带跳的随机时滞微分方程的截断EM格式的收敛性分析

The Convergence of the Truncated Euler-Maruyama Method for Stochastic Delay Differential Equations with Poisson Jumps

摘要: 本文证明了带泊松跳的随机时滞微分方程的截断EM格式的收敛性。通过讨论截断EM格式的随机C-稳定性和随机B相容性，研究了数值格式的收敛性及其收敛阶为$\frac{1}{2}$ ，从而避免了讨论数值解高阶矩的有界性。最后，通过一个例子说明了截断EM格式对带泊松跳的SDDEs的收敛性与理论结果的一致性。

Abstract: The convergence of the truncated Euler-Maruyama (EM) method for the stochastic delay differential equations (SDDEs) with poisson jumps are established in this paper. By discussing the stochastic C-stability and stochastic B-consistency of the truncated EM scheme, the convergence and convergence rate which is $\frac{1}{2}$ has been researched, which avoiding the explore of the boundedness of the high-order moments of the numerical solution. Finally, an example is given to illustrate the consistence with the theoretical results on the convergence of the truncated EM to the SDDEs with poisson jumps.

1. 引言

带泊松跳的随机时滞微分方程，不仅考虑了系统上的随机因素和延迟因素，还考虑了泊松跳，能更准确地描述某些实际系统，并广泛应用于经济学、生物学、物理学等领域。因此。研究带泊松跳的随机时滞微分方程的解的性质具有重要意义。由于随机时滞微分方程的复杂性，除了少数情况外，通常很难获得解的精确数学表达式。因此，研究用于求解随机时滞微分方程的精确解的近似数值方法尤为重要[1]。

Higman等人研究了一些针对带泊松跳的随机微分方程(SDEs)的隐式方法[1]，他们探讨了这些SDEs在漂移系数满足非全局利普希茨条件，而扩散系数和泊松跳变的系数则是全局利普希茨条件下的不同隐式方法的收敛速度。然而，显式方法具有避免求解某些非线性系统的优势，近年来对SDEs显式方法的研究蓬勃发展。截断EM方法在Mao论文[2] [3]中被提出。Higham、Mao和Stuart [4]在局部利普希茨条件以及精确解和数值解的高阶矩有界性的假设下，研究了随机微分方程Euler-Maruyama (EM)方案的强收敛性。后续数值解的收敛性分析大多基于这一结论。非利普希茨条件下的基本收敛定理[5]是Milstein 1987年提出的全局利普希茨条件下基本收敛定理的扩展[6]。在单步数值格式的局部截断误差满足一定收敛性条件，且数值解的高阶矩有界的假设下，获得了数值方案的强收敛性和收敛阶。在Mao论文[2]中，他针对随机微分方程提出了截断EM方案，自此以后，截断 EM 方法得到了广泛研究[7] [8]。

我们可以看到，上述提到的研究数值解收敛性的两种方法都需要研究数值解的高阶矩的有界性。Beyn、Isaac和Kruse在2016年提出具有随C-稳定性和随机B-相容性的数值方案具有强收敛性(简称CBC收敛定理)，该定理避免了研究数值解的高阶矩的有界性，优化了分析过程，并有效地分析了随机微分方程的数值解的收敛性。自此以后，投影EM格式、分步后退EM格式、Milstein格式等格式的收敛性分析都采用了CBC收敛定理[9] [10]。最近，Zhan和Li [11]基于CBC收敛定理研究了带泊松跳的SDEs的截断EM方法。与高等人在[12]中研究的带泊松跳的SDDEs的收敛性和收敛速度相比，本文基于CBC收敛定理，研究了带泊松跳的随机微分延迟方程的截断EM方案的收敛性，这将简化分析过程。

2. 前提

考虑下面的带有泊松跳的随机时滞微分方程

$\{\begin{array}{ll}\text{d}Y\left(t\right)=f\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}t+g\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}W\left(t\right)+h\left(Y\left(t^{-}\right),Y\left({\left(t-\tau \right)}^{-}\right)\right)\text{d}N\left(t\right),\hfill & t>0;\hfill \\ Y\left(t\right)=\xi \left(t\right),\hfill & t\in \left[-\tau ,0\right],\hfill \end{array}$ (1)

其中初始值为$f,h:R^d\to R^d$ ；$g:R^d\to R^{d\times m}$ ，$\xi \left(t\right)\in C\left(\left[-\tau ,0\right];R^d\right)$ ，$W\left(t\right)$ 是一个m维布朗运动，$N\left(t\right)$ 是一个强度为$\lambda >0$ 的标量泊松过程，补偿泊松过程$\tilde{N}\left(t\right)=N\left(t\right)-\lambda t$ 是一个鞅。

在进一步讨论之前，我们提出一些假设。在本文中，我们用L和C来表示一般常数，它们的值会随着行数的变化而变化。

假设2.1 假设初始值$\xi$ 满足

$\left|\xi \left(u\right)-\xi \left(v\right)\right|\le K_1{\left|u-v\right|}^{\beta },\tau \le v<u\le 0$

其中$K_1>0$ ，$\beta \in \left[1/2,1\right]$ 。

假设2.2 存在常数$q\in \left(1,\infty \right),\eta \in \left(1/2,\infty \right)$ 和$L\in [0,\infty )$ ，使得对于任意$x_1,x_2,y_1,y_2\in R^d$ ，

$\begin{array}{l}\langle x_1-x_2,f\left(x_1,y_1\right)-f\left(x_2,y_2\right)\rangle +\eta {\left|g\left(x_1,y_1\right)-g\left(x_2,y_2\right)\right|}^2\\ +\eta \lambda {\left|h\left(x_1,y_1\right)-h\left(x_2,y_2\right)\right|}^2\le L\left({\left|x_1-x_2\right|}^2+{\left|y_1-y_2\right|}^2\right),\end{array}$ (2)

$\begin{array}{l}\left|f\left(x_1,y_1\right)-f\left(x_2,y_2\right)\right|\vee \left|g\left(x_1,y_1\right)-g\left(x_2,y_2\right)\right|\\ \le L\left(1+{\left|x_1\right|}^{q-1}+{\left|x_2\right|}^{q-1}+{\left|y_1\right|}^{q-1}+{\left|y_2\right|}^{q-1}\right)\left(\left|x_1-x_2\right|+\left|y_1-y_2\right|\right),\end{array}$ (3)

$\left|h\left(x_1,y_1\right)-h\left(x_2,y_2\right)\right|\le L\left(\left|x_1-x_2\right|+\left|y_1-y_2\right|\right),$ (4)

在$\left|f\left(0,0\right)\right|\vee \left|g\left(0,0\right)\right|\vee \left|h\left(0,0\right)\right|\le L$ 时成立。

假设2.3 假设存在常数$L>0$ 和$p\ge 2$ 成立

$\langle x,f\left(x,y\right)\rangle +\frac{\overline{p}-1}{2}{\left|g\left(x,y\right)\right|}^2\le L\left(1+{\left|x\right|}^2+{\left|y\right|}^2\right).$

引理2.1 如果假设2.1和2.2成立，$\overline{p}$ 为假设2.3中定义，那么存在方程(1)的唯一全局解$Y\left(t\right)$ 满足

$\underset{t\in \left[-\tau ,T\right]}{\sup }E{\left|Y\left(t\right)\right|}^{\overline{p}}\le C.$

证明：假设同李敏[9] [10]的引理2.1相同，所以我们在这里省略。

3. 带泊松跳的随机时滞微分方程的截断EM收敛性

在本节中，我们先给出随机C稳定性和随机B相容性的定义，及利用随机C稳定性和随机B相容性给出收敛性定理，接着给出截断EM格式，并分析其收敛性。首先，定义了随机单步方法。

在区间$[-\tau ,0]$ 上定义一个均匀网格，其网格步长为

$\Delta =\frac{\tau }{m},t_i=i\Delta ,i\ge -m.$

这意味着网格点$t_{-m}=\tau ,t_0=0$ ，此外，存在正整数N满足$T=t_N+\epsilon$ ，从而我们得到网格点的有序排列$t_{-m}\le \cdots \le t_0\le t_1\le \cdots \le t_N\le T$ ，进一步，对于任意网格点$t_i$ ，有$t_i-\tau =t_{i-m}$ 。

定义3.1 考虑一般的单步法

$Y_{i+1}=\Phi \left(Y_i,Y_{i-m},t_i,\Delta \right),0\le i\le N-1\text{,}$

其中$Y\left(0\right)=\xi \left(0\right)$ ，$Y_i,Y_{i-m}$ 为$Y\left(t_i\right),Y\left(t_{i-m}\right)$ 的数值逼近，假设映射$\Phi$ 满足以下条件：对于任意$\left(t,h\right)\in \left[0,T\right]\times \left[0,T\right]$ ，其中$t+h\le T$ ，以及任意$x,y\in L^2\left(\Omega ,{ℱ}_t,P;R^d\right)$ 均有

$\Phi \left(x,y,t,\Delta \right)\in L^2\left(\Omega ,{ℱ}_{t+\Delta },P;R^d\right),$

则单步法也被称为$\left(\Phi ,t,\Delta \right)$ 。

定义3.2 如果存在$\eta \in \left(1/2,\infty \right)$ 以及$C\ge 0$ 使得

$\begin{array}{l}{\Vert E\left[\Phi \left(x_1,y_1,t,\Delta \right)-\Phi \left(x_2,y_2,t,\Delta \right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}^2+2\eta {\Vert E^{id}\left[\Phi \left(x_1,y_1,t,\Delta \right)-\Phi \left(x_2,y_2,t,\Delta \right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}^2\\ \le \left(1+C\Delta \right){\Vert x_1-x_2\Vert }_{L^2\left(\Omega ;R^d\right)}^2+C\Delta {\Vert y_1-y_2\Vert }_{L^2\left(\Omega ;R^d\right)}^2,\end{array}$ (5)

对于任意$x_1,x_2,y_1,y_2\in L^2\left(\Omega ,{ℱ}_t,P;R^d\right)$ 并且在$t+\Delta \le T$ 下$\left(t,h\right)\in \left[0,T\right]\times \left[0,T\right]$ 成立，那么称单步法$\left(\Phi ,t,\Delta \right)$ 是随机C稳定的，其中$E^{id}\left[x|{ℱ}_t\right]=x-E\left[x|{ℱ}_t\right]$ 。

定义3.3 如果存在常数C和$\kappa$ 使得

${\Vert E\left[Y\left(t+\Delta \right)-\Phi (Y\left(t\right),Y\left(t-\tau \right),t,\Delta |{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\kappa +1},$ (6)

${\Vert E^{id}\left[Y\left(t+\Delta \right)-\Phi \left(Y\left(t\right),Y\left(t-\tau \right),t,\Delta \right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\kappa +\frac{1}{2}},$ (7)

成立，其中$Y\left(t\right)$ 是方程(1)的精确解，那么则称单步法$\left(\Phi ,t,\Delta \right)$ 是随机B相容的。

定理3.1 如果单步法是随机C稳定并且$\kappa$ 阶随机B相容的，那么

${\max }_{i=0,1,2,\cdots ,N}{\Vert Y\left(t_n\right)-Y_n\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\kappa },$

成立，其中$Y\left(t_n\right)$ 是方程(1)的精确解，$Y_n$ 是随机单步法生成的数值解。

证明：证明同李敏[9]中的定理3.1相同，所以我们这里省略。

为了定义截断EM数值解，我们选择严格单增连续函数$\mu :R_{+}\to R_{+}$ 使得当$r\to \infty$ 时$\mu \left(r\right)\to \infty$ 以及

$\underset{\left|x\right|\vee \left|y\right|\le r}{\sup }\left(\left|f\left(x,y\right)\right|\vee \left|g\left(x,y\right)\right|\right)\le \mu \left(r\right),r\ge 1,$

成立。记${\mu }^{-1}$ 是$\mu$ 的反函数，很明显${\mu }^{-1}$ 是一个$\left[\mu \left(0\right),\infty \right)\to R_{+}$ 的严格单增连续函数。选择一个常数${\Delta }^{*}\in \left(0,1\right]$ 和一个严格单增函数$\phi \left(0,{\Delta }^{*}\right]\to \left(0,\infty \right)$ 使得

$\phi \left({\Delta }^{*}\right)\ge \mu \left(1\right),\underset{\Delta \to 0}{\lim }\phi \left(\Delta \right)=\infty ,{\Delta }^{\frac{1}{4}}h\left(\Delta \right)\le 1,\forall \Delta \in \left(0,{\Delta }^{*}\right].$

对于给定的步长$\Delta \in \left(0,{\Delta }^{*}\right]$ 定义一个从$R^n$ 到闭球$x\in R^n:\left|x\right|\le {\mu }^{-1}\left(h\left(\Delta \right)\right)$ 的映射${\pi }_{\Delta }$ ，其定义为

${\pi }_{\Delta }\left(x\right)=\left(\left|x\right|\wedge {\mu }^{-1}\left(\phi \left(\Delta \right)\right)\right)\frac{x}{\left|x\right|},$

其中我们定义当$x=0$ 时，$x/\left|x\right|=0$ 。定义截断函数，对于$x,y\in R^n$ ：

$f_{\Delta }\left(x,y\right)=f\left({\pi }_{\Delta }\left(x\right),{\pi }_{\Delta }\left(y\right)\right),g_{\Delta }\left(x,y\right)=g\left({\pi }_{\Delta }\left(x\right),{\pi }_{\Delta }\left(y\right)\right).$

很明显$\left|f_{\Delta }\left(x,y\right)\right|\vee \left|g_{\Delta }\left(x,y\right)\right|\le \phi \left(\Delta \right)$ ，$\forall x,y\in R^n$ 。

下面我们可以将随机时滞微分方程的截断数值格式$\left(\Phi ,t,\Delta \right)$ 定义如下

$\begin{array}{c}{Y}_{i+1}=\Phi \left(Y_i,Y_{i-m_i},t_i,h\right)\\ ={\pi }_{\Delta }\left(Y_i\right)+f_{\Delta }\left(Y_i,Y_{i-m}\right)\Delta t_i+g_{\Delta }\left(Y_i,Y_{i-m}\right)\Delta W\left(t_i\right)+h\left(Y_i,Y_{i-m}\right)\Delta N\left(t_i\right),0\le i\le N-1.\end{array}$ (8)

其中$Y_0=\xi \left(0\right)$ ，$\Delta t_i=t_{i+1}-t_i$ ，$\Delta W\left(t_i\right)=W\left(t_{i+1}\right)-W\left(t_i\right)$ ，$\Delta N\left(t_i\right)=N\left(t_{i+1}\right)-N\left(t_i\right)$ 。

在以下部分，我们将通过分析数值格式的随机C稳定性和随机B相容性来研究其收敛性。

3.1. 带泊松跳的SDDE方程的截断EM格式的随机C稳定性

深入探究之前，我们引入一些引理，这些引理将在研究TEM格式的随机C稳定性中发挥关键作用。

引理3.1 对于所有满足截断EM数值格式定义条件的$\mu ,h$ 和$\Delta$ ，对于任意的$x_1,x_2\in R^n$ ，$\Delta \in \left(0,{\Delta }^{*}\right]$ ，总成立

$\frac{\left|{\pi }_{\Delta }\left(x_1\right)-{\pi }_{\Delta }\left(x_2\right)\right|}{\left|x_1-x_2\right|}\le C.$

下面我们给出截断EM格式的随机C稳定性的证明。

定理3.2 若假设2.2成立，其中$L\in \left(0,\infty \right),q\in \left(1,\infty \right)$ 且$\eta \in \left(1/2,\infty \right)$ ，则对于满足$\Delta \in \left(0,{\Delta }^{*}\right)$ ，

${\mu }^{-1}\left(h\left(\Delta \right)\right)\le {\Delta }^{-\frac{1}{2\left(q-1\right)}}$ 的截断EM格式是随机C稳定的。

证明：注意到对于任意$x_1,x_2,y_1,y_2\in L^2\left(\Omega ,ℱ,P;R^d\right)$ ，我们有

$E\left[\Phi \left(x_1,y_1,t,\Delta \right)-\Phi \left(x_2,y_2,t,\Delta \right)|{ℱ}_t\right]={\pi }_{\Delta }\left(x_1\right)-{\pi }_{\Delta }\left(x_2\right)+\left(f_{\Delta }\left(x_1,y_1\right)-f_{\Delta }\left(x_2,y_2\right)\right)\Delta ,$

$\begin{array}{l}{E}^{id}\left[\Phi \left(x_1,y_1,t,\Delta \right)-\Phi \left(x_2,y_2,t,\Delta \right)|{ℱ}_t\right]\\ =\left(g_{\Delta }\left(x_1,y_1\right)-g_{\Delta }\left(x_2,y_2\right)\right)\Delta W\left(t\right)+\left(h\left(x_1,y_1\right)-h\left(x_2,y_2\right)\right)\Delta N\left(t\right),\end{array}$

很明显

$\begin{array}{l}{\left|{\pi }_{\Delta }\left(x_1\right)-{\pi }_{\Delta }\left(x_2\right)+\left(f_{\Delta }\left(x_1,y_1\right)-f_{\Delta }\left(x_2,y_2\right)\right)\Delta \right|}^2\\ ={\left|{\pi }_{\Delta }\left(x_1\right)-{\pi }_{\Delta }\left(x_2\right)\right|}^2+2\Delta \langle {\pi }_{\Delta }\left(x_1\right)-{\pi }_{\Delta }\left(x_2\right),f_{\Delta }\left(x_1,y_1\right)-f_{\Delta }\left(x_2,y_2\right)\rangle +{\Delta }^2{\left|f_{\Delta }\left(x_1,y_1\right)-f_{\Delta }\left(x_2,y_2\right)\right|}^2.\end{array}$

此外

$\begin{array}{l}\left|f_{\Delta }\left(x_1,y_1\right)-f_{\Delta }\left(x_2,y_2\right)\right|\\ \le L\left(1+{\left|{\pi }_{\Delta }\left(x_1\right)\right|}^{q-1}+{\left|{\pi }_{\Delta }\left(x_2\right)\right|}^{q-1}+{\left|{\pi }_{\Delta }\left(y_1\right)\right|}^{q-1}+{\left|{\pi }_{\Delta }\left(y_2\right)\right|}^{q-1}\right)\left(\left|{\pi }_{\Delta }\left(x_1\right)-{\pi }_{\Delta }\left(x_2\right)\right|+\left|{\pi }_{\Delta }\left(y_1\right)-{\pi }_{\Delta }\left(y_1\right)\right|\right)\\ \le L\left(1+4{\left({\mu }^{-1}h\left(\Delta \right)\right)}^{q-1}\right)\left(\left|x_1-x_2\right|+\left|y_1-y_1\right|\right)\le L\left(1+4{\Delta }^{-\frac{1}{2}}\right)\left(\left|x_1-x_2\right|+\left|y_1-y_1\right|\right),\end{array}$

其中用到了假设2.2，引理4.1以及${\mu }^{-1}\left(h\left(\Delta \right)\right)\le {\Delta }^{-\frac{1}{2\left(q-1\right)}}$ ，因此

${\Delta }^2{\left|f_{\Delta }\left(x_1,y_1\right)-f_{\Delta }\left(x_2,y_2\right)\right|}^2\le {\Delta }^{2}2L^2{\left(1+4{\Delta }^{-\frac{1}{2}}\right)}^2\left({\left|x_1-x_2\right|}^2+{\left|y_1-y_1\right|}^2\right)\le C\Delta \left({\left|x_1-x_2\right|}^2+{\left|y_1-y_1\right|}^2\right).$

此外

$E{\left(h\left(x_1,y_1\right)-h\left(x_2,y_2\right)\right)}^2{\left(\Delta N\left(t\right)\right)}^2=\lambda \Delta E{\left|h\left(x_1,y_1\right)-h\left(x_2,y_2\right)\right|}^2,$

利用$\Delta \in \left(0,1\right]$ 和不等式(2)可得

$\begin{array}{l}{\left|{\pi }_{\Delta }\left(x_1\right)-{\pi }_{\Delta }\left(x_2\right)+\Delta \left(f_{\Delta }\left(x_1,y_1\right)-f_{\Delta }\left(x_2,y_2\right)\right)\right|}^2+2\eta \Delta {\left|g_{\Delta }\left(x_1,y_1\right)-g_{\Delta }\left(x_2,y_2\right)\right|}^2\\ +2\lambda \eta \Delta {\left|\left(h\left(x_1,y_1\right)-h\left(x_2,y_2\right)\right)\right|}^2\le \left(1+C\Delta \right){\left|x_1-x_2\right|}^2+C\Delta {\left|y_1-y_2\right|}^2.\end{array}$

这意味着

$\begin{array}{l}{\Vert E\left[\Phi \left(x_1,y_1,t,\Delta t\right)-\Phi \left(x_2,y_2,t,\Delta t\right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}^2+2\eta {\Vert E^{id}\left[\Phi \left(x_1,y_1,t,\Delta t\right)-\Phi \left(x_2,y_2,t,\Delta t\right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}^2\\ \le \left(1+C\Delta \right){\left|x_1-x_2\right|}^2+C\Delta {\left|y_1-y_2\right|}^2.\end{array}$

3.2. 带泊松跳的SDDE方程的截断EM格式的随机B相容性

正如在研究随机C稳定性时所做的一样，我们将引入一些引理，这些引理将在研究数值方案的随机B一致性中发挥关键作用。

引理3.2(BGD不等式) 对于$p\ge 2$ ，存在常数L使得

$\begin{array}{l}E\left[\underset{0\le t\le T}{\sup }{\left|{\displaystyle {\int }_0^{t}h}\left(Y\left(s\right),Y\left(s-\tau \right)\text{d}\tilde{N}\left(s\right)\right)\right|}^p\right]\\ \le LE{\displaystyle {\int }_0^T{\left|h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\right|}^p\lambda \text{d}s}+LE{\left[{\displaystyle {\int }_0^T{\left|h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\right|}^2\lambda \text{d}s}\right]}^{\frac{p}{2}}.\end{array}$ (9)

引理3.3 假设假设2.2和2.3对于$\overline{p}\ge pq$ 成立，其中$p\in \left[2,\infty \right)$ ，q在假设2.2中引入，而$\overline{p}$ 在假设2.3中引入，则对于所有的$t_1,t_2\in \left[-\tau ,T\right]$ ，有

${\Vert Y\left(t_2\right)-Y\left(t_1\right)\Vert }_{L^p\left(\Omega ;R^d\right)}\le C\left(1+2\underset{-\tau \le s\le T}{\sup }{\Vert Y\left(s\right)\Vert }_{L^{pq}\left(\Omega ;R^d\right)}^q\right){\left|t_2-t_1\right|}_{}^{\frac{1}{p}},$

证明：对于$-\tau \le t_1\le t_2\le T$ ，

$\begin{array}{c}{\Vert Y\left(t_1\right)-Y\left(t_2\right)\Vert }_{L^p\left(\Omega ;R^d\right)}\le {\displaystyle {\int }_{t_1}^{t_2}{\Vert f\left(Y\left(s\right),Y\left(s-\tau \right)\right)\Vert }_{L^p\left(\Omega ;R^d\right)}\text{d}s}+{\Vert {\displaystyle {\int }_{t_1}^{t_2}g\left(Y\left(s\right),Y\left(s-\tau \right)\right)\text{d}W\left(s\right)}\Vert }_{L^p\left(\Omega ;R^d\right)}\\ +{\Vert {\displaystyle {\int }_{t_1}^{t_2}h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\text{d}N\left(s\right)}\Vert }_{L^p\left(\Omega ;R^d\right)}.\end{array}$

此外，由$\overline{p}\ge pq$ 和引理2.1得

$\underset{s\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(s\right)\Vert }_{L^{pq}\left(\Omega ;R^d\right)}\le \underset{s\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(s\right)\Vert }_{L^{\overline{p}}\left(\Omega ;R^d\right)}<\infty .$

那么，由不等式(4)可得

$\left|f\left(x,y\right)\right|\vee \left|g\left(x,y\right)\right|\le L\left(1+{\left|x\right|}^q+{\left|y\right|}^q\right).$

由[13]的命题5.4可得

${\displaystyle {\int }_{t_1}^{t_2}{\Vert f\left(Y\left(s\right),Y\left(s-\tau \right)\right)\Vert }_{L^p\left(\Omega ;R^d\right)}\text{d}s}\le L\left(1+2\underset{\tau \le s\le T}{\sup }{\Vert Y\left(s\right)\Vert }_{L^{pq}\left(\Omega ,R^d\right)}^q\right)\left|t_2-t_1\right|,$

同样的，使用BDG不等式，我们有

$\begin{array}{c}{\Vert {\displaystyle {\int }_{t_1}^{t_2}g\left(Y\left(s\right),Y\left(s-\tau \right)\right)\text{d}W\left(s\right)}\Vert }_{L^p\left(\Omega ;R^d\right)}\le L{\left({\displaystyle {\int }_{t_1}^{t_2}{\Vert g\left(Y\left(s\right),Y\left(s-\tau \right)\right)\Vert }_{L^p\left(\Omega ,R^d\right)}^2\text{d}s}\right)}^{\frac{1}{2}}\\ \le L\left(1+2\underset{\tau \le s\le T}{\sup }{\Vert Y\left(s\right)\Vert }_{L^{pq}\left(\Omega ,R^d\right)}^q\right){\left|t_2-t_1\right|}^{\frac{1}{2}}.\end{array}$

此外

$\begin{array}{l}{\Vert {\displaystyle {\int }_{t_1}^{t_2}h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\text{d}N\left(s\right)}\Vert }_{L^p\left(\Omega ;R^d\right)}\\ \le {\Vert {\displaystyle {\int }_{t_1}^{t_2}h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\text{d}\tilde{N}\left(s\right)}\Vert }_{L^p\left(\Omega ;R^d\right)}+{\Vert {\displaystyle {\int }_{t_1}^{t_2}\lambda h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\text{d}s}\Vert }_{L^p\left(\Omega ;R^d\right)}.\end{array}$

注意到$\tilde{N}\left(t\right)$ 是一个鞅，根据BDG不等式(9)可得

$\begin{array}{l}{\Vert {\displaystyle {\int }_{t_1}^{t_2}h}\left(Y\left(s\right),Y\left(s-\tau \right)\right)\text{d}\tilde{N}\left(s\right)\Vert }_{L^p\left(\Omega ;R^d\right)}\\ \le {\left(E\left({\displaystyle {\int }_{t_1}^{t_2}{\left|h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\right|}^p\lambda \text{d}s}\right)\right)}^{\frac{1}{p}}+L{\left(E{\left({\displaystyle {\int }_{t_1}^{t_2}{\left|h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\right|}^2\lambda \text{d}s}\right)}^{\frac{p}{2}}\right)}^{\frac{1}{p}}\\ \le L\left(1+2\underset{s\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(s\right)\Vert }_{L^{pq}\left(\Omega ;R^d\right)}^q\right){\left|t_2-t_1\right|}_{}^{\frac{1}{p}}+L{\left({\displaystyle {\int }_{t_1}^{t_2}{\Vert h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\Vert }_{L^p\left(\Omega ;R^d\right)}^2\text{d}s}\right)}^{\frac{1}{2}}\\ \le L\left(1+2\underset{s\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(s\right)\Vert }_{L^{pq}\left(\Omega ;R^d\right)}^q\right){\left|t_2-t_1\right|}_{}^{\frac{1}{p}}+L\left(1+2\underset{s\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(s\right)\Vert }_{L^{pq}\left(\Omega ;R^d\right)}^q\right){\left|t_2-t_1\right|}_{}^{\frac{1}{2}}\\ \le L\left(1+2\underset{s\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(s\right)\Vert }_{L^{pq}\left(\Omega ;R^d\right)}^q\right){\left|t_2-t_1\right|}_{}^{\frac{1}{p}}.\end{array}$

此外，根据假设2.2我们可得

$\begin{array}{c}{\Vert {\displaystyle {\int }_{t_1}^{t_2}\lambda h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\text{d}s}\Vert }_{L^p\left(\Omega ;R^d\right)}\le \lambda {\displaystyle {\int }_{t_1}^{t_2}{\Vert h\left(Y\left(s\right),Y\left(s-\tau \right)\right)\Vert }_{L^p\left(\Omega ;R^d\right)}\text{d}s}\\ \le L\left(1+2\underset{\tau \le s\le T}{\sup }{\Vert Y\left(s\right)\Vert }_{L^p\left(\Omega ,R^d\right)}\right)\left|t_2-t_1\right|,\end{array}$

因此

${\Vert Y\left(t_1\right)-Y\left(t_2\right)\Vert }_{L^p\left(\Omega ;R^d\right)}\le C\left(1+2\underset{-\tau \le s\le T}{\sup }{\Vert Y\left(s\right)\Vert }_{L^{pq}\left(\Omega ;R^d\right)}^q\right){\left|t_2-t_1\right|}_{}^{\frac{1}{p}}.$

引理3.4 如果假设2.2和2.3成立，假设存在$\epsilon >0$ 满足$\overline{p}\epsilon \ge 1\left(1+\epsilon \right)\left(q-1\right)$ 和$\overline{p}\ge 2\left(1+\epsilon \right)q$ ，其中$q$ 和$\overline{p}$ 在假设中引入，则

$\begin{array}{l}{\displaystyle {\int }_t^{t+\Delta t}{\Vert f\left(Y\left(s\right),Y\left(s-\tau \right)\right)-f\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\text{d}s}\le C{\Delta }^{1+\frac{1}{2\left(1+\epsilon \right)}},\\ {\Vert {\displaystyle {\int }_t^{t+\Delta t}g}\left(Y\left(s\right),Y\left(s-\tau \right)\right)-g\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}W\left(s\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\frac{1}{2}+\frac{1}{2\left(1+\epsilon \right)}},\\ {\Vert {\displaystyle {\int }_t^{t+\Delta t}h}\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}N\left(s\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\frac{1}{2}+\frac{1}{2\left(1+\epsilon \right)}}.\end{array}$

证明：根据Itô等距公式

$\begin{array}{l}{\Vert {\displaystyle {\int }_t^{t+\Delta }h}\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}N\left(s\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\\ ={\left(\lambda {\displaystyle {\int }_t^{t+\Delta }{\Vert h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}^2\text{d}s}\right)}^{\frac{1}{2}}\\ \le {\Vert {\displaystyle {\int }_t^{t+\Delta }h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}\tilde{N}\left(s\right)}\Vert }_{L^2\left(\Omega ;R^d\right)}^2\\ +\lambda {\Vert {\displaystyle {\int }_t^{t+\Delta }h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}s}\Vert }_{L^2\left(\Omega ;R^d\right)}^2\\ \le {\left(\lambda {\displaystyle {\int }_t^{t+\Delta }{\Vert h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}^2\text{d}s}\right)}^{\frac{1}{2}}\\ +\lambda \Delta {\displaystyle {\int }_t^{t+\Delta }{\Vert h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}^2\text{d}s}.\end{array}$

由$\overline{p}\epsilon \ge 1\left(1+\epsilon \right)\left(q-1\right),\overline{p}\ge 2\left(1+\epsilon \right)q$和引理2.1，我们有

$\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2\frac{1+\epsilon }{\epsilon }\left(q-1\right)}\left(\Omega ;R^d\right)}<\infty ,\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{1\left(1+\epsilon \right)q}\left(\Omega ;R^d\right)}<\infty .$ (10)

因此，根据假设2.2，利用共轭参数$\rho =1+\epsilon ,{\rho }^{\prime }=\frac{1+\epsilon }{\epsilon }\left(\epsilon >0\right)$ 的赫尔德不等式和引理2.1可得

$\begin{array}{l}{\Vert h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\\ \le L{\Vert \left(1+4\underset{t\in \left[-\tau ,T\right]}{\sup }{\left|Y\left(t\right)\right|}^{q-1}\right)\left|Y\left(s\right)-Y\left(t\right)\right|\Vert }_{L^2\left(\Omega ;R^d\right)}\\ +L{\Vert \left(1+4\underset{t\in \left[-\tau ,T\right]}{\sup }{\left|Y\left(t\right)\right|}^{q-1}\right)\left|Y\left(s-\tau \right)-Y\left(t-\tau \right)\right|\Vert }_{L^2\left(\Omega ;R^d\right)}\\ \le L\left(1+4\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2{\rho }^{\prime }\left(q-1\right)}\left(\Omega ;R^d\right)}^{q-1}\right){\Vert Y\left(s\right)-Y\left(t\right)\Vert }_{L^{2\rho }\left(\Omega ;R^d\right)}\\ +L\left(1+4\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2{\rho }^{\prime }\left(q-1\right)}\left(\Omega ;R^d\right)}^{q-1}\right){\Vert Y\left(s-\tau \right)-Y\left(t-\tau \right)\Vert }_{L^{2\rho }\left(\Omega ;R^d\right)}.\end{array}$

因此，我们考虑${\Vert Y\left(s-\tau \right)-Y\left(t-\tau \right)\Vert }_{L^{2\rho }\left(\Omega ;R^d\right)}$ 的不同情况：

1) 当$s-\tau >0,t-\tau <0$ 时，使用假设2.1和引理4.3，

$\begin{array}{l}{\Vert Y\left(s-\tau \right)-Y\left(t-\tau \right)\Vert }_{L^{2\rho }\left(\Omega ;R^d\right)}\\ ={\Vert Y\left(s-\tau \right)-Y\left(0\right)+Y\left(0\right)-Y\left(t-\tau \right)\Vert }_{L^{2\rho }\left(\Omega ;R^d\right)}\\ \le {\Vert Y\left(s-\tau \right)-Y\left(0\right)\Vert }_{L^{2\rho }\left(\Omega ;R^d\right)}+{\Vert \xi \left(0\right)-\xi \left(t-\tau \right)\Vert }_{L^{2\rho }\left(\Omega ;R^d\right)}\\ \le C\left(1+2\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2\rho q}\left(\Omega ;R^d\right)}^q\right){\left|s-t\right|}_{}^{\frac{1}{2}}+K_1{\left|t-\tau \right|}^{\beta }\\ \le C\left(1+2\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2\rho q}\left(\Omega ;R^d\right)}^q\right){\left|\Delta \right|}_{}^{\frac{1}{2}}.\end{array}$

2) 当$s-\tau >0,t-\tau >0$ ，使用引理4.3，

$\begin{array}{c}{\Vert Y\left(s-\tau \right)-Y\left(t-\tau \right)\Vert }_{L^{2\rho }\left(\Omega ;R^d\right)}\le C\left(1+2\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2\rho q}\left(\Omega ;R^d\right)}^q\right){\left|s-t\right|}_{}^{\frac{1}{2}}\\ \le C\left(1+2\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2\rho q}\left(\Omega ;R^d\right)}^q\right){\left|\Delta \right|}_{}^{\frac{1}{2}}.\end{array}$

3) 当$s-\tau <0,t-\tau <0$ ，使用假设2.1，

${\Vert Y\left(s-\tau \right)-Y\left(t-\tau \right)\Vert }_{L^{2\rho }\left(\Omega ;R^d\right)}={\Vert \xi \left(s-\tau \right)-\xi \left(t-\tau \right)\Vert }_{L^{2\rho }\left(\Omega ;R^d\right)}\le C{\left|\Delta \right|}_{}^{\frac{1}{2}}.$

结合引理2.1可得

$\begin{array}{l}{\Vert h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\\ \le L\left(1+4\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2{\rho }^{\prime }\left(q-1\right)}\left(\Omega ;R^d\right)}^{q-1}\right)\left(1+2\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2\rho q}\left(\Omega ;R^d\right)}^q\right){\left|\Delta \right|}_{}^{\frac{1}{2\left(1+\epsilon \right)}}\\ +L\left(1+4\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2{\rho }^{\prime }\left(q-1\right)}\left(\Omega ;R^d\right)}^{q-1}\right)\left(1+2\underset{t\in \left[-\tau ,T\right]}{\sup }{\Vert Y\left(t\right)\Vert }_{L^{2\rho q}\left(\Omega ;R^d\right)}^q\right){\left|\Delta \right|}_{}^{\frac{1}{2}}\\ \le C{\Delta }^{\frac{1}{2\left(1+\epsilon \right)}}.\end{array}$

因此

${\Vert {\displaystyle {\int }_t^{t+\Delta t}h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}N\left(s\right)}\Vert }_{L^2\left(\Omega ;R^n\right)}\le C{\Delta }^{\frac{1}{2}+\frac{1}{2\left(\epsilon +1\right)}}.$

同样的，使用Itô等距公式

$\begin{array}{l}{\Vert {\displaystyle {\int }_t^{t+\Delta t}g\left(Y\left(s\right),Y\left(s-\tau \right)\right)-g\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}W\left(s\right)}\Vert }_{L^2\left(\Omega ;R^d\right)}\\ ={\left({\displaystyle {\int }_t^{t+\Delta t}{\Vert g\left(Y\left(s\right),Y\left(s-\tau \right)\right)-g\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}^2\text{d}s}\right)}^{\frac{1}{2}}.\end{array}$

和前面的证明相同，我们可以得到

${\Vert g\left(Y\left(s\right),Y\left(s-\tau \right)\right)-g\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\frac{1}{2\left(1+\epsilon \right)}},$ ${\Vert f\left(Y\left(s\right),Y\left(s-\tau \right)\right)-f\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\frac{1}{2\left(1+\epsilon \right)}},$

因此

${\displaystyle {\int }_t^{t+\Delta t}{\Vert f\left(Y\left(s\right),Y\left(s-\tau \right)\right)-f\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^n\right)}\text{d}s}\le C{\Delta }^{1+\frac{1}{2\left(\epsilon +1\right)}},$

${\Vert {\displaystyle {\int }_t^{t+\Delta t}g\left(Y\left(s\right),Y\left(s-\tau \right)\right)-g\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}W\left(s\right)}\Vert }_{L^2\left(\Omega ;R^n\right)}\le C{\Delta }^{\frac{1}{2}+\frac{1}{2\left(\epsilon +1\right)}}.$

定理3.3 假设假设2.2~2.3成立，假设存在$\epsilon >0$ 满足$\overline{p}\epsilon \ge 2\left(1+\epsilon \right)\left(q-1\right)$ 和$\overline{p}\ge 2\left(1+\epsilon \right)q$ ，其中$q,\overline{p}$

在假设 2.2中引入，此外，假设${\mu }^{-1}\left(h\left(\Delta \right)\right)\le {\Delta }^{-\frac{1}{2\left(q-1\right)}}$ 成立，则截断EM数值格式$\Phi$ 是$\frac{1}{2\left(1+\epsilon \right)}$ 阶随机B相容的。

证明：由方程(1)和方程(8)得

$\begin{array}{l}Y\left(t+\Delta t\right)-\Phi \left(Y\left(t\right),Y\left(t-\tau \right),t,\Delta \right)\\ =Y\left(t\right)-{\pi }_{\Delta }\left(Y\left(t\right)\right)+{\displaystyle {\int }_t^{t+\Delta }f\left(Y\left(s\right),Y\left(s-\tau \right)\right)-f\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}s}+f\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Delta \\ -f_{\Delta }\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Delta +{\displaystyle {\int }_t^{t+\Delta }g\left(Y\left(s\right),Y\left(s-\tau \right)\right)-g\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}W\left(s\right)}\\ +\left(g\left(Y\left(t\right),Y\left(t-\tau \right)\right)-g_{\Delta }\left(Y\left(t\right),Y\left(t-\tau \right)\right)\right)\Delta W\left(t\right)\\ +{\displaystyle {\int }_t^{t+\Delta }h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}N\left(s\right)}.\end{array}$

于是

$\begin{array}{l}{\Vert E\left[Y\left(t+\Delta \right)-\Phi \left(Y\left(t\right),Y\left(t-\tau \right),t,\Delta \right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\\ \le {\Vert Y\left(t\right)-{\pi }_{\Delta }\left(Y\left(t\right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}+\Delta {\Vert f\left(Y\left(t\right),Y\left(t-\tau \right)\right)-f_{\Delta }\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\\ +{\displaystyle {\int }_t^{t+\Delta }{\Vert E\left[f\left(Y\left(s\right),Y\left(s-\tau \right)\right)-f\left(Y\left(t\right),Y\left(t-\tau \right)\right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\text{d}s}\\ +{\left\{E{\displaystyle {\int }_t^{t+\Delta }h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h{\left(Y\left(t\right),Y\left(t-\tau \right)\right)}^2\lambda \text{d}s}\right\}}^{\frac{1}{2}}:\\ =I_{41}+I_{42}+I_{43}+I_{44}.\end{array}$

对于$I_{42}$ ，由不等式${\Vert E\left[X|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\le {\Vert X\Vert }_{L^2\left(\Omega ;R^d\right)}$ 和引理4.4得

$\begin{array}{l}{\displaystyle {\int }_t^{t+\Delta }{\Vert E\left[f\left(Y\left(s\right),Y\left(s-\tau \right)\right)-f\left(Y\left(t\right),Y\left(t-\tau \right)\right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\text{d}s}\\ \le {\displaystyle {\int }_t^{t+\Delta }{\Vert f\left(Y\left(s\right),Y\left(s-\tau \right)\right)-f\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\text{d}s}\le C{\Delta }^{1+\frac{1}{2\left(1+\epsilon \right)}},\end{array}$

和$I_{44}$ 相同，由引理4.4可得$I_{44}\le C{\Delta }^{\frac{1}{2}+\frac{1}{2\left(1+\epsilon \right)}}$ 。对于$I_{43}$ ，由李敏[9]的引理2.3可得：

${\Vert f\left(Y\left(t\right),Y\left(t-\tau \right)\right)-f_{\Delta }\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\left(\Delta \right)}_{}^{\frac{1}{2}}.$

对于$I_{41}$ ，和${\Vert Y\left(t\right)-{\pi }_{\Delta }\left(Y\left(t\right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}$ 相同，我们可以得到$I_{41}\le C{\Delta }^{\frac{3}{2}}$ 。

我们来估算另一项

$\begin{array}{l}{\Vert E^{id}\left[Y\left(t+\Delta t\right)-\Phi \left(Y\left(t\right),Y\left(t-\tau \right),t,\Delta \right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\\ \le {\displaystyle {\int }_t^{t+\Delta }{\Vert E^{id}\left[f\left(Y\left(s\right),Y\left(s-\tau \right)\right)-f\left(Y\left(t\right),Y\left(t-\tau \right)\right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\text{d}s}\\ +{\Vert {\displaystyle {\int }_t^{t+\Delta }g\left(Y\left(s\right),Y\left(s-\tau \right)\right)-g\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}W\left(s\right)}\Vert }_{L^2\left(\Omega ;R^d\right)}\\ +{\Vert \left(g\left(Y\left(t\right),Y\left(t-\tau \right)\right)-g_{\Delta }\left(Y\left(t\right),Y\left(t-\tau \right)\right)\right)\Delta W\left(t\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\\ +{\Vert {\displaystyle {\int }_t^{t+\Delta }h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}N\left(s\right)}\Vert }_{L^2\left(\Omega ;R^d\right)},\end{array}$

由不等式${\Vert E^{id}\left[X|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\le {\Vert X\Vert }_{L^2\left(\Omega ;R^d\right)}$ 和引理4.4可得

$\begin{array}{l}{\displaystyle {\int }_t^{t+\Delta }{\Vert E^{id}\left[f\left(Y\left(s\right),Y\left(s-\tau \right)\right)-f\left(Y\left(t\right),Y\left(t-\tau \right)\right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\text{d}s}\\ \le {\displaystyle {\int }_t^{t+\Delta }{\Vert f\left(Y\left(s\right),Y\left(s-\tau \right)\right)-f\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\text{d}s}\le C{\Delta }^{\frac{3}{2}},\end{array}$

${\Vert {\displaystyle {\int }_t^{t+\Delta }g\left(Y\left(s\right),Y\left(s-\tau \right)\right)-g\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}W\left(s\right)}\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\frac{1}{2}+\frac{1}{2\left(1+\epsilon \right)}},$

${\Vert {\displaystyle {\int }_t^{t+\Delta }h\left(Y\left(s\right),Y\left(s-\tau \right)\right)-h\left(Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}N\left(s\right)}\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\frac{1}{2}+\frac{1}{2\left(1+\epsilon \right)}}.$

由

$\begin{array}{l}{\Vert \left(g\left(Y\left(t\right),Y\left(t-\tau \right)\right)-g_{\Delta }\left(Y\left(t\right),Y\left(t-\tau \right)\right)\right)\Delta W\left(t\right)\Vert }_{L^2\left(\Omega ;R^d\right)}^2\\ =\Delta {\Vert g\left(Y\left(t\right),Y\left(t-\tau \right)\right)-g_{\Delta }\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}^2,\end{array}$ $\begin{array}{l}{\Vert \left(h\left(Y\left(t\right),Y\left(t-\tau \right)\right)-h_{\Delta }\left(Y\left(t\right),Y\left(t-\tau \right)\right)\right)\Delta N\left(t\right)\Vert }_{L^2\left(\Omega ;R^d\right)}^2\\ =\lambda \Delta {\Vert h\left(Y\left(t\right),Y\left(t-\tau \right)\right)-h_{\Delta }\left(Y\left(t\right),Y\left(t-\tau \right)\right)\Vert }_{L^2\left(\Omega ;R^d\right)}^2,\end{array}$

和不等式(12)相同，我们可以得到

${\Vert \left(g\left(Y\left(t\right),Y\left(t-\tau \right)\right)-g_{\Delta }\left(Y\left(t\right),Y\left(t-\tau \right)\right)\right)\Delta W\left(t\right)\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\frac{1}{2}}.$

由上面可以得到

${\Vert E\left[Y\left(t+\Delta t\right)-\Phi \left(Y\left(t\right),Y\left(t-\tau \right),t,\Delta \right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{1+\frac{1}{2\left(1+\epsilon \right)}},$

${\Vert E^{id}\left[Y\left(t+\Delta t\right)-\Phi \left(Y\left(t\right),Y\left(t-\tau \right),t,\Delta \right)|{ℱ}_t\right]\Vert }_{L^2\left(\Omega ;R^d\right)}\le C{\Delta }^{\frac{1}{2}+\frac{1}{2\left(1+\epsilon \right)}}.$

利用定理3.1，可以很容易地得到截断EM数值格式对随机延迟微分方程的收敛性和收敛率。

定理3.4 假设假设2.1~2.3成立，假设$\epsilon >0$ 满足$\overline{p}\epsilon \ge 2\left(1+\epsilon \right)\left(q-1\right)$ 和$\overline{p}\ge 2\left(1+\epsilon \right)q$ ，此外

${\mu }^{-1}\left(h\left(\Delta \right)\right)\le {\Delta }^{-\frac{1}{2\left(q-1\right)}}$ ，那么截断EM格式是$\frac{1}{2\left(1+\epsilon \right)}$ 收敛的。

4. 数值实验

考虑下面带有泊松跳的随机时滞微分方程

$\text{d}Y\left(t\right)=-Y{\left(t\right)}^5\text{d}t+\left(Y{\left(t\right)}^2+Y{\left(t-1\right)}^2\right)\text{d}W\left(t\right)+Y{\left(t\right)}^3\text{d}N\left(t\right)，$ (11)

其中初始值为$\xi \left(0\right)=1$ ，泊松密度为$\lambda =2$ 。因此$f\left(x,y\right)=-x^5$ ，$g\left(x,y\right)=x^2+y^2$ ，$h\left(x,y\right)=x^3$ 。我们可以定义截断EM数值解如下

$\begin{array}{l}{Y}_{\Delta }\left(t_{k+1}\right)={\pi }_{\Delta }\left(Y_{\Delta }\left(t_k\right)\right)+\left[-{\left(\left(Y_{\Delta }\%\left(t_k\right)\wedge {\Delta }^{-\frac{1}{50}}\right)\frac{Y_{\Delta }\left(t_k\right)}{\left|Y_{\Delta }\left(t_k\right)\right|}\right)}^5\right]\Delta \\ +\left[{\left(\left(Y_{\Delta }\left(t_k\right)\wedge {\Delta }^{-\frac{1}{50}}\right)\frac{Y_{\Delta }\left(t_k\right)}{\left|Y_{\Delta }\left(t_k\right)\right|}\right)}^2+{\left(\left(Y_{\Delta }\left(t_{k-N}\right)\wedge {\Delta }^{-\frac{1}{50}}\right)\frac{Y_{\Delta }\left(t_{k-N}\right)}{\%}\left|Y_{\Delta }\left(t_{k-N}\right)\right|\right)}^2\right]\Delta {\omega }_k\\ +\left[-{\left(\left(Y_{\Delta }\left(t_k\right)\wedge {\Delta }^{-\frac{1}{50}}\right)\frac{Y_{\Delta }\left(t_k\right)}{\left|Y_{\Delta }\left(t_k\right)\right|}\right)}^3\right]\Delta N_k,t>0;k=0,1,2,\cdots ,M-1.\end{array}$

其中$Y_{\Delta }\left(t_k\right)=1,k=-N,-N+1,\cdots ,0$ ，我们可以将步长为${\Delta }^{*}={10}^{-6}$ 的数值解视为在$\left[0,1\right]$ 区间上离散布朗运动路径的精确解的近似值${\tilde{Y}}_{{\Delta }^{*}}$ ，同时，我们在$M=1000$ 条样本路径上，将${\tilde{Y}}_{{\Delta }^{*}}$ 与

$\Delta ={10}^{-3},\Delta =5\times {10}^{-4},\Delta =3\times {10}^{-4},\Delta =2\times {10}^{-4}$ 的数值近似进行比较。这里的均方误差表示为

$Erro{r}_{\Delta }={\left(\frac{1}{M}{\displaystyle \sum _{i=1}^M{\left|Y_{\Delta }^i\left(T\right)-{\tilde{Y}}_{{\Delta }^{*}}\left(T\right)\right|}^2}\right)}^{\frac{1}{2}}$ 。其中$Y_{\Delta }^i\left(T\right)$ 表示在$t=T$ 时，沿第i条样本路径、步长为$\Delta$ 的修正部分

截断函数的数值解。收敛率为

$Order=\log \left(\frac{Erro{r}_h}{Erro{r}_{\frac{h}{2}}}\right)/\log \left(2\right)$

TEM方法的收敛性和收敛速度如图1所示。从这个例子我们可以看出，TEM方法对SDDEs的数值解是收敛的，且收敛速度几乎为1/2，这验证了理论结果。

Figure 1. Convergence rate of SDDE Equation (11) with Poisson jumps

图1. 带泊松跳的SDDE方程(11)的收敛率

5. 总结

针对时滞随机微分方程截断EM格式，本文通过验证该格式的随机C稳定性与随机B相容性，得到了该数值格式的收敛性并得到了1/2的收敛率。通过利用随机C稳定性与随机B相容性讨论收敛性，避免了研究数值解的高阶矩的有界性，优化了分析过程，并有效地分析了随机微分方程的数值解的收敛性。随机C稳定性对数值格式有较强的要求，如果数值格式不满足随机C稳定行，是否存在避免研究数值解的高阶矩的有界性的收敛性分析方法？需要我们进行进一步的研究探讨。

基金项目

国家自然科学基金(No. 12301521)，山西省自然科学基金(No. 20210302124081, No. 202303021211026)，山西省留学归国人员资助项目(No.2023-038)。