbb {E}}_{t_n}^{x}[{Y}_{t}^{t_n,x}\varDelta W_{t_n,t}^{\top }]}{d t}\right| _{t=t_n}\\&\quad =\displaystyle \sum _{i=1}^{k}\alpha _{k,i}{\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+i}}^{t_n,x}\varDelta W_{n,i}^{\top }\right. \\&\qquad \left. +{\bar{Y}}_{t_{n+i+1}}^{t_n,x}\varDelta W_{n,i+1}^{\top }+{\bar{Y}}_{t_{n+i+2}}^{t_n,x}\varDelta W_{n,i+2}^{\top }+{\bar{Y}}_{t_{n+i+3}}^{t_n,x}\varDelta W_{n,i+3}^{\top }\right] +{\bar{R}}_{z,n}^{k}\\&\quad =(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+1}}^{t_n,x}\varDelta W_{n,1}^{\top }\right] +(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+2}}^{t_n,x}\varDelta W_{n,2}^{\top }\right] \\&\qquad +\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) {\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+j}}^{t_n,x}\varDelta W_{n,j}^{\top }\right] \\&\qquad +(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+1}}^{t_n,x}\varDelta W_{n,k+1}^{\top }\right] \\&\qquad +(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+2}}^{t_n,x}\varDelta W_{n,k+2}^{\top }\right] \\&\qquad +\alpha _{k,k}{\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+3}}^{t_n,x}\varDelta W_{n,k+3}^{\top }\right] +{\bar{R}}_{z,n}^{k}, \end{aligned} \end{aligned}$$

(20)

where $\alpha _{k,i}$ are given in Table 3, ${\bar{R}}_{y,n}^{k}$ and ${\bar{R}}_{z,n}^{k}$ are truncation errors. By inserting (19) and (20) into (16) and (17), respectively, we obtain

$$\begin{aligned} \begin{aligned}&\alpha _{k,0}{\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n}}^{t_n,x}\right] +(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+1}}^{t_n,x}\right] +(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+2}}^{t_n,x}\right] \\&\qquad +\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) {\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+j}}^{t_n,x}\right] \\&\qquad +(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+1}}^{t_n,x}\right] +(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+2}}^{t_n,x}\right] \\&\qquad +\alpha _{k,k}{\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+3}}^{t_n,x}\right] =-f(t_n,x,Y_{t_n},Z_{t_n})+{R}_{y,n}^{k} \end{aligned} \end{aligned}$$

(21)

and

$$\begin{aligned} \begin{aligned}&(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+1}}^{t_n,x}\varDelta W_{n,1}^{\top }\right] +(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+2}}^{t_n,x}\varDelta W_{n,2}^{\top }\right] \\&\qquad +\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) {\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+j}}^{t_n,x}\varDelta W_{n,j}^{\top }\right] \\&\qquad +(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+1}}^{t_n,x}\varDelta W_{n,k+1}^{\top }\right] \\&\qquad +(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+2}}^{t_n,x}\varDelta W_{n,k+2}^{\top }\right] \\&\qquad +\alpha _{k,k}{\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{t_{n+k+3}}^{t_n,x}\varDelta W_{n,k+3}^{\top }\right] =Z_{t_n}+{R}_{z,n}^{k}, \end{aligned} \end{aligned}$$

(22)

where ${R}_{y,n}^{k}=-{\bar{R}}_{y,n}^{k}$ and ${R}_{z,n}^{k}=-{\bar{R}}_{z,n}^{k}.$

We denote the numerical approximations of $Y_t$ and $Z_t$ at $t_n$ by $Y^n$ and $Z^n,$ respectively. Furthermore, for $\bar{a}$ and $\bar{b}$ in (18) we choose $\bar{a}(t,\bar{X}_{t}^{t_n,x})=a(t_n,x)$ and $\bar{b}(t,\bar{X}_{t}^{t_n,x})=b(t_n,x)$ for $t\in [t_n,T].$ Finally, from (21) to (22), the semi-discrete scheme can be obtained as

Scheme 1

Assume that $Y^{N_T-i}$ and $Z^{N_T-i}$ are known for $i=0,1,\ldots ,k+2.$ For $n=N_T-k-3,\ldots , 0,$ $X^{n,j},$ $Y^{n}=Y^{n}(X^n)$ and $Z^{n}=Z^{n}(X^n)$ can be solved by

$$\begin{aligned} X^{n,j}=&X^n + a(t_n,X^n)\varDelta t_{n,j}+ b(t_n,X^n)\varDelta W_{n,j},\quad j=1,\ldots ,k+3, \end{aligned}$$

(23)

$$\begin{aligned} Z^n=&(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{X^n}\left[ {\bar{Y}}_{}^{n+1}\varDelta W_{n,1}^{\top }\right] +(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{X^n}\left[ {\bar{Y}}_{}^{n+2}\varDelta W_{n,2}^{\top }\right] \nonumber \\&+\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) {\mathbb {E}}_{t_n}^{X^n}\left[ {\bar{Y}}_{}^{n+j}\varDelta W_{n,j}^{\top }\right] \nonumber \\&+(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{X^n}\left[ {\bar{Y}}_{}^{n+k+1}\varDelta W_{n,k+1}^{\top }\right] \nonumber \\&+(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{X^n}\left[ {\bar{Y}}_{}^{n+k+2}\varDelta W_{n,k+2}^{\top }\right] +\alpha _{k,k}{\mathbb {E}}_{t_n}^{X^n}\left[ {\bar{Y}}_{}^{n+k+3}\varDelta W_{n,k+3}^{\top }\right] , \end{aligned}$$

(24)

$$\begin{aligned} \alpha _{k,0}Y^n=&-(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{X^n}\left[ \bar{Y}^{n+1}\right] -(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{X^n}\left[ \bar{Y}^{n+2}\right] \nonumber \\&-\displaystyle \sum _{j=3}^{k}(\alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}){\mathbb {E}}_{t_n}^{X^n}\left[ \bar{Y}^{n+j}\right] \nonumber \\&-(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{X^n}\left[ \bar{Y}^{n+k+1}\right] \nonumber \\&-(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{X^n}\left[ \bar{Y}^{n+k+2}\right] \nonumber \\&-\alpha _{k,k}{\mathbb {E}}_{t_n}^{X^n}\left[ \bar{Y}^{n+k+3}\right] -f(t_n,X^n,Y^n,Z^n). \end{aligned}$$

(25)

Remark 3

• 1. $\bar{Y}^{n+j}$ is the value of $Y^{n+j}$ at the space point $X^{n,j}$ for $j=1,\ldots ,k+3.$

• 2. The latter implicit equation can be solved by using iterative methods, e.g., Newton’s method or Picard scheme.

• 3. By Theorem 2 and (12) it holds [6]

  $$\begin{aligned} {\bar{R}}_{y,n}^{k}=\mathcal {O}(\varDelta t)^k~\text{ and }~{\bar{R}}_{z,n}^{k}=\mathcal {O}(\varDelta t)^k \end{aligned}$$

  provided that $\mathcal {L}^{k+4}_{t,x}u(t,x)$ is bounded, where ${\bar{R}}_{y,n}^{k}$ and ${\bar{R}}_{z,n}^{k}$ are defined in (19) and (20), respectively.

• 4. Similar to the scheme proposed in [38], one can obtain high-order accurate numerical solutions for (24) and (25), although the Euler scheme is used for (23). This is the main advantages because the usage of the Euler scheme reduces dramatically the total computational complexity, and one is only interested in the solution of (24) and (25) in many applications.

5 The Fully Discrete Multi-step Scheme for Decoupled FBSDEs

To solve $(X^n, Y^n,Z^n)$ numerically, next we consider the space discretization. We define firstly the partition of the real space as $\mathcal {R}^{n}_h=\{x_i|x_i\in \mathbb {R}^n\}$ with

$$\begin{aligned} h^n=\max _{x \in \mathbb {R}^n} dist(x,\mathcal {R}_h^n), \end{aligned}$$

where $dist(x,\mathcal {R}_h^n)$ is the distance from x to $\mathcal {R}_h^x.$ Furthermore, for each x we define the neighbor grid set (local subset) $\mathcal {R}_{h,x}^n$ satisfying

• 1. $dist(x,\mathcal {R}_h^n) < dist(x,\mathcal {R}_h^n)/\ \mathcal {R}_{h,x}^n,$

• 2. the number of elements in $\mathcal {R}_{h,x}^n$ is finite and uniformly bounded.

Based on the space discretization, we can solve $Y^n(x)$ and $Z^n(x)$ for each grid point $x\in \mathcal {R}^n_h,$ $n=N_t-k-3,\ldots ,0,$ by

$$\begin{aligned} \begin{aligned} Z^n=&(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{}^{n+1}\varDelta W_{n,1}^{\top }\right] +(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{}^{n+2}\varDelta W_{n,2}^{\top }\right] \\&+\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) {\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{}^{n+j}\varDelta W_{n,j}^{\top }\right] \\&+(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{}^{n+k+1}\varDelta W_{n,k+1}^{\top }\right] \\&+(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{}^{n+k+2}\varDelta W_{n,k+2}^{\top }\right] +\alpha _{k,k}{\mathbb {E}}_{t_n}^{x}\left[ {\bar{Y}}_{}^{n+k+3}\varDelta W_{n,k+3}^{\top }\right] , \end{aligned} \end{aligned}$$

(26)

and

$$\begin{aligned} \begin{aligned} \alpha _{k,0}Y^n=&-(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x}\left[ \bar{Y}^{n+1}\right] -(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x}\left[ \bar{Y}^{n+2}\right] \\&-\displaystyle \sum _{j=3}^{k}(\alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}){\mathbb {E}}_{t_n}^{x}\left[ \bar{Y}^{n+j}\right] \\&-(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ \bar{Y}^{n+k+1}\right] -(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ \bar{Y}^{n+k+2}\right] \\&-\alpha _{k,k}{\mathbb {E}}_{t_n}^{x}\left[ \bar{Y}^{n+k+3}\right] -f(t_n,x,Y^n,Z^n). \end{aligned} \end{aligned}$$

(27)

Note that $\bar{Y}^{n+j}$ is the value of $Y^{n+j}$ at the space point $X^{n,j}$ generated by

$$\begin{aligned} X^{n,j}=x + a(t_n,x)\varDelta t_{n,j}+ b(t_n,x)\varDelta W_{n,j},\quad j=1,\ldots ,k+3. \end{aligned}$$

However, $X^{n,j}$ does not belong to $\mathcal {R}_h^{n+j}.$ This is to say that the value of $Y^{n+j}$ at $X^{n,j}$ needs to be approximated based on the values of $Y^{n+j}$ on $\mathcal {R}_h^{n+j},$ this can be done by using a local interpolation. By $LI^n_{h,X}F$ we denote the interpolated value of the function F at space point $X\in \mathbb {R}^n$ by using the values of F only in the neighbor grid set, namely $\mathcal {R}^n_{h,X}.$ Including the interpolations, (26) and (27) become

$$\begin{aligned} \begin{aligned} Z^n=&(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x}\left[ LI^{n+1}_{h,X^{n,j}}{Y}_{}^{n+1}\varDelta W_{n,1}^{\top }\right] \\&+(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x}\left[ LI^{n+2}_{h,X^{n,j}}{Y}_{}^{n+2}\varDelta W_{n,2}^{\top }\right] \\&+\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) {\mathbb {E}}_{t_n}^{x}\left[ LI^{n+j}_{h,X^{n,j}}{Y}_{}^{n+j}\varDelta W_{n,j}^{\top }\right] \\&+(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ {LI^{n+k+1}_{h,X^{n,j}}Y}_{}^{n+k+1}\varDelta W_{n,k+1}^{\top }\right] \\&+(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ LI^{n+k+2}_{h,X^{n,j}}{Y}_{}^{n+k+2}\varDelta W_{n,k+2}^{\top }\right] \\&+\alpha _{k,k}{\mathbb {E}}_{t_n}^{x}\left[ {LI^{n+k+3}_{h,X^{n,j}}Y}_{}^{n+k+3}\varDelta W_{n,k+3}^{\top }\right] +R_{z,n}^{k,LI_h}, \end{aligned} \end{aligned}$$

(28)

and

$$\begin{aligned} \begin{aligned} \alpha _{k,0}Y^n=&-(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x}\left[ LI^{n+1}_{h,X^{n,j}}Y^{n+1}\right] \\&-(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x}\left[ LI^{n+2}_{h,X^{n,j}}Y^{n+2}\right] \\&-\displaystyle \sum _{j=3}^{k}(\alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}){\mathbb {E}}_{t_n}^{x}\left[ LI^{n+j}_{h,X^{n,j}}Y^{n+j}\right] \\&-(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ LI^{n+k+1}_{h,X^{n,j}}Y^{n+k+1}\right] \\&-(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x}\left[ LI^{n+k+2}_{h,X^{n,j}}Y^{n+k+2}\right] \\&-\alpha _{k,k}{\mathbb {E}}_{t_n}^{x}\left[ LI^{n+k+3}_{h,X^{n,j}}Y^{n+k+3}\right] -f(t_n,x,Y^n,Z^n)+R_{y,n}^{k,LI_h}. \end{aligned} \end{aligned}$$

(29)

Furthermore, to approximate the conditional expectations in (28) and (29) we employ the Gauss-Hermite quadrature rule which is an extension of the Gaussian quadrature method for approximating the value of integrals of the form $\int _{-\infty }^{\infty }\exp (-\mathbf{x}^2)g(\mathbf{x})\,d\mathbf{x}$ by

$$\begin{aligned} \int _{-\infty }^{\infty }\dots \int _{-\infty }^{\infty }g(\mathbf{x})\exp (-\mathbf{x}^{\top }{} \mathbf{x})\,d\mathbf{x} \approx \displaystyle \sum _{\mathbf{j}=1}^{L}w_{\mathbf{j}}g(\mathbf{a_j}), \end{aligned}$$

(30)

where $\mathbf{x}=(x_1,\ldots ,x_n)^{\top },$ $\mathbf{x}^{\top } \mathbf{x}= \displaystyle \sum _{j=1}^{n} x^2_j,$ L is the number of used sample points, $\mathbf{j}=\left( j_1,j_2,\ldots ,j_n\right) ,$ $\displaystyle \sum _{\mathbf{j=1}}^{L}=\displaystyle \sum _{j_1=1,\ldots ,j_n=1}^{L,\ldots ,L},$ $\mathbf{a_j}=(a_{j_1},\ldots ,a_{j_n})$ and $\omega _\mathbf{j}=\displaystyle \prod _{i=1}^{n}\omega _{j_i},$ $\{a_{j_i}\}_{j_i=1}^L$ are the roots of the Hermite polynomial $H_L(x)$ of degree L and $\{\omega _{j_i}\}_{j_i=1}^L$ are corresponding weights [1]. For a standard n-dimensional standard normal distributed random variable X we know that

$$\begin{aligned} \mathbb {E}\left[ g(X)\right]&=\frac{1}{(2\pi )^{\frac{d}{2}}}\int _{-\infty }^{\infty }g(\mathbf{x})\exp \left( -\frac{\mathbf{x}^{\top }{} \mathbf{x}}{2}\right) \,dx\\&=\frac{1}{(\pi )^{\frac{d}{2}}}\int _{-\infty }^{\infty }g(\sqrt{2}{} \mathbf{x})\exp \left( -\mathbf{x}^{\top }{} \mathbf{x}\right) \,dx\\&{\mathop {=}\limits ^{(30)}} \frac{1}{(\pi )^{\frac{d}{2}}}\displaystyle \sum _{\mathbf{j=1}}^{L}\omega _{\mathbf{j}}g(\mathbf{a_j})+R^{GH}_L, \end{aligned}$$

where $R^{GH}_L$ is the truncation error of the Gauss-Hermite quadrature rule for g.

In the conditional expectations of the form ${\mathbb {E}}_{t_n}^{x}\left[ LI^{n+j}_{h,X^{n,j}}{Y}_{}^{n+j}\varDelta W_{n,j}^{\top }\right]$ and ${\mathbb {E}}_{t_n}^{x}\left[ LI^{n+j}_{h,X^{n,j}}Y^{n+j}\right]$ in (28) and (29), $LI^{n+j}_{h,X^{n,j}}Y^{n+j}$ is the interpolated value of $\bar{Y}^{n+j},$ which is a function of $X^{n,j}$ and can be represented by (Theorem 3)

$$\begin{aligned} \bar{Y}^{n+j}=Y^{n+j}\left( X^{n+j}\right) =Y^{n+j}\left( X^{n+j}+a(t_n,X^n)\varDelta t_{n,j}+b(t_n,X^n)\varDelta W_{n,j}\right) , \end{aligned}$$

where $\varDelta W_{n,j} \sim \sqrt{\varDelta t_{n,j}}N(0,I_n).$ Straightforwardly, we can approximate those conditional expectations as

$$\begin{aligned} \mathbb {E}^{\mathbf{x},h}_{t_n}\left[ \bar{Y}^{n+j}\right]= & {} \frac{1}{\pi ^{\frac{d}{2}}}\sum _{\mathbf{j=1}}^{L}\omega _{\mathbf{j}}Y^{n+j} \left( \mathbf{x}+a(t_n,\mathbf{x})\varDelta t_{n,j}+b(t_n,\mathbf{x})\varDelta t_{n,j}\sqrt{2\varDelta t_{n,j}}{} \mathbf{a_j}\right) \\&+R^{GH}_L(Y) \end{aligned}$$

and

$$\begin{aligned} \mathbb {E}^{\mathbf{x},h}_{t_n}\left[ \bar{Y}^{n+j}\varDelta W_{t_n, j}^{\top }\right]= & {} \frac{1}{\pi ^{\frac{d}{2}}}\sum _{\mathbf{j=1}}^{L}\omega _{\mathbf{j}}Y^{n+j} \left( \mathbf{x}+a(t_n,\mathbf{x})\varDelta t_{n,j}+b(t_n,\mathbf{x})\varDelta t_{n,j}\sqrt{2\varDelta t_{n,j}}\mathbf{a_j}\right) \mathbf{a_j}\\&+R^{GH}_L(YW), \end{aligned}$$

where $\mathbb {E}^{\mathbf{x},h}_{t_n}\left[ \cdot \right]$ denotes the approximation of $\mathbb {E}^{\mathbf{x}}_{t_n}\left[ \cdot \right] .$ Finally, by inserting these approximations into (28) and (29) we obtain

$$\begin{aligned} \begin{aligned} Z^n=&(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+1}_{h,X^{n,j}}{Y}_{}^{n+1}\varDelta W_{n,1}^{\top }\right] \\&+(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+2}_{h,X^{n,j}}{Y}_{}^{n+2}\varDelta W_{n,2}^{\top }\right] \\&+\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) {\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+j}_{h,X^{n,j}}{Y}_{}^{n+j}\varDelta W_{n,j}^{\top }\right] \\&+(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h}\left[ {LI^{n+k+1}_{h,X^{n,j}}Y}_{}^{n+k+1}\varDelta W_{n,k+1}^{\top }\right] \\&+(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+2}_{h,X^{n,j}}{Y}_{}^{n+k+2}\varDelta W_{n,k+2}^{\top }\right] \\&+\alpha _{k,k}{\mathbb {E}}_{t_n}^{x,h}\left[ {LI^{n+k+3}_{h,X^{n,j}}Y}_{}^{n+k+3}\varDelta W_{n,k+3}^{\top }\right] +R_{z,n}^{k,LI_h}+R_{z,n}^{k,\mathbb {E}}, \end{aligned} \end{aligned}$$

(31)

and

$$\begin{aligned} \alpha _{k,0}Y^n=&-(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+1}_{h,X^{n,j}}Y^{n+1}\right] \nonumber \\&-(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+2}_{h,X^{n,j}}Y^{n+2}\right] \nonumber \\&-\displaystyle \sum _{j=3}^{k}(\alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+j}_{h,X^{n,j}}Y^{n+j}\right] \nonumber \\&-(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+1}_{h,X^{n,j}}Y^{n+k+1}\right] \nonumber \\&-(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+2}_{h,X^{n,j}}Y^{n+k+2}\right] \nonumber \\&-\alpha _{k,k}{\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+3}_{h,X^{n,j}}Y^{n+k+3}\right] -f(t_n,x,Y^n,Z^n)+R_{y,n}^{k,LI_h}+R_{y,n}^{k,\mathbb {E}}. \end{aligned}$$

(32)

Remark 4

• 1. The estimate of $R_{y,n}^{k,\mathbb {E}}$ or $R_{z,n}^{k,\mathbb {E}}$ reads [1, 31, 43]

  $$\begin{aligned} \mathcal {O}\left( \frac{L!}{2^L(2L)!}\right) . \end{aligned}$$

• 2. For the local interpolation errors $R_{y,n}^{k,LI_h}$ or $R_{z,n}^{k,LI_h}$ the following estimate holds

  $$\begin{aligned} \mathcal {O}\left( h^{r+1}\right) \end{aligned}$$

  (33)

  when using r-degree polynomial interpolation in the k-step scheme, and provided that a,  b,  f and g are sufficiently smooth such that $\mathcal {L}^{k+4}_{t,x}u(t,x)$ is bounded and $u(t,\cdot ) \in C_b^{r+1},$ see [1, 5, 6, 43].

• 3. To balance the temporal discretization error $R^k_{y,n}=\mathcal {O}(\varDelta t)^k$ and $R^k_{z,n}=\mathcal {O}(\varDelta t)^k,$ one needs to control well both the interpolation and integration error mentioned in last two points.

• 4. For a k-step scheme we need to know the support values of $Y^{N_T-i}$ and $Z^{N_T-i}, i=0,\ldots ,k+2.$ One can use the following three ways to deal with this problem: before running the multi-step scheme, we choose a quite smaller $\varDelta t$ and run one-step scheme until $N_T-k-2;$ Alternatively, one can prepare these initial values “iteratively”, namely we compute $Y^{N_T-1}$ and $Z^{N_T-1}$ based on $Y^{N_T}$ and $Z^{N_T}$ with $k=1,$ and the compute $Y^{N_T-2}$ and $Z^{N_T-2}$ based on $Y^{N_T}, Y^{N_T-1}, Z^{N_T}, Z^{N_T-1}$ with $k=2$ and so on; Finally, one can use the Runge–Kutta scheme proposed in [7] with small $\varDelta t$ to initialize our proposed multi-step scheme.

By removing all the error terms, from (31) and (32) we obtain the fully discrete scheme as follows.

Scheme 2

Assume that $Y^{N_T-i}$ and $Z^{N_T-i}$ on $\mathcal {R}^{N_T-i}_h$ are known for $i=0,1,\ldots ,k+2.$ For $n=N_T-k-3,\ldots , 0$ and $x\in \mathcal {R}_h^n,$ $Y^{n}=Y^{n}(x)$ and $Z^{n}=Z^{n}(x)$ can be solved by

$$\begin{aligned} X^{n,j}=&x + a(t_n,x)\varDelta t_{n,j}+ b(t_n,x)\varDelta W_{n,j},\quad j=1,\ldots ,k+3,\\ Z^n=&(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+1}_{h,X^{n,j}}{Y}_{}^{n+1}\varDelta W_{n,1}^{\top }\right] \\&+(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+2}_{h,X^{n,j}}{Y}_{}^{n+2}\varDelta W_{n,2}^{\top }\right] \\&+\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) {\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+j}_{h,X^{n,j}}{Y}_{}^{n+j}\varDelta W_{n,j}^{\top }\right] \\&+(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h}\left[ {LI^{n+k+1}_{h,X^{n,j}}Y}_{}^{n+k+1}\varDelta W_{n,k+1}^{\top }\right] \\&+(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+2}_{h,X^{n,j}}{Y}_{}^{n+k+2}\varDelta W_{n,k+2}^{\top }\right] \\&+\alpha _{k,k}{\mathbb {E}}_{t_n}^{x,h}\left[ {LI^{n+k+3}_{h,X^{n,j}}Y}_{}^{n+k+3}\varDelta W_{n,k+3}^{\top }\right] ,\\ \alpha _{k,0}Y^n=&-(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+1}_{h,X^{n,j}}Y^{n+1}\right] -(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+2}_{h,X^{n,j}}Y^{n+2}\right] \\&-\displaystyle \sum _{j=3}^{k}(\alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+j}_{h,X^{n,j}}Y^{n+j}\right] \\&-(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+1}_{h,X^{n,j}}Y^{n+k+1}\right] \\&-(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+2}_{h,X^{n,j}}Y^{n+k+2}\right] \\&-\alpha _{k,k}{\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+3}_{h,X^{n,j}}Y^{n+k+3}\right] -f(t_n,x,Y^n,Z^n). \end{aligned}$$

We note that Scheme 2 is a k-step scheme. For a fixed k,  one needs to perform three procedures for approximating $Y^n$ and $Z^n$ at every point in $\mathcal {R}_h^n$ on each time level: (1) solve $X^{n,j}$ by using the Euler scheme; (2) solve $Z^n$ explicitly with the second equation in Scheme 2; (3) finally, solve $Y^n$ implicitly by using e.g., the Newton iteration provided that the driver function f(t, x, y, z) is Lipschitz continuous with respect to y,  and a prescribed tolerance.

6 Numerical Schemes for Coupled FBSDEs

The authors in [38] extended their scheme proposed for solving decoupled FBSDEs to the one which can solve fully coupled FBSDEs. Similarly, our Scheme 2 can be extended to solve (2) in a fully coupled case.

Scheme 3

Assume that $Y^{N_T-i}$ and $Z^{N_T-i}$ on $\mathcal {R}^{N_T-i}_h$ are known for $i=0,1,\ldots ,k+2.$ For $n=N_T-k-3,\ldots , 0$ and $x\in \mathcal {R}_h^n,$ $Y^{n}=Y^{n}(x)$ and $Z^{n}=Z^{n}(x)$ can be solved by

• 1. set $Y^{n,0}=Y^{n+1}(x)$ and $Z^{n,0}=Z^{n+1}(x),$ and set $l=0;$

• 2. for $l=0,1,\ldots ,$ solve $Y^{n,l+1}=Y^{n,l+1}(x)$ and $Z^{n,l+1}=Z^{n,l+1}(x)$ by

  $$\begin{aligned} X^{n,j}=&x + a(t_n,x,Y^{n,l}(x),Z^{n,l}(x))\varDelta t_{n,j}+ b(t_n,x,Y^{n,l}(x),Z^{n,l}(x))\varDelta W_{n,j},\\&\qquad j=1,\ldots ,k+3,\\ Z^n=&(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x,h} \left[ LI^{n+1}_{h,X^{n,j}}{Y}_{}^{n+1}\varDelta W_{n,1}^{\top }\right] \\&+(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+2}_{h,X^{n,j}}{Y}_{}^{n+2}\varDelta W_{n,2}^{\top }\right] \\&+\displaystyle \sum _{j=3}^{k}\left( \alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}\right) {\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+j}_{h,X^{n,j}}{Y}_{}^{n+j}\varDelta W_{n,j}^{\top }\right] \\&+(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h} \left[ {LI^{n+k+1}_{h,X^{n,j}}Y}_{}^{n+k+1}\varDelta W_{n,k+1}^{\top }\right] \\&+(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h} \left[ LI^{n+k+2}_{h,X^{n,j}}{Y}_{}^{n+k+2}\varDelta W_{n,k+2}^{\top }\right] \\&+\alpha _{k,k}{\mathbb {E}}_{t_n}^{x,h}\left[ {LI^{n+k+3}_{h,X^{n,j}}Y}_{}^{n+k+3}\varDelta W_{n,k+3}^{\top }\right] ,\\ \alpha _{k,0}Y^n&=-(\alpha _{k,0}+\alpha _{k,1}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+1}_{h,X^{n,j}}Y^{n+1}\right] \\&\quad -(\alpha _{k,0}+\alpha _{k,1}+\alpha _{k,2}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+2}_{h,X^{n,j}}Y^{n+2}\right] \\&\quad -\displaystyle \sum _{j=3}^{k}(\alpha _{k,j-3}+\alpha _{k,j-2}+\alpha _{k,j-1}+\alpha _{k,j}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+j}_{h,X^{n,j}}Y^{n+j}\right] \\&\quad -(\alpha _{k,k-2}+\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+1}_{h,X^{n,j}}Y^{n+k+1}\right] \\&\quad -(\alpha _{k,k-1}+\alpha _{k,k}){\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+2}_{h,X^{n,j}}Y^{n+k+2}\right] \\&\quad -\alpha _{k,k}{\mathbb {E}}_{t_n}^{x,h}\left[ LI^{n+k+3}_{h,X^{n,j}}Y^{n+k+3}\right] -f(t_n,x,Y^n,Z^n). \end{aligned}$$

  until $\max \left( \left| Y^{n,l+1}-Y^{n,l}\right| ,\left| Z^{n,l+1}-Z^{n,l}\right| \right) <\epsilon _0,$

• 3. let $Y^n=Y^{n,l+1}$ and $Z^n=Z^{n,l+1}.$

Remark 5

• 1. Scheme 3 coincides with Scheme 2 if a and b do not depend on Y and Z.

• 2. We only assume that the coupled FBSDEs are uniquely solvable, the lacking analysis will be the task of future work.

We know that the mesh $\mathcal {R}_h^n$ is essentially unbounded. However, in real computations, one is usually interested in certain approximations of $(Y_t, Z_t)$ at $(t_n, x),$ where x belongs to a bounded domain. For example, the value of an option at the current asset price is usually asked. Therefore, only a bounded sub-mesh of $\mathcal {R}_h^n$ is used on each time level.

7 Numerical Experiments

In this section we use some numerical examples to show that Schemes 2 and 3 can achieve ninth-order convergence rate for solving FBSDEs. The uniform partitions in both time and space will be used, that is, the time interval [0, T] will be uniformly divided into $N_T$ parts with $\varDelta t=\frac{T}{N_T}$ such that $t_n=n\varDelta t, n=0,1,\ldots ,N_T;$ the space partition is $\mathcal {R}_h^n=R_h$ for all n with

$$\begin{aligned} \mathcal {R}_h=\mathcal {R}_{1,h}\times \mathcal {R}_{2,h} \times \cdots \mathcal {R}_{n,h}, \end{aligned}$$

where $\mathcal {R}_{j,h}$ is the partition of $\mathbb {R}$

$$\begin{aligned} \mathcal {R}_{j,h}=\left\{ x_i^j: x_i^j=ih, i=0,\pm 1,\ldots ,\pm \infty \right\} ,\quad j=1,2,\ldots ,n. \end{aligned}$$

In our numerical experiments we choose the local Lagrange interpolation for $LI^n_{h,x}$ based on the set of some neighbor grids near x,  i.e., $\mathcal {R}_{h,x} \subset \mathcal {R}_h$ such that (33) holds. Following [38], we set sufficiently many Gauss-Hermite quadrature points such that the quadrature error could be negligible. Note that the truncation error is defined in (12), in order to thus balance the temporal and space truncation error in our numerical examples, we force $h^{r+1}=(\varDelta t)^{k+1},$ where r is the degree of the Lagrangian interpolation polynomials. For example, one can firstly specify a value of r,  and then adjust the value of h such that $h=\varDelta t^{\frac{k+1}{r+1}}.$ For the numerical results in this paper, r is set to be a value from the set $\{10, 11, \cdots , 21\}$ to control the errors. Furthermore, we will consider k from 3 such that at least one combination of four $\alpha _{k,i}\,s$ is included, however, up to that $k=9.$ Finally, CR and RT are used to denote the convergence rate and the running time in second, respectively. For the comparative purpose, we take examples considered in [38]. Numerical experiment were performed in MATLAB with an Intel(R) Core(TM) i5-8350 CPU @ 1.70 GHz and 15 G RAM.

Example 1

The first example reads

$$\begin{aligned} {\left\{ \begin{array}{ll} dX_t=&{}\frac{1}{1+2\exp (t+X_t)}\,dt + \frac{\exp (t+X_t)}{1+\exp (t+X_t)}\,dW_t,\quad X_0=1,\\ -dY_t=&{}\left( -\frac{2Y_t}{1+2\exp (t+X_t)}-\frac{1}{2}\left( \frac{Y_tZ_t}{1+\exp (t+X_t)}-Y_t^2Z_t\right) \right) \,dt-Z_t\,dW_t,\\ Y_T=&{}\frac{\exp (T+X_T)}{1+\exp (T+X_T)}, \end{array}\right. } \end{aligned}$$

with the analytic solution

$$\begin{aligned} {\left\{ \begin{array}{ll} Y_t=&{}\frac{\exp (t+X_t)}{1+\exp (t+X_t)},\\ Z_t=&{}\frac{(\exp (t+X_t))^2}{(1+\exp (t+X_t))^3}. \end{array}\right. } \end{aligned}$$

Obviously, in this example, the generator a and b does not depend on $Y_t$ and $Z_t,$ i.e., a decoupled FBSDE. All the convergence rates, running time and absolute errors are reported in Table 5. In the proposed scheme, $k+3$ points are needed for the iterations, i.e., one need at least 12 points when $k=9.$ Furthermore, we can not choose a large value for $N_T$ due to the accuracy of double precision. Therefore, to show the convergence rate up to ninth order we consider $N_T=\{16, 20, 24, 28, 32 \}$ in this example.

Table 5 Errors, running time and convergence rates for Example 1, $T=1$

From Table 5 we see that the quite high accuracy of Scheme 2 for solving decoupled FBSDEs. Scheme 2 is a k-order scheme up to that $k=9,$ and more efficient for taking a larger value for k,  which is consistent with the theory [6], see also Table 4. By using the scheme proposed in [38], one can obtain the accuracy of the order $1\text{ e }-14$ for 113 seconds by using $k=6$ and $N_T=128,$ see Table 3 in [38]. In Table 5, we show that the same accuracy can be achieved only for around 4 seconds by using $k=9$ and $N_T=32.$

For the second example we consider the coupled FBSDE (taken from [38]) to test Scheme 3, in which an iterative process is required with longer computational time.

Example 2

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}dX_t=-\frac{1}{2}\sin (t+X_t)\cos (t+X_t)(Y^2_t+Z_t)\,ds\\ &{}\qquad +\frac{1}{2}\cos (t+X_t)+(Y_t\sin (t+X_t)+Z_t+1)\,dW_s,\quad X_0=1.5,\\ &{}-dY_t=\left( Y_t Z_t-\cos (t+X_t)\right) \,dt-Z_t\,dW_t,\\ &{}Y_T=\sin (T+X_T). \end{array}\right. } \end{aligned}$$

has the analytic solution

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}Y_t=\sin (t+X_t),\\ &{}Z_t=\cos ^2(t+X_t). \end{array}\right. } \end{aligned}$$

In this coupled FBSDE, the diffusion coefficient b depends on X, Y and Z,  i.e., quite general. For the same reasons as those explained for Example 1, we set $N_T=\{13, 15, 17, 19, 21\}$ in order to show the convergence rate up to ninth order.

Table 6 Errors, running time and convergence rates for Example 2, $T=1$

From the results listed in Table 6, we can draw same conclusions as those having been for Example 1.

Finally, we illustrate the accuracy of the proposed scheme for a two-dimensional example, which is also taken from [38] and reads

Example 3

$$\begin{aligned} \left\{ \begin{array}{l} \begin{pmatrix} dX^1_t\\ dX^2_t \end{pmatrix}=\begin{pmatrix} \frac{1}{2}\sin ^2(t+X^1_t)\\ \frac{1}{2}\sin ^2(t+X^2_t) \end{pmatrix}\,dt + \begin{pmatrix} \frac{1}{2}\cos ^2(t+X^1_t)\\ \frac{1}{2}\cos ^2(t+X^2_t) \end{pmatrix}\,dW_t,\quad \begin{pmatrix} X^1_0\\ X^2_0 \end{pmatrix}=\begin{pmatrix} 0\\ 0 \end{pmatrix},\\ \begin{pmatrix} dY^1_t\\ dY^2_t \end{pmatrix}=\begin{pmatrix} &{}-\frac{3}{2}\cos (t+X^1_t)\sin (t+X^2_t)-\frac{3}{2}\sin (t+X^1_t)\cos (t+X^2_t)-Z^2_t\\ &{}+\frac{1}{2}Y_t^1\left( \frac{1}{4}\cos ^4(t+X^2_t)+\frac{1}{4}\cos ^4(t+X^1_t)\right) -\frac{1}{4}(Y^2_t)^3\\ &{}\frac{3}{2}\sin (t+X^1_t)\cos (t+X^2_t)+\frac{3}{2}\cos (t+X^1_t)\sin (t+X^2_t)-Z^1_t\\ &{}+\frac{1}{2}Y_t^2 \left( \frac{1}{4}\cos ^4(t+X^2_t)+\frac{1}{4}\cos ^4(t+X^1_t)\right) -\frac{1}{4}Y^1_t(Y^2_t)^2 \end{pmatrix}\,dt \\ \quad - \begin{pmatrix} Z^1_t\\ Z^2_t \end{pmatrix}\,dW_t,\\ \begin{pmatrix} Y^1_T\\ Y^2_T \end{pmatrix}=\begin{pmatrix} \sin (T+X^1_T)\sin (T+X^2_T)\\ \cos (T+X^1_T)\cos (T+X^2_T) \end{pmatrix} \end{array}\right. \end{aligned}$$

with the analytic solution

$$\begin{aligned} \left\{ \begin{array}{l} \begin{pmatrix} Y^1_t\\ Y^2_t \end{pmatrix}=\begin{pmatrix} \sin (t+X^1_t)\sin (t+X^2_t)\\ \cos (t+X^1_t)\cos (t+X^2_t) \end{pmatrix}, \\ \begin{pmatrix} Z_t^1\\ Z_t^2 \end{pmatrix}=\begin{pmatrix} &{}\frac{1}{2}\cos (t+X^1_t)\sin (t+X^2_t)\cos ^2(t+X^2_t)\\ &{}\qquad \qquad +\frac{1}{2}\sin (t+X^1_t)\cos (t+X^2_t)\cos ^2(t+X^1_t)\\ &{}-\frac{1}{2}\sin (t+X^1_t)\cos ^3(t+X^2_t)-\frac{1}{2}\cos ^3(t+X^1_t)\sin (t+X^2_t) \end{pmatrix}. \end{array}\right. \end{aligned}$$

The numerical approximations are reported in Table 7, which show that our multi-step scheme is still quite highly accurate for solving a two-dimensional FBSDE.

Table 7 Errors, running time and convergence rates for Example 3, $T=1$

We observe that the convergence rates are roughly consistent with the theoretical results, the slight deviation comes from the quadratures and especially the two-dimensional interpolations. Obviously, the high efficiency and accuracy have been shown in this two-dimensional example. Note that the parallel computing toolbox in MATLAB has been used in this example, more precisely, the parallel for-Loops (parfor) is used for the two-dimensional interpolation on the grid points. Theoretically, our schemes can be also used for very high-dimensional problems. Although our semi-discrete scheme allows for ninth order of convergence in time, it is quite challenging to approximate the resulting conditional expectations with the same accuracy due to the curse of dimensionality. Recently, several approaches, see e.g., [12, 16, 17, 29] have been proposed for solving extremely high-dimensional problems. Those works open up possibility in practical applications, and preventing the curse of dimensionality. Thus, a high order accurate method for numerically solving very high-dimensional FBSDEs is considered as our future work.

8 Conclusion

In this work, by using the FDMs with the combinations of some the multi-steps we have adopted the high-order multi-step method in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36(4) (2014), pp.A1731-A1751] for numerically solving FBSDEs. First of all, our new schemes allow for higher convergence rate up to ninth order. Secondly, they also keep the key feature that is the numerical solution of backward component maintains the higher-order accuracy by using the Euler method to the forward component. This feature makes our schemes be promising in solving problems in practice. The effectiveness and higher-order accuracy have been confirmed by the numerical experiments. A rigorous stability analysis for the proposed schemes is the task of future work.