双尺度随机时滞微分方程的平均原理

The Averaging Principle of Two-Scale Stochastic Delay Differential Equation

摘要: 本文研究了分数布朗运动驱动的非自治双尺度随机时滞微分方程的平均原理。首先，通过广义Stieltjes积分和随机平均原理，推导了非自治双尺度系统的均方收敛定理。然后，结合均方收敛定理和停时理论，分别得到了原系统和平均系统的矩估计。最后，证明了当时间尺度参数趋于零时，慢变量方程的解过程在均方意义下收敛于平均方程的解过程。

Abstract: The main goal of this article is to study an average principle of a class of non-autonomous two time-scale stochastic differential delay equations driven by fractional Brownian motion. Firstly, the mean square convergence theorem for non-autonomous scale systems was derived by means of Generalized Stieltjes integral and Stochastic Average Principle. Then, combining the mean square convergence theorem and the Stopping-time theory, the moment estimates of the original system and the average system were obtained, respectively. Finally, it showed that when the time scale pa-rameters approach zero, the solution process of the slow variable equation converges to the solution process of the mean equation in the mean square sense.

1. 引言

双尺度随机时滞微分方程最早可以追溯到Itô和Gikhman创立随机微积分理论的年代。由于在化学 [1] 、力学 [2] 、电子电路 [3] 和大气海洋学 [4] 等实际应用领域中表现出来的广泛应用，双尺度随机微分方程俨然已经成为一个重要的研究领域，越来越多的学者开始重视双尺度随机微分方程的研究。对于多尺度随机微分方程的定性分析，平均原理提供了极大帮助。多尺度随机微分方程的平均原理就是在一定的条件下，将快变量过程视为随机噪声并将其消除从而得到一个极限过程，使得极限过程的解逼近原始方程中慢变量的解。平均原理广泛应用于力学、物理、控制等领域，是简化动力系统从而得到微分方程近似解的有力工具。

关于随机动力系统的平均原理有较长的历史，其奠基性工作由前苏联数学家Bogoliubov在文献 [5] 中完成。紧接着，Gikhman [6] ，Volosov [7] 和Besjes [8] 研究了非线性常微分方程的平均问题。随机平均原理首先由Stratonovich提出，此后，Khasminskii [9] 将平均原理发展到具有快–慢尺度的随机常微分方程的研究中，证明了在较弱的收敛意义下成立的平均原理。值得一提的是，Freidlin和Wentzell [10] [11] 给出了依概率收敛的平均原理。另外，Golec和Ladde [12] 以及Givon [13] 等人又进一步将结果推广到均方意义下收敛的情形。然后，Cerrai和Freidlin [14] 研究了一类无限维随机反应扩散模型的平均原理。近年来，Fu和Duan [15] 深入研究了带有高斯噪声的双尺度随机偏微分方程，得到在Lipschitz条件下的均方收敛的平均结果。随后，Fu和Liu [16] 又进一步将结果推广为：慢方程的解强收敛于平均方程的解。Xu [17] 将以布朗运动为驱动的随机系统推广到非高斯噪声(带跳)的情形并得到了强收敛结果。之后，Xu [18] 等人又进一步证得由分数布朗运动驱动的双尺度随机微分方程的平均原理。

然而，许多研究都是基于系统未来的状态与过去的状态是相互独立的假设下展开的，这与实际情况并不相符。事实上，滞后现象不可避免地出现在随机动力系统中，也就是说事物的发展趋势既依赖于当前的状态，还依赖于过去的历史状态。时滞是实际应用中许多系统的重要特征，在生物学、信号传输、随机控制等应用领域的很多数学模型中，都发挥着不可或缺的作用，许多学者都进行了研究 [19] [20] [21] 。文献 [19] 给出了随机时滞微分方程和带跳的随机时滞微分方程的平均原理，证明了平均方程的解在p阶矩意义下和依概率意义下收敛于随机微分方程的解。文献 [20] 讨论了基于Hurst参数 $H\in \left(1/2,1\right)$ 的分数布朗运动驱动的随机时滞微分方程的平均原理，原始方程的解依概率收敛和均方意义收敛于相应平均方程的解。

双尺度随机微分方程的平均原理已经取得了很大的进展，但对于双尺度随机时滞微分方程的平均法的研究却很少。文献 [22] 利用随机平均原理研究了由布朗运动驱动的自治双尺度随机时滞微分方程，为一类自治双尺度随机时滞微分方程建立了一个平均原理的强极限定理。文献 [23] 则在文献 [22] 的基础上研究了Hurst参数 $H\in \left(1/2,1\right)$ 的乘法分数布朗噪声驱动的自治双时间尺度随机时滞微分方程的平均原理。

受此启发，本文在文献 [20] [23] 的研究基础上，研究由分数布朗运动驱动的非自治双尺度随机时滞微分方程的平均原理，其中该非自治系统的慢变量过程由Hurst参数 $H\in \left(1/2,1\right)$ 的分数布朗运动驱动，快变量过程由布朗运动驱动。借助广义Stieltjes积分和停时理论，通过随机平均原理得到非自治双尺度随机时滞微分方程的平均方程，并证得平均方程和原方程的慢分量在均方意义下解的近似等价性。

本文的其余部分组织如下：第2节预备知识。第3节是本文的主要结果及其证明过程，即证明慢方程的解是强收敛于平均方程的解。第4节总结了本文。

2. 预备知识

假设是一个带流的完备概率空间，其中 $\Omega =\left[0,T\right]$ 为实轴上的有限区间，T是正常数， $\mathbb{P}$ 是概率测度，满足通常条件，即是右连续的且包含所有的 $\mathbb{P}$ 零测集。符号 $\left|·\right|$ 表示 ${ℝ}^n$ 中的Euclidean范数， $\Vert ·\Vert$ 表示矩阵范数。 $ℂ:=C\left(\left[-\tau ,0\right];{ℝ}^n\right)$ 表示所有从 $\left[-\tau ,0\right]$ 映到 ${ℝ}^n$ 的右连续且具有左极限的函数f构成的Banach空间，其范数为最大范数 ${\Vert ·\Vert }_{\infty }$ 。对于 $h(·)\in C\left(\left[-\tau ,\infty \right)\right)$ 和 $t\ge 0$ ，定义一个随机过程 $h:=\left\{h\left(t\right),t\in \left[-\tau ,\infty \right)\right\}$ ， $h_t\in ℂ$ 定义为 $h_t\left(\theta \right)=h\left(t+\theta \right)$ ， $\theta \in \left[-\tau ,0\right]$ 。 $W:=\left\{W_t,t\in \left[0,T\right]\right\}$ 是定义在完备概率空间上的m维布朗运动， $B^H:=\left\{B_t^H,t\in \left[0,T\right]\right\}\left(1/2<H<1\right)$ 是一个定义在完备概率空间上的n维分数布朗运动。在本文中，符号C代表常数，其值在变化。

定义2.1 [23] 令 $a,b\in ℝ$ 。阶数为 $\alpha \left(\alpha >0\right)$ 的左侧和右侧Riemann-Liouville分数阶积分被定义为

$\begin{array}{l}{I}_{\alpha +}^{\alpha }f\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_a^t{\left(t-\omega \right)}^{\alpha -1}f\left(\omega \right)\text{d}\omega },\\ I_{b-}^{\alpha }f\left(t\right)=\frac{{\left(-1\right)}^{-\alpha }}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_t^b{\left(\omega -x\right)}^{\alpha -1}f\left(\omega \right)\text{d}\omega },\end{array}$

其中 ${\left(-1\right)}^{\alpha }={\text{e}}^{-i\pi \alpha }$ ， $\Gamma \left(\alpha \right)={\displaystyle {\int }_0^{\infty }{r}^{\alpha -1}{\text{e}}^{-r}\text{d}r}$ 是Euler Gamma函数。

假设 $1/2<H<1$ ， $1-H<\alpha <1$ ，记 ${\text{W}}^{\alpha ,1}\left(\left[\text{0,}T\right];{ℝ}^n\right)$ 表示由可测函数 $f:\left[0,T\right]\to {ℝ}^n$ ，且

${\Vert f\Vert }_{\alpha ,1}:={\displaystyle {\int }_0^T{s}^{-\alpha }\left|f\left(s\right)\right|\text{d}s}+{\displaystyle {\int }_0^T{\displaystyle {\int }_0^s{\left(s-\omega \right)}^{-\alpha -1}\left|f\left(s\right)-f\left(\omega \right)\right|\text{d}\omega \text{d}s}}<\infty$ (1)

构成的空间。假设 $1/2<H<1$ ， $1-H<\alpha <1$ ，记 ${\text{W}}_0^{\alpha ,\infty }\left(\left[\text{0,}T\right];{ℝ}^n\right)$ 表示由可测函数 $f:\left[0,T\right]\to {ℝ}^n$ ，且

${\Vert f\Vert }_{\alpha ,\infty }:=\underset{t\in \left[0,\infty \right]}{\sup }\left(\left|f\left(t\right)\right|+{\displaystyle {\int }_s^t{\left(r-\omega \right)}^{-\alpha -1}\left|f\left(t\right)-f\left(\omega \right)\right|\text{d}\omega }\right)<\infty$ (2)

构成的空间。对任意 $\lambda \ge 0$ 定义等价范数

${\Vert f\Vert }_{\alpha ,\infty }:=\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}\left(\left|f\left(t\right)\right|+{\displaystyle {\int }_s^t{\left(r-\omega \right)}^{-\alpha -1}\left|f\left(t\right)-f\left(\omega \right)\right|\text{d}\omega }\right)<\infty .$ (3)

定义2.3 [24] 令 $a,b\in ℝ$ 。阶数为 $\alpha \left(\alpha >0\right)$ 的广义Riemann-Stieltjes分数阶积分被定义为

$\begin{array}{l}{\displaystyle {\int }_0^{T}f\left(r\right)\text{d}g\left(r\right)}={\left(-1\right)}^{\alpha }{\displaystyle {\int }_0^T{D}_{0+}^{\alpha }f\left(r\right)D_{T-}^{1-\alpha }{g}_{T-}\left(r\right)\text{d}r},\\ {\displaystyle {\int }_s^{t}f\left(r\right)\text{d}g\left(r\right)}={\displaystyle {\int }_0^{T}f\left(r\right)1_{\left(s,t\right)}\text{d}g}\left(r\right).\end{array}$

其中对于任意的 $0\le a<b\le T$ ， $g_{b-}\left(t\right):=g\left(t\right)-g\left(b\right)$ ，且对于任意的 $a<t<b$ ，阶数为 $\alpha \left(\alpha >0\right)$ 的左侧和右侧Weyl型分数阶积分被定义为

$D_{\alpha +}^{\alpha }f\left(t\right):=\frac{1}{\Gamma \left(1-\alpha \right)}\left[\frac{f\left(t\right)}{{\left(t-a\right)}^{\alpha }}+\alpha {\displaystyle {\int }_a^t\frac{f\left(t\right)-f\left(s\right)}{{\left(t-s\right)}^{\alpha +1}}\text{d}s}\right],$

$D_{b-}^{1-\alpha }{g}_{b-}\left(t\right)=\frac{{\left(-1\right)}^{1-\alpha }}{\Gamma \left(1-\alpha \right)}\left[\frac{g\left(b\right)-g\left(t\right)}{{\left(b-t\right)}^{1-\alpha }}+\left(1-\alpha \right){\displaystyle {\int }_t^b\frac{g\left(t\right)-g\left(s\right)}{{\left(s-t\right)}^{2-\alpha }}\text{d}s}\right].$

其中 $\Gamma (·)$ 是Gamma函数。

假设 $1/2<H<1$ ， $1-H<\alpha <1$ ，设 ${\text{W}}_T^{\alpha ,0}\left(\left[\text{0,}T\right];{ℝ}^n\right)$ 是可测函数 $g:\left[0,T\right]\to {ℝ}^m$ 构成的空间，且具有范数

${\Vert g\Vert }_{\alpha ,0,T}:=\underset{0\le s<t\le T}{\sup }\left(\frac{\left|g\left(t\right)-g\left(s\right)\right|}{{\left(t-s\right)}^{1-\alpha }}+{\displaystyle {\int }_s^t\frac{\left|g\left(u\right)-g\left(s\right)\right|}{{\left(u-s\right)}^{2-\alpha }}\text{d}u}\right)<\infty .$

引理2.4 [25] 假设 $H\in \left(1/2,1\right)$ 且对任意的 $\alpha \in \left(1-H,1/2\right)$ ，函数 $f\in {\text{W}}^{\alpha ,1}\left(\left[\text{0,}T\right];{ℝ}^n\right)$ 和 $g\in {\text{W}}^{1-\alpha \text{,}\infty }\left(\left[0,T\right];{ℝ}^n\right)$ ，且对于任意 $t\in \left[0,T\right]$ ，积分 ${\displaystyle {\int }_0^{t}f\text{d}{B}_s^H}$ 存在，则有以下不等式成立：

$\left|{\displaystyle {\int }_0^{t}f\text{d}{B}_s^H}\right|\le {\Lambda }_{\alpha }\left(B_s^H\right){\Vert f\Vert }_{\alpha ,1}.$

其中 ${\Lambda }_{\alpha }\left(B^H\right):=\frac{1}{\Gamma \left(1-\alpha \right)\Gamma \left(\alpha \right)}{\Vert B^H\Vert }_{\alpha ,0,T}$ 。

进一步，根据Fernique定理，对于任何 $0<\vartheta <2$ ，有 $\mathbb{E}\left[{\text{e}}^{{\left({\Lambda }_{\alpha }\left(B^H\right)\right)}^{\vartheta }}\right]<\infty$ 。 (4)

假设 $\tau >0$ ， $s,t\in \left[-\tau ,T\right]$ ，记 ${\text{W}}_{\text{0}}^{\alpha \text{,}\infty }\left(\left[s,t\right];{ℝ}^n\right)$ 表示可测函数 $f:\left[s,t\right]\to {ℝ}^n$ 构成的空间且满足以下可积条件

${\Vert f\Vert }_{\alpha ,\infty \left(s,t\right)}:=\underset{r\in \left[s,t\right]}{\sup }\left(\left|f\left(r\right)\right|+{\displaystyle {\int }_s^r{\left(r-u\right)}^{-\alpha -1}\left|f\left(r\right)-f\left(u\right)\right|\text{d}u}\right)<\infty .$ (5)

这里 ${\Vert f\Vert }_{\infty }:=\underset{r\in \left[s,t\right]}{\sup }{\text{e}}^{-\lambda r}\left|f\left(r\right)\right|$ ， ${\Vert f\left(r\right)\Vert }_{\alpha \left(s\right)}:=\left|f\left(r\right)\right|+{\displaystyle {\int }_s^r{\left(r-u\right)}^{-\alpha -1}\left|f\left(r\right)-f\left(u\right)\right|\text{d}u}$ 。

对于任意 $\lambda \ge 0$ 定义等价范数 ${\Vert f\Vert }_{\alpha ,\lambda \left(s,t\right)}:=\underset{r\in \left[s,t\right]}{\sup }{\text{e}}^{-\lambda r}\left(\left|f\left(r\right)\right|+{\displaystyle {\int }_s^r{\left(r-u\right)}^{-\alpha -1}\left|f\left(r\right)-f\left(u\right)\right|\text{d}u}\right)<\infty$ 。 (6)

对于任意 $0<\lambda \le 1$ ，记 $C^{\lambda }\left(\left[s,t\right];{ℝ}^n\right)$ 表示由λ-阶Hölder连续函数 $f:\left[s,t\right]\to {ℝ}^n$ 且

${\Vert f\Vert }_{\alpha ,\lambda \left(s,t\right)}^p:=\underset{r\in \left[s,t\right]}{\sup }{\text{e}}^{-\lambda r}{\left(\left|f\left(r\right)\right|+{\displaystyle {\int }_s^r{\left(r-u\right)}^{-\alpha -1}\left|f\left(r\right)-f\left(u\right)\right|\text{d}u}\right)}^p,p\ge 1.$ (7)

构成的空间。这里 ${\Vert f\Vert }_{\alpha ,\infty \left(\tau \right)}^p:={\Vert f\Vert }_{\alpha ,\infty \left(-\tau ,T\right)}^p$ ， ${\Vert f\Vert }_{\alpha ,\lambda \left(\tau \right)}^p:={\Vert f\Vert }_{\alpha ,\lambda \left(-\tau ,T\right)}^p$ 和 ${\Vert f\Vert }_{\alpha \left(\tau \right)}^p:={\Vert f\Vert }_{\alpha ,\infty \left(-\tau \right)}^p$ 。当 $\tau =0$ 时，我们将省略对应范数的τ。

引理2.5 [26] 对于任意非负的a和b，且 $a+b<1$ ，并且对于任意的 $\lambda \ge 1$ ，存在正常数C，使得

${\displaystyle {\int }_0^t{\text{e}}^{-\lambda \left(t-r\right)}{\left(t-r\right)}^{-a}{r}^{-b}\text{d}r}\le C{\lambda }^{a+b-1}.$

当 $0\le a<1$ ， $b\le 0$ 以及任意的 $\lambda \ge 1$ 时，有 ${\displaystyle {\int }_0^t{r}^{-b}{\left(t-r\right)}^{-a}{r}^{-b}\text{d}r}\le \Gamma \left(1-a\right)t^{-b}{\lambda }^{a-1}$ 。

引理2.6 [25] 对于可测函数 $f:\left[0,T\right]\to {ℝ}^n$ ，存在正常数C，使得

${\Vert {\displaystyle {\int }_0^{t}f\left(r\right)\text{d}r}\Vert }_{\alpha }\le C{\displaystyle {\int }_0^t\left|f\left(r\right)\right|{\left(t-r\right)}^{-\alpha }\text{d}r},$

且 ${\Vert {\displaystyle {\int }_0^{t}f\left(r\right)\text{d}{B}_r^H}\Vert }_{\alpha }\le C{\Lambda }_{\alpha }\left(B^H\right){\displaystyle {\int }_0^t\left({\left(t-r\right)}^{-2\alpha }+r^{-\alpha }\right)\left(\left|f\left(r\right)\right|+{\displaystyle {\int }_0^t{\left(r-u\right)}^{-\alpha -1}\left|f\left(r\right)-f\left(u\right)\right|\text{d}u}\right)\text{d}r}$ 。

本文考虑如下形式的双尺度随机时滞微分方程：

$\{\begin{array}{l}\text{d}{X}^{\epsilon }\left(t\right)=f\left(t,X_t^{\epsilon },Y_t^{\epsilon }\right)\text{d}t+{\sigma }_1\left(t,X^{\epsilon }\left(t-\tau \right)\right)\text{d}{B}_t^H,t\in \left[0,T\right],X_0^{\epsilon }=\xi \in ℂ\\ \text{d}{Y}^{\epsilon }\left(t\right)=\frac{1}{\epsilon }g\left(X_t^{\epsilon },Y^{\epsilon }\left(t\right),Y^{\epsilon }\left(t-\tau \right)\right)\text{d}t+\frac{1}{\sqrt{\epsilon }}{\sigma }_2\left(X_t^{\epsilon },Y^{\epsilon }\left(t\right),Y^{\epsilon }\left(t-\tau \right)\right)\text{d}{W}_t,t\in \left[0,T\right],Y_0^{\epsilon }=\eta \in ℂ\end{array}$ (8)

其中 $t\in \left[0,T\right]$ ， $\tau >0$ 为正常数时滞， $0<\epsilon \ll 1$ 是一个小的正参数，代表系统中的时间尺度比率。因此，根据这个时间尺度，变量 $X_t^{\epsilon }$ 称为慢分量，变量 $Y_t^{\epsilon }$ 称为快分量。我们对系数做如下定义： $f:\left[0,T\right]\times ℂ\times ℂ\to {ℝ}^n$ ， ${\sigma }_1:\left[0,T\right]\times {ℝ}^n\to {ℝ}^{n\times m}$ ， $g:ℂ\times {ℝ}^n\times {ℝ}^n\to {ℝ}^n$ ， ${\sigma }_2:ℂ\times {ℝ}^n\times {ℝ}^n\to {ℝ}^{n\times m}$ 。

现在，假设方程(8)的各个系数满足以下条件：

(H1) f是可测函数，对于任意的 $x_i,y_i\in ℂ\left(i=1,2\right)$ 和 $s,t\in \left[0,T\right]$ ，存在常数 ${\theta }_i\ge 0,\left(i=1,2,3\right)\text{,}0<\kappa \le 1$ 和 $L_i\ge 0(i=1,2)$ ，使得

$\begin{array}{l}\left|f\left(t,x_1,y_1\right)\right|\le L_1\left(1+\left|x_1\right|+\left|y_1\right|\right);\\ \left|f\left(t,x_1,y_1\right)-f\left(t,x_2,y_1\right)\right|\le L_2\left(1+{\left|y_1\right|}^{{\theta }_1}\right)\left|x_1-x_2\right|;\\ \left|f\left(t,x_1,y_1\right)-f\left(t,x_1,y_2\right)\right|\le L_2\left|y_1-y_2\right|\left(1+{\left|x_1\right|}^{{\theta }_2}+{\left|y_1\right|}^{{\theta }_3}+{\left|y_2\right|}^{{\theta }_3}\right);\\ \left|f\left(t,x_1,y_1\right)-f\left(s,x_1,y_1\right)\right|\le L_2{\left|t-s\right|}^{\kappa }\left(1+{\left|x_1\right|}^{{\theta }_2}+{\left|y_1\right|}^{{\theta }_3}\right).\end{array}$

(H2) 对任意的 $x_1,x_2\in {ℝ}^n$ ， $s,t\in \left[0,T\right]$ ，存在常数 $\beta \in \left(0,1\right]$ ， $L_i>0,i=3,4,5,6$ 使得

$\begin{array}{l}\left|{\nabla }_x{\sigma }_1(t,x_1)\right|\le L_3;\left|{\sigma }_1\left(t,x_1\right)\right|\le L_6\left(1+{\left|x\right|}^{\gamma }\right);\\ \left|{\nabla }_x{\sigma }_1\left(t,x_1\right)-{\nabla }_x{\sigma }_1\left(t,x_2\right)\right|+\left|{\sigma }_1\left(t,x_1\right)-{\sigma }_1\left(t,x_2\right)\right|\le L_4\left|x_1-x_2\right|;\\ \left|{\nabla }_x{\sigma }_1\left(t,x_1\right)-{\nabla }_x{\sigma }_1\left(s,x_1\right)\right|+\left|{\sigma }_1\left(t,x_1\right)-{\sigma }_1\left(s,x_1\right)\right|\le L_5{\left|t-s\right|}^{\beta }.\end{array}$

(H3) 假设 $\nabla g=\left({\nabla }^{\left(1\right)}g,{\nabla }^{\left(2\right)}g,{\nabla }^{\left(3\right)}g\right)$ 和 $\nabla {\sigma }_2=\left({\nabla }^{\left(1\right)}{\sigma }_2,{\nabla }^{\left(2\right)}{\sigma }_2,{\nabla }^{\left(3\right)}{\sigma }_2\right)$ 是有界的。

(H4) 对于任意的 $x\in ℂ$ ， $y_1,y_2,z_1,z_2\in {ℝ}^n$ ，存在 ${\lambda }_i>0,i=1,2$ ， ${\lambda }_1>{\lambda }_2>0$ 且与x无关，使得

$\begin{array}{l}2\langle y_1-y_2,g\left(x,y_1,z_1\right)-g\left(x,y_2,z_2\right)\rangle +{\Vert {\sigma }_2\left(x,y_1,z_1\right)-{\sigma }_2\left(x,y_2,z_2\right)\Vert }^2\le -{\lambda }_1{\left|y_1-y_2\right|}^2+{\lambda }_2{\left|z_1-z_2\right|}^2,\\ 2\langle y,g\left(x,y,z\right)\rangle +\Vert {\sigma }_2\left(x,y,z\right)\Vert \le -{\lambda }_3{\left|y\right|}^2+{\lambda }_4{\left|z\right|}^2+C\left(1+{\Vert x\Vert }_{\infty }^2\right).\end{array}$

(H5) 对于初值 $X_0^{\epsilon }=\xi \in ℂ$ ，存在 ${\lambda }_5>0$ ，使得 $\left|\xi \left(t\right)-\xi \left(s\right)\right|\le {\lambda }_5\left|t-s\right|$ ， $s,t\in \left[-\tau ,0\right]$ 。

不难证明，在条件(H1)~(H3)和(H5)下，方程(8)存在唯一解：

$\{\begin{array}{l}{X}^{\epsilon }\left(t\right)=\xi \left(0\right)+{\displaystyle {\int }_0^{t}f\left(s,X_s^{\epsilon },Y_s^{\epsilon }\right)\text{d}s}+{\displaystyle {\int }_0^t{\sigma }_1\left(s,X^{\epsilon }\left(s-\tau \right)\right)\text{d}{B}_s^H},\\ Y^{\epsilon }\left(t\right)=\eta \left(0\right)+\frac{1}{\epsilon }{\displaystyle {\int }_0^{t}g\left(X_s^{\epsilon },Y^{\epsilon }\left(s\right),Y^{\epsilon }\left(s-\tau \right)\right)\text{d}s}+\frac{1}{\sqrt{\epsilon }}{\displaystyle {\int }_0^t{\sigma }_2\left(X_s^{\epsilon },Y^{\epsilon }\left(s\right),Y^{\epsilon }\left(s-\tau \right)\right)\text{d}{W}_s},\\ X_0^{\epsilon }=\xi \in ℂ,Y_0^{\epsilon }=\eta \in ℂ,t\in \left[0,T\right].\end{array}$ (9)

接下来，我们推导当 $\epsilon \to 0+$ 时，慢分量 $X_t^{\epsilon }$ 收敛于以下平均方程(利用随机平均原理，我们可以得到方程(8)的平均方程)的解：

$\text{d}\bar{X}\left(t\right)=\bar{f}\left(t,{\bar{X}}_t\right)\text{d}t+{\sigma }_1\left(t,\bar{X}\left(t-\tau \right)\right)\text{d}{B}_t^H\text{,}t\in \left[0,T\right]\text{,}{\bar{X}}_0=\xi \in ℂ.$ (10)

其中 ${\mu }^x$ 是相应冻结方程的转移半群 ${\left(P_t^x\right)}_{t\ge 0}$ 的唯一不变测度。显然，在条件(H1)~(H5)下，方程(10)也有唯一解 ${\left(\bar{X}\left(t\right)\right)}_{t\ge -\tau }$ [27] 。

下面，我们来介绍冻结方程。对于任意 $t\in \left[0,T\right]$ ，固定慢变量 $x\in ℂ$

$\text{d}Y\left(t\right)=g\left(x,Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}t+{\sigma }_2\left(x,Y\left(t\right),Y\left(t-\tau \right)\right)\text{d}{W}_t,Y_0=\eta \in ℂ.$ (11)

在条件(H3)下，方程(11)有唯一的解 ${\left(Y\left(t\right)\right)}_{t\ge -\tau }$ [22] 。

在本部分最后，给出以下引理，这是证明本文结论的重要工具：

定义2.7 设 ${\mathcal{F}}_t$ 为 $\left\{Y_r^{\chi ,\eta },r\le t\right\}$ 生成的 $\sigma$ -域。对 $0\le \zeta \le s\le T$ ，有

$J\left(s,\zeta ,\chi ,\eta \right)=E\left[\langle f\left(k\delta ,\chi ,Y_s^{\chi ,\eta }\right)-\bar{f}\left(k\delta ,\chi \right),f\left(k\delta ,\chi ,Y_{\zeta }^{\chi ,\eta }\right)-\bar{f}\left(k\delta ,\chi \right)\rangle \right],$ (12)

此时，存在独立于 $s,\zeta$ 的常数 $C,\rho >0$ ，使得 $J\left(s,\zeta ,\chi ,\eta \right)\le C\left(1+{\Vert \chi \Vert }_{\infty }^2+{\Vert \eta \Vert }_{\infty }^2\right){\text{e}}^{-\frac{\rho }{2}\left(s-\zeta \right)}$ 。

证明 由式(12)，调用 $Y_t^{\chi ,\eta }$ 的马尔科夫性质，Hölder不等式、条件(H1)和文献 [22] ，可以证明。

3. 主要结果

在本节，我们给出本文的主要结论：

定理3.1 假设原始方程(8)和平均方程(10)都满足假设条件(H1)~(H5)，则慢分量 $X_t^{\epsilon }$ 均方收敛于平均方程的解 ${\bar{X}}_t$ 。即对任意的 $t\in \left[0,T\right]$ ，对于所有的 $\epsilon \in \left(0,1\right)$ ，有

$\underset{\epsilon \to 0}{\lim }\mathbb{E}\left[{\Vert X^{\epsilon }-\bar{X}\Vert }_{\alpha ,\infty \left(\tau \right)}^2\right]=0.$

证明 定理3.1的证明包括以下步骤：

Step1：我们给出方程(8)的解 $\left(X^{\epsilon },Y^{\epsilon }\right)$ 的估计。

首先，我们对快慢过程有如下估计：

定理3.2 假设条件(H1)~(H5)成立，那么对任意 $t\in \left[0,T\right]$ ，存在正常数C，使得当 $\epsilon \in \left(0,1\right)$ 时，有

$\mathbb{E}\left[{\Vert X^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}^2\right]\le C.$

证明 设 $\Lambda :={\Lambda }_{\alpha }\left(B^H\right)\vee 1$ ，对任意的 $\lambda \ge 1$ ，首先，我们估计 ${\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}$ ，由条件(H1)~(H2)、引理2.4和引理2.5，有以下估计

$\begin{array}{c}{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}:=\underset{s\in \left[-\tau ,t\right]}{\sup }{\text{e}}^{-\lambda s}\left|X^{\epsilon }\left(s\right)\right|\le \underset{s\in \left[-\tau ,0\right]}{\sup }{\text{e}}^{-\lambda s}\left|\xi \left(s\right)\right|+\underset{s\in \left[0,t\right]}{\sup }{\text{e}}^{-\lambda s}\left|X^{\epsilon }\left(s\right)\right|\\ \le \underset{s\in \left[-\tau ,0\right]}{\sup }{\text{e}}^{-\lambda s}\left|\xi \left(s\right)\right|+\underset{s\in \left[0,t\right]}{\sup }{\text{e}}^{-\lambda s}\left(\left|{\displaystyle {\int }_0^{s}f\left(r,X_r^{\epsilon },Y_r^{\epsilon }\right)\text{d}r}\right|+\left|{\displaystyle {\int }_0^s{\sigma }_1\left(r,X^{\epsilon }\left(r-\tau \right)\right)\text{d}{B}_r^H}\right|\right)\\ \le {\Vert \xi \Vert }_{\alpha ,\infty \left(-\tau ,0\right)}+C\underset{0\le s\le t}{\sup }{\displaystyle {\int }_0^s{\text{e}}^{-\lambda \left(s-r\right)}\left(1+{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}\right)\text{d}r}\\ +{\Lambda }_{\alpha }\left(B^H\right)\underset{0\le s\le t}{\sup }\left({\displaystyle {\int }_0^s{\text{e}}^{-\lambda \left(s-u+\tau \right)}\left[\left(1+{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}\right)u^{-\alpha }+{\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}\right]\text{d}u}\right)\\ \le K\Lambda \left(1+{\lambda }^{\alpha -1}{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}+{\lambda }^{-1}{\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}\right).\end{array}$ (13)

其中常数 $K>1$ 且与 ${\Vert \xi \Vert }_{\alpha ,\infty \left(-\tau ,0\right)}$ 有关。

对于 ${\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}$ ，有

$\begin{array}{c}{\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}:=\underset{s\in \left[-\tau ,t\right]}{\sup }{\text{e}}^{-\lambda s}{\displaystyle {\int }_{-\tau }^s{\left(s-r\right)}^{-\alpha -1}\left|X^{\epsilon }\left(s\right)-X^{\epsilon }\left(r\right)\right|\text{d}r}\\ \le \underset{s\in \left[-\tau ,0\right]}{\sup }{\text{e}}^{-\lambda s}{\displaystyle {\int }_{-\tau }^s{\left(s-r\right)}^{-\alpha -1}\left|\xi \left(s\right)-\xi \left(r\right)\right|\text{d}r}+\underset{s\in \left[0,t\right]}{\sup }{\text{e}}^{-\lambda s}{\displaystyle {\int }_{-\tau }^0{\left(-r\right)}^{-\alpha -1}\left|\xi \left(0\right)-\xi \left(r\right)\right|\text{d}r}\\ +\underset{s\in \left[0,t\right]}{\sup }{\text{e}}^{-\lambda s}{\displaystyle {\int }_{-\tau }^0{\left(s-r\right)}^{-\alpha -1}\left|X^{\epsilon }\left(r\right)-\xi \left(0\right)\right|\text{d}r}+\underset{s\in \left[0,t\right]}{\sup }{\text{e}}^{-\lambda s}{\displaystyle {\int }_0^s{\left(s-r\right)}^{-\alpha -1}\left|X^{\epsilon }\left(s\right)-X^{\epsilon }\left(r\right)\right|\text{d}r}\\ =:{\displaystyle \underset{i=1}{\overset{4}{\sum }}{B}_i}.\end{array}$ (14)

接下来，我们分别估计 $B_1$ ， $B_2$ ， $B_3$ 和 $B_4$ 。对于 $B_1$ 和 $B_2$ ，由条件(H5)，

$B_1+B_2\le {\Vert \xi \Vert }_{\alpha ,\infty \left(-\tau ,0\right)}.$ (15)

对于 $B_3$ ，由引理2.4，可以得到

$\begin{array}{c}{B}_3\le \underset{s\in \left[0,t\right]}{\sup }\frac{{\text{e}}^{-\lambda s}}{s^{\alpha }}\left|X^{\epsilon }\left(s\right)-\xi \left(0\right)\right|\le \underset{s\in \left[0,t\right]}{\sup }\frac{1}{s^{\alpha }}{\displaystyle {\int }_0^s{\text{e}}^{-\lambda \left(s-r\right)}\left(1+{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}\right)\text{d}r}\\ +{\Lambda }_{\alpha }\left(B^H\right)C_{\alpha ,\beta }\underset{s\in \left[0,t\right]}{\sup }\frac{1}{s^{\alpha }}{\displaystyle {\int }_0^s{\text{e}}^{-\lambda \left(s-u+\tau \right)}\left[\left(1+{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}\right)u^{-\alpha }+{\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}\right]\text{d}u}\\ \le {\Lambda }_{\alpha }\left(B^H\right)C_{\alpha ,\beta }\underset{s\in \left[0,t\right]}{\sup }{\displaystyle {\int }_0^s{\text{e}}^{-\lambda \left(s-u+\tau \right)}\left[\left(1+{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}\right)u^{-2\alpha }+{\left(s-u\right)}^{-\alpha }{\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}\right]\text{d}u}\\ \le K\Lambda \left(1+{\lambda }^{2\alpha -1}{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}+{\lambda }^{-\alpha }{\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}\right).\end{array}$ (16)

对于 $B_4$ ，由引理2.6，可得

$\begin{array}{c}{B}_4\le \underset{s\in \left[0,t\right]}{\sup }{\text{e}}^{-\lambda s}{\displaystyle {\int }_0^s{\left(s-r\right)}^{-\alpha -1}{\displaystyle {\int }_r^{s}f\left(u,X_u^{\epsilon },Y_u^{\epsilon }\right)\text{d}u}\text{d}r}+\underset{s\in \left[0,t\right]}{\sup }{\text{e}}^{-\lambda s}{\displaystyle {\int }_0^s{\left(s-r\right)}^{-\alpha -1}\left|{\displaystyle {\int }_r^s{\sigma }_1\left(u,X^{\epsilon }\left(u-\tau \right)\right)\text{d}{B}_u^H}\right|\text{d}r}\\ \le \underset{s\in \left[0,t\right]}{\sup }{\displaystyle {\int }_0^s{\text{e}}^{-\lambda \left(s-r\right)}{\left(s-r\right)}^{-\alpha }\left(1+{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}\right)\text{d}r}\\ +C{\Lambda }_{\alpha }\left(B^H\right)\underset{s\in \left[0,t\right]}{\sup }\left({\displaystyle {\int }_0^s{\text{e}}^{-\lambda \left(s-u+\tau \right)}{\left(s-u\right)}^{-2\alpha }\left(1+{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}\right)\text{d}u+{\displaystyle {\int }_0^s{\text{e}}^{-\lambda \left(s-u+\tau \right)}{\left(s-u\right)}^{-\alpha }{\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}\text{d}u}}\right).\end{array}$ (17)

最后，将式(15)~(17)代入不等式(14)，得到

${\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}\le K\Lambda \left(1+{\lambda }^{2\alpha -1}{\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}+{\lambda }^{-\alpha }{\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}\right).$ (18)

选取 $\lambda ={\left(4K\Lambda \right)}^{1/\left(1-\alpha \right)}$ ，由式(13)，得到

${\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}\le \frac{4}{3}K\Lambda \left(1+{\lambda }^{-1}{\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}\right).$ (19)

将上式代入不等式(18)，进行简单变换，得到

${\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),t}\le \frac{3}{2}K\Lambda +2{\left(K\Lambda \right)}^{1/\left(1-\alpha \right)}\le C_{T,\left|\xi \right|}{\Lambda }^{1/\left(1-\alpha \right)}.$

再将其代入不等式(19)，则 ${\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),t}\le C{\Lambda }^{1/\left(1-\alpha \right)}$ 。

因此，可以得到

${\Vert X^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}\le {\text{e}}^{\lambda T}\left({\Vert X^{\epsilon }\Vert }_{\infty ,\lambda \left(\tau \right),T}+{\Vert X^{\epsilon }\Vert }_{1,\lambda \left(\tau \right),T}\right)\le C{\text{e}}^{{\Lambda }^{1/\left(1-\alpha \right)}}\left(1+{\Lambda }_{\alpha }^{1/\left(1-\alpha \right)}\left(B^H\right)\right).$

由于 $0<1/\left(1-\alpha \right)<2$ 和引理2.4中的 $\mathbb{E}\left[{\text{e}}^{{\left({\Lambda }_{\alpha }\left(B^H\right)\right)}^{\vartheta }}\right]<\infty$ ，可以得到

$\mathbb{E}\left[{\Vert X^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}^2\right]\le C.$

定理3.2得证。

此外，同理可证

${\Vert {\hat{X}}^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}+{\Vert \bar{X}\Vert }_{\alpha ,\infty \left(\tau \right)}\le C{\text{e}}^{C{\Lambda }_{\alpha }^{1/\left(1-\alpha \right)}\left(B^H\right)}\left(1+{\Lambda }_{\alpha }^{1/\left(1-\alpha \right)}\left(B^H\right)\right).$ (20)

$\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}^2+{\Vert \bar{X}\Vert }_{\alpha ,\infty \left(\tau \right)}^2\right]\le C.$ (21)

其次，慢过程有如下连续性：

定理3.3 假设条件(H1)~(H5)成立，那么任意 $t\in \left[0,T\right]$ ，存在正常数C，使得对足够小的 $\delta$ 和 $\epsilon \in \left(0,1\right)$ ，有

$\underset{t\in \left[0,T\right]}{\sup }\mathbb{E}\left[{\Vert X_t^{\epsilon }-X_{t\left(\delta \right)}^{\epsilon }\Vert }_{\infty }^2\right]\le C{\delta }^{1-2\alpha }.$

证明 首先，我们估计 $X^{\epsilon }\left(t\right)-X^{\epsilon }\left(s\right)$ 。对于任意的 $t,s\in \left[0,T\right]$ ，存在正常数C，使得

$\begin{array}{c}\left|X^{\epsilon }\left(t\right)-X^{\epsilon }\left(s\right)\right|\le \left|{\displaystyle {\int }_s^{t}f\left(r,X_r^{\epsilon },Y_r^{\epsilon }\right)\text{d}r}\right|+\left|{\displaystyle {\int }_s^t{\sigma }_1\left(r,X^{\epsilon }\left(r-\tau \right)\right)\text{d}{B}_r^H}\right|\\ \le C\left(1+{\Vert X^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}\right)\left|t-s\right|\\ +C{\Lambda }_{\alpha }\left(B^H\right)\left({\displaystyle {\int }_{s-\tau }^{t-\tau }\left(\left(1+{\Vert X^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}\right){\left(r-s\right)}^{-\alpha }\text{d}r+{\displaystyle {\int }_{s-\tau }^v\left|X^{\epsilon }\left(v\right)-X^{\epsilon }\left(u\right)\right|{\left(v-u\right)}^{-\alpha -1}\text{d}u\text{d}v}\right)}\right)\\ \le C\left(1+{\Lambda }_{\alpha }\left(B^H\right)\right)\left(1+{\Vert X^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}\right){\left|t-s\right|}^{1-\alpha }.\end{array}$

其次，小区间长度为 $\delta :=\frac{\tau }{N}<1$ (N为某一固定的正整数)。对任意的 $t\in \left[0,T\right]$ 和 $\theta \in \left[-\tau ,0\right]$ ，存在常数 $k,m\ge 0$ ，若 $t\in \left[k\delta ,\left(k+1\right)\delta \right]$ 和 $\theta \in \left[-\left(m+1\right)\delta ,-m\delta \right]$ ，则 $t+\theta \in \left[\left(k-m-1\right)\delta ,\left(k+1-m\right)\delta \right]$ 和 $t\left(\delta \right)+\theta \in \left[\left(k-m-1\right)\delta ,\left(k-m\right)\delta \right]$ 。对于 $E\left[{\Vert X_t^{\epsilon }-X_{t\left(\delta \right)}^{\epsilon }\Vert }_{\infty }^2\right]$ ，有

$\begin{array}{c}\mathbb{E}\left[{\Vert X_t^{\epsilon }-X_{t\left(\delta \right)}^{\epsilon }\Vert }_{\infty }^2\right]\le \mathbb{E}\left[{\displaystyle {\sum }_{m=0}^{N-1}\underset{\theta \in \left[-\left(m+1\right)\delta ,-m\delta \right]}{\sup }{\left|X^{\epsilon }\left(t+\theta \right)-X^{\epsilon }\left(t\left(\delta \right)+\theta \right)\right|}^2}\right]\\ \le N\underset{m=0,\cdots ,N-1}{\max }\mathbb{E}\left[\underset{\theta \in \left[-\left(m+1\right)\delta ,-m\delta \right]}{\sup }{\left|X^{\epsilon }\left(t+\theta \right)-X^{\epsilon }\left(t\left(\delta \right)+\theta \right)\right|}^2\right]\\ =:N\underset{m=0,\cdots ,N-1}{\max }J\left(t,m,\delta \right).\end{array}$

为了完成定理3.3的证明，我们只需证明 $J\left(t,m,\delta \right)\le C{\delta }^{2\left(1-\alpha \right)}$ 。

因此，我们讨论以下三种情况：

情况1：当 $m\le k-1$ 时，根据Hölder不等式，引理3.1，条件(H1)和(H2)，存在正常数C，使得

$\begin{array}{c}J\left(t,m,\delta \right)\le C\mathbb{E}\left[\underset{\theta \in \left[-\left(m+1\right)\delta ,-m\delta \right]}{\sup }{\left|{\displaystyle {\int }_{k\delta +\theta }^{t+\theta }f\left(r,X_r^{\epsilon },Y_r^{\epsilon }\right)\text{d}r+{\displaystyle {\int }_{k\delta +\theta }^{t+\theta }{\sigma }_1\left(r,X^{\epsilon }\left(r-\tau \right)\right)\text{d}{B}_r^H}}\right|}^2\right]\\ \le C\delta \mathbb{E}\left[{\displaystyle {\int }_{k\delta -\left(m+1\right)\delta }^{t-m\delta }{\left|f\left(r,X_r^{\epsilon },Y_r^{\epsilon }\right)\right|}^2\text{d}r}\right]+C\mathbb{E}\left[{\left|{\displaystyle {\int }_{k\delta -\left(m+1\right)\delta }^{t-\left(m+1\right)\delta }{\sigma }_1\left(r,X^{\epsilon }\left(r-\tau \right)\right)\text{d}{B}_r^H}\right|}^2\right]\\ +C\mathbb{E}\left[\underset{\theta \in \left[-\left(m+1\right)\delta ,-m\delta \right]}{\sup }{\left|{\displaystyle {\int }_{t-\left(m+1\right)\delta }^{t+\theta }{\sigma }_1\left(r,X^{\epsilon }\left(r-\tau \right)\right)\text{d}{B}_r^H}\right|}^2\right]\\ +C\mathbb{E}\left[\underset{\theta \in \left[-\left(m+1\right)\delta ,-m\delta \right]}{\sup }{\left|{\displaystyle {\int }_{k\delta -\left(m+1\right)\delta }^{k\delta +\theta }{\sigma }_1\left(r,X^{\epsilon }\left(r-\tau \right)\right)\text{d}{B}_r^H}\right|}^2\right]\\ \le C{\delta }^2+C{\delta }^{2\left(1-\alpha \right)}\mathbb{E}\left[{\left(1+{\Lambda }_{\alpha }\left(B^H\right)\right)}^2{\left(1+{\Vert X^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}\right)}^2\right]\le C{\delta }^{2\left(1-\alpha \right)}.\end{array}$

情况2：当 $m\ge k-1$ 时，由条件(H5)，存在正常数C，使得

${\left|X^{\epsilon }\left(t+\theta \right)-X^{\epsilon }\left(t\left(\delta \right)+\theta \right)\right|}^2={\left|\xi \left(t+\theta \right)-\xi \left(t\left(\delta \right)+\theta \right)\right|}^2\le C{\delta }^2.$

情况3：当 $m=k$ 时，由Hölder不等式，条件(H1)和(H2)，存在正常数C，使得

$\begin{array}{c}J\left(t,m,\delta \right)=\mathbb{E}\left[\underset{\theta \in \left[-\left(m+1\right)\delta ,-m\delta \right]}{\sup }{\left|X^{\epsilon }\left(t+\theta \right)-X^{\epsilon }\left(t_{\delta }+\theta \right)\right|}^2\right]\\ \le C{\delta }^2+C\mathbb{E}\left[\underset{\theta \in \left[-\left(m+1\right)\delta ,-m\delta \right]}{\sup }{\left|X^{\epsilon }\left(t+\theta \right)-X^{\epsilon }\left(t\left(\delta \right)+\theta \right)\right|}^2{1}_{\left\{t+\theta >0\right\}}\right]\\ \le C{\delta }^2+C\mathbb{E}\left[\underset{\theta \in \left[-t,-k\delta \right]}{\sup }{\left|{\displaystyle {\int }_0^{t+\theta }f\left(r,X_r^{\epsilon },Y_r^{\epsilon }\right)\text{d}r}\right|}^2\right]+C\mathbb{E}\left[\underset{\theta \in \left[-t,-k\delta \right]}{\sup }{\left|{\displaystyle {\int }_0^{t+\theta }{\sigma }_1\left(r,X^{\epsilon }\left(r-\tau \right)\right)\text{d}{B}_r^H}\right|}^2\right]\\ \le C{\delta }^{2\left(1-\alpha \right)}.\end{array}$

综上所述， $\underset{t\in \left[0,T\right]}{\sup }\mathbb{E}\left[{\Vert X_t^{\epsilon }-X_{t\left(\delta \right)}^{\epsilon }\Vert }_{\infty }^2\right]\le C{\delta }^{1-2\alpha }$ 。

定理3.3得证。

参考文献 [22] 中定理3.1和文献 [23] 中引理3.10的证明，易得如下结论：

定理3.4 假设条件(H1)~(H5)成立，那么对任意 $t\in \left[0,T\right]$ ，存在正常数C，使得

$\underset{t\in \left[0,T\right]}{\sup }\mathbb{E}\left[{\Vert Y_t^{\epsilon }\Vert }_{\infty }^2\right]\le C.$

Step2：根据文献 [9] ，我们先构造辅助过程 $\left({\hat{X}}_t^{\epsilon },{\hat{Y}}_t^{\epsilon }\right)\in ℂ\times ℂ$ ，再对区间 $\left[0,T\right]$ 进行分割，令小区间长度为 $\delta :=\frac{\tau }{N}<1$ 。对于过程 ${\hat{Y}}_t^{\epsilon }$ ，令它的初值 ${\hat{Y}}_0^{\epsilon }=Y_0^{\epsilon }=\eta \in ℂ$ ，对 $t\in \left[k\delta ,\text{min}\left\{\left(k+1\right)\delta ,T\right\}\right]$ ， $k\ge 0$ 并满足方程

${\hat{Y}}^{\epsilon }\left(s\right)={\hat{Y}}^{\epsilon }\left(k\delta \right)+\frac{1}{\epsilon }{\displaystyle {\int }_{k\delta }^{t}g\left(X_{k\delta }^{\epsilon },{\hat{Y}}^{\epsilon }\left(s\right),{\hat{Y}}^{\epsilon }\left(s-\tau \right)\right)\text{d}s}+\frac{1}{\sqrt{\epsilon }}{\displaystyle {\int }_{k\delta }^t{\sigma }_2\left(X_{k\delta }^{\epsilon },{\hat{Y}}^{\epsilon }\left(s\right),{\hat{Y}}^{\epsilon }\left(s-\tau \right)\right)\text{d}{W}_s},$

其中 $X_{k\delta }^{\epsilon }$ 和 $Y_{k\delta }^{\epsilon }$ 分别为在 $k\delta$ 时的慢变量和快变量的解过程。令 $\lfloor ·\rfloor$ 表示取整函数，并定义 $t\in \left[0,T\right]$ 时，对于过程 ${\hat{X}}_t^{\epsilon }$ 满足方程

${\hat{X}}^{\epsilon }\left(t\right)=\xi +{\displaystyle {\int }_0^{t}f\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)\text{d}s}+{\displaystyle {\int }_0^t{\sigma }_1\left(s,X^{\epsilon }\left(s-\tau \right)\right)\text{d}{B}_s^H}.$ (22)

其中 $s\left(\delta \right)=\lfloor \frac{s}{\delta }\rfloor \delta$ 表示处于点s右端并且距离其最近的分点。

现在，我们来估计 $X_t^{\epsilon }-{\hat{X}}_t^{\epsilon }$ 和 $Y_t^{\epsilon }-{\hat{Y}}_t^{\epsilon }$ 这两项。

参考文献 [22] 中定理4.2和文献 [23] 中引理3.9的证明，易得如下结论：

定理3.5 假设(H1)~(H5)成立，那么对任意 $t\in \left[0,\tau \right)$ ，存在正常数C和独立于 $\epsilon$ 的 $\beta >0$ ，使得对足够小的 $\delta$ 和 $\epsilon \in \left(0,1\right)$ ，有

$\underset{t\in \left[0,T\right]}{\sup }\mathbb{E}\left[{\Vert Y_t^{\epsilon }-{\hat{Y}}_t^{\epsilon }\Vert }_{\infty }^2\right]\le C{\epsilon }^{-1}{\delta }^{1-2\alpha }{\text{e}}^{\beta \frac{\delta }{\epsilon }}.$

接下来，我们估计 ${\hat{X}}^{\epsilon }-X^{\epsilon }$ ，得到以下引理：

定理3.6 假设条件(H1)~(H5)成立，那么对任意 $t\in \left[0,T\right]$ ，存在正常数C和独立于 $\epsilon$ 的 $\beta >0$ ，使得对足够小的 $\delta$ 和 $\epsilon \in \left(0,1\right)$ ，有

$E\left[{\Vert {\hat{X}}^{\epsilon }-X^{\epsilon }\Vert }_{\alpha ,\lambda \left(\tau \right)}^2\right]\le C{\delta }^2\left(1+{\delta }^{-1-2\alpha }\left(1+{\epsilon }^{-1}{\text{e}}^{\beta \frac{\delta }{\epsilon }}\right)\right).$

证明 根据方程(8)和方程(22)，

$\begin{array}{c}\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-X^{\epsilon }\Vert }_{\alpha ,\lambda \left(\tau \right)}^2\right]\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\left|{\hat{X}}^{\epsilon }\left(t\right)-X^{\epsilon }\left(t\right)\right|}^2\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\left({\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha -1}\left|{\hat{X}}^{\epsilon }\left(t\right)-X^{\epsilon }\left(t\right)-{\hat{X}}^{\epsilon }\left(s\right)+X^{\epsilon }\left(s\right)\right|\text{d}s}\right)}^2\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\left({\displaystyle {\int }_{-\tau }^0{\left(t-s\right)}^{-\alpha -1}\left|{\hat{X}}^{\epsilon }\left(t\right)-X^{\epsilon }\left(t\right)-{\hat{X}}^{\epsilon }\left(s\right)+X^{\epsilon }\left(s\right)\right|\text{d}s}\right)}^2\right]\\ =:D_1+D_2+D_3.\end{array}$

首先，对于第一项 $D_1$ 和第二项 $D_2$ ，由条件(H1)~(H2)，引理2.6，定理3.3和定理3.5可得

$\begin{array}{c}{D}_1+D_2\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\Vert {\displaystyle {\int }_0^t\left(f\left(s,X_s^{\epsilon },Y_s^{\epsilon }\right)-f\left(s,X_s^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)\right)\text{d}s}\Vert }_{\alpha }^2\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\Vert {\displaystyle {\int }_0^t\left(f\left(s,X_s^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-f\left(s,X_{s\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)\right)\text{d}s}\Vert }_{\alpha }^2\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\Vert {\displaystyle {\int }_0^t\left(f\left(s,X_{s\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-f\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)\right)\text{d}s}\Vert }_{\alpha }^2\right]\\ \le C{\displaystyle {\int }_0^T\left(\mathbb{E}\left[{\Vert Y^{\epsilon }-{\hat{Y}}_{s\left(\delta \right)}^{\epsilon }\Vert }_{\infty }^2+{\Vert X^{\epsilon }-X_{s\left(\delta \right)}^{\epsilon }\Vert }_{\infty }^2\right]\right)\text{d}s}+C{\delta }^2\\ \le C{\delta }^2\left(1+{\delta }^{-1-2\alpha }\left(1+{\epsilon }^{-1}{\text{e}}^{\beta \frac{\delta }{\epsilon }}\right)\right).\end{array}$

其次，对于第三项 $D_3$ ，有

$\begin{array}{c}{D}_3\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\left({\displaystyle {\int }_{-\tau }^0{\left(t-s\right)}^{-\alpha -1}\left|{\hat{X}}^{\epsilon }\left(t\right)-X^{\epsilon }\left(t\right)-{\hat{X}}^{\epsilon }\left(0\right)+X^{\epsilon }\left(0\right)\right|\text{d}s}\right)}^2\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\left({\displaystyle {\int }_{-\tau }^0{\left(-s\right)}^{-\alpha -1}\left|{\hat{X}}^{\epsilon }\left(0\right)-X^{\epsilon }\left(0\right)-\xi \left(0\right)+\xi \left(0\right)\right|\text{d}s}\right)}^2\right]\\ \le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\left({\displaystyle {\int }_{-\tau }^0{\left(t-s\right)}^{-\alpha -1}\left|{\hat{X}}^{\epsilon }\left(t\right)-X^{\epsilon }\left(t\right)\right|\text{d}s}\right)}^2\right]\end{array}$

$\begin{array}{c}\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{t}^{-2\alpha }{\text{e}}^{-\lambda t}{\left|{\displaystyle {\int }_0^t\left(f\left(s,X_s^{\epsilon },Y_s^{\epsilon }\right)-f\left(s,X_s^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)\right)\text{d}s}\right|}^2\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{t}^{-2\alpha }{\text{e}}^{-\lambda t}{\left|{\displaystyle {\int }_0^t\left(f\left(s,X_s^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-f\left(s,X_{s\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)\right)\text{d}s}\right|}^2\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{t}^{-2\alpha }{\text{e}}^{-\lambda t}{\left|{\displaystyle {\int }_0^t\left(f\left(s,X_{s\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-f\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)\right)\text{d}s}\right|}^2\right]\\ \le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{t}^{1-2\alpha }{\text{e}}^{-\lambda t}{\displaystyle {\int }_0^t\left({\Vert X_s^{\epsilon }-X_{s\left(\delta \right)}^{\epsilon }\Vert }_{\infty }^2+{\Vert Y_s^{\epsilon }-{\hat{Y}}_s^{\epsilon }\Vert }_{\infty }^2\right)\text{d}s}\right]+C_{\alpha ,T}{\delta }^2\\ \le C{\delta }^2\left(1+{\delta }^{-1-2\alpha }\left(1+{\epsilon }^{-1}{\text{e}}^{\beta \frac{\delta }{\epsilon }}\right)\right).\end{array}$

综上所述，

$\mathbb{E}\left[{\Vert {\hat{X}}_s^{\epsilon }-X^{\epsilon }\Vert }_{\alpha ,\lambda \left(\tau \right)}^2\right]\le C{\delta }^2\left(1+{\delta }^{-1-2\alpha }\left(1+{\epsilon }^{-1}{\text{e}}^{\beta \frac{\delta }{\epsilon }}\right)\right).$

定理3.6得证。

Step3：对 ${\hat{X}}^{\epsilon }-\bar{X}$ 进行估计。

对 $\forall R>1$ ，我们定义以下停时 ${\tau }_R$ ：

${\tau }_R:=inf\left\{t\ge 0:{\Vert B^H\Vert }_{|\alpha ,0,t}\ge R\right\}\wedge T.$ (17)

定理3.7 假设(H1)~(H5)成立，那么对任意 $t\in \left[0,T\right]$ ，存在正常数C和独立于 $\epsilon$ 的 $\beta >0$ ，使得对足够小的 $\delta$ 和 $\epsilon \in \left(0,1\right)$ ，有

$\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2\right]\le C{\delta }^2\left(1+{\delta }^{-1-2\alpha }\left(1+{\epsilon }^{-1}{\text{e}}^{\beta \frac{\delta }{\epsilon }}\right)\right)+C\left(\delta +\epsilon {\delta }^{-1}\right)+C\sqrt{R^{-1}E\left[{\Vert B^H\Vert }_{\alpha ,0,T}^2\right]}.$

证明由方程(8)和方程(22)，有

$\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2\right]\le \mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_{\left\{{\tau }_R<T\right\}}\right]+\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_{\left\{{\tau }_R\ge T\right\}}\right].$ (23)

对于不等式(23)右侧的第一项，根据切比雪夫不等式，可得

$\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_{\left\{{\tau }_R<T\right\}}\right]\le {\left(\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^4\right]\right)}^{1/2}{\left(\mathbb{P}\left({\tau }_R<T\right)\right)}^{1/2}.$

很容易得到 $\mathbb{P}\left({\tau }_R<T\right)\le \mathbb{P}\left({\Vert B^H\Vert }_{\alpha ,0,T}\ge R\right)\le R^{-1}\mathbb{E}\left[{\Vert B^H\Vert }_{\alpha ,0,T}^2\right]$ 。

根据文献 [25] 中的引理7.5中 ${\lim }_{R\to \infty }{R}^{-1}\mathbb{E}\left[{\Vert B^H\Vert }_{\alpha ,0,T}^2\right]=0$ ，可以得到

$\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_{\left\{{\tau }_R<T\right\}}\right]\le C\sqrt{R^{-1}\mathbb{E}\left[{\Vert B^H\Vert }_{\alpha ,0,T}^2\right]}.$

对于不等式(23)右侧的第二项，

$\begin{array}{c}\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_D\right]\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\left|{\hat{X}}^{\epsilon }\left(t\right)-\bar{X}\left(t\right)\right|}^2{1}_D\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\left({\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha -1}\left|{\hat{X}}^{\epsilon }\left(t\right)-\bar{X}\left(t\right)-{\hat{X}}^{\epsilon }\left(s\right)+\bar{X}\left(s\right)\right|\text{d}s}\right)}^2{1}_D\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\left({\displaystyle {\int }_{-\tau }^0{\left(t-s\right)}^{\alpha +1}\left|{\hat{X}}^{\epsilon }\left(t\right)-\bar{X}\left(t\right)-{\hat{X}}^{\epsilon }\left(s\right)+\bar{X}\left(s\right)\right|\text{d}s}\right)}^2{1}_D\right]\\ =:B_1+B_2+B_3,\end{array}$

其中 $D:=\left\{{\Vert B^H\Vert }_{\alpha ,0,T}\le R\right\}$ 。

对于前两项 $B_1$ 和 $B_2$ ，有

$\begin{array}{c}{B}_1+B_2\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\Vert {\displaystyle {\int }_0^t\left(f\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-\bar{f}\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon }\right)\right)\text{d}s}\Vert }_{\alpha }^2{1}_D\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\Vert {\displaystyle {\int }_0^t\left(\bar{f}\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon }\right)-\bar{f}\left(s,X_{s\left(\delta \right)}^{\epsilon }\right)\right)\text{d}s}\Vert }_{\alpha }^2{1}_D\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\Vert {\displaystyle {\int }_0^t\left(\bar{f}\left(s,X_{s\left(\delta \right)}^{\epsilon }\right)-\bar{f}\left(s,X_s^{\epsilon }\right)\right)\text{d}s}\Vert }_{\alpha }^2{1}_D\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\Vert {\displaystyle {\int }_0^t\left(\bar{f}\left(s,X_s^{\epsilon }\right)-\bar{f}\left(s,{\hat{X}}_s^{\epsilon }\right)\right)\text{d}s}\Vert }_{\alpha }^2{1}_D\right]\end{array}$

$\begin{array}{c}+C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\Vert {\displaystyle {\int }_0^t\left(\bar{f}\left(s,{\hat{X}}_s^{\epsilon }\right)-\bar{f}\left(s,{\bar{X}}_s^{\epsilon }\right)\right)\text{d}s}\Vert }_{\alpha }^2{1}_D\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\Vert {\displaystyle {\int }_0^t\left({\sigma }_1\left(s,{\hat{X}}^{\epsilon }\left(s-\tau \right)\right)-{\sigma }_1\left(s,\bar{X}\left(s-\tau \right)\right)\right)\text{d}{B}_s^H}\Vert }_{\alpha }^2{1}_D\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\Vert {\displaystyle {\int }_0^t\left({\sigma }_1\left(s,X^{\epsilon }\left(s-\tau \right)\right)-{\sigma }_1\left(s,{\hat{X}}^{\epsilon }\left(s-\tau \right)\right)\right)\text{d}{B}_s^H}\Vert }_{\alpha }^2{1}_D\right]\\ =:{\displaystyle \underset{i=1}{\overset{7}{\sum }}{C}_i}.\end{array}$

第一项 $C_1$ 可写成

$\begin{array}{c}{C}_1\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\left|{\displaystyle {\int }_0^t\left(f\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-\bar{f}\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon }\right)\right)\text{d}s}\right|}^2\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\left({\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha -1}\left|{\displaystyle {\int }_s^t\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|\text{d}s}\right)}^2\right]\\ =:C_{11}+C_{12}.\end{array}$

对于 $C_{11}$ ，由条件(H1)和基本不等式，有

$\begin{array}{c}{C}_{11}\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\left|{\displaystyle {\sum }_{k=0}^{\lfloor \frac{t}{\delta }\rfloor -1}{\displaystyle {\int }_{k\delta }^{\left(k+1\right)\delta }\left(f\left(k\delta ,X_{k\delta }^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-\bar{f}\left(k\delta ,X_{k\delta }^{\epsilon }\right)\right)\text{d}s}}\right|}^2\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\left|{\displaystyle {\int }_{\lfloor \frac{t}{\delta }\rfloor \delta }^t\left(f\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-\bar{f}\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon }\right)\right)\text{d}s}\right|}^2\right]\end{array}$

$\begin{array}{c}\le C{\delta }^2+C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }\lfloor \frac{t}{\delta }\rfloor {\displaystyle {\sum }_{k=0}^{\lfloor \frac{t}{\delta }\rfloor -1}{\left|{\displaystyle {\int }_{k\delta }^{\left(k+1\right)\delta }\left(f\left(k\delta ,X_{k\delta }^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-\bar{f}\left(k\delta ,X_{k\delta }^{\epsilon }\right)\right)\text{d}s}\right|}^2}\right]\\ \le C{\delta }^2+\frac{C}{{\delta }^2}\underset{0\le k\le \lfloor \frac{T}{\delta }\rfloor -1}{\max }\mathbb{E}\left[{\left|{\displaystyle {\int }_{k\delta }^{\left(k+1\right)\delta }\left(f\left(k\delta ,X_{k\delta }^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-\bar{f}\left(k\delta ,X_{k\delta }^{\epsilon }\right)\right)\text{d}s}\right|}^2\right]\\ \le C{\delta }^2+C\frac{{\epsilon }^2}{{\delta }^2}\underset{0\le k\le \lfloor \frac{T}{\delta }\rfloor -1}{\max }{\displaystyle {\int }_0^{\frac{\delta }{\epsilon }}{\displaystyle {\int }_{\zeta }^{\frac{\delta }{\epsilon }}{J}_k\left(s,\zeta \right)\text{d}s\text{d}\zeta }}.\end{array}$

对于 $C_{12}$ ，根据Hölder不等式和 $\alpha \in \left(1-H,1/2\right)$ ，可以得到

$\begin{array}{c}{C}_{12}\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\left({\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha -1}\left|{\displaystyle {\int }_s^t\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_s^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|\text{d}s}\right)}^2\right]\\ \le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha -\frac{1}{2}}\text{d}s}\times {\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha -\frac{3}{2}}{\left|{\displaystyle {\int }_s^t\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2\text{d}s}\right]\\ \le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha -\frac{3}{2}}{\left|{\displaystyle {\int }_s^t\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2{1}_{l^c}\text{d}s}\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha -\frac{3}{2}}{\left|{\displaystyle {\int }_s^t\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2{1}_l\text{d}s}\right]\\ =:C_{121}+C_{122}.\end{array}$

其中 $1$ 是一个指示函数， $l:=\left\{t<\left(\lfloor \frac{s}{\delta }\rfloor +2\right)\delta \right\}$ 和 $l^c:=\left\{t\ge \left(\lfloor \frac{s}{\delta }\rfloor +2\right)\delta \right\}$ 。

由(H1)和条件 $\lfloor {\lambda }_1\rfloor -\lfloor {\lambda }_2\rfloor \le {\lambda }_1-{\lambda }_2+1$ ，当 ${\lambda }_1\ge {\lambda }_2\ge 0$ 时，则

$\begin{array}{c}{C}_{121}\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\left(t-s\right)}_{}^{-\alpha -\frac{3}{2}}{\left|{\displaystyle {\int }_s^{\left(\lfloor \frac{s}{\delta }\rfloor +1\right)\delta }\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2{1}_{l^c}\text{d}s}\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\left(t-s\right)}_{}^{-\alpha -\frac{3}{2}}{\left|{\displaystyle {\int }_{t\left(\delta \right)}^t\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2{1}_{l^c}\text{d}s}\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle \underset{0}{\overset{t}{\int }}{\left(t-s\right)}_{}^{-\alpha -\frac{3}{2}}\left(\lfloor \frac{t}{\delta }\rfloor -\lfloor \frac{s}{\delta }\rfloor -1\right){\displaystyle {\sum }_{k=\lfloor \frac{s}{\delta }\rfloor +1}^{\lfloor \frac{t}{\delta }\rfloor -1}{\left|{\displaystyle {\int }_{k\delta }^{\left(k+1\right)\delta }\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2}{1}_{l^c}\text{d}s}\right]\end{array}$

$\begin{array}{c}\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\left(t-s\right)}_{}^{-\alpha -\frac{3}{2}}}\left(\left(\lfloor \frac{s}{\delta }\rfloor +1\right)\delta -s\right){\left|{\displaystyle {\int }_s^{\left(\lfloor \frac{s}{\delta }\rfloor +1\right)\delta }\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2{1}_{l^c}\text{d}s\right]\\ \begin{array}{c}+\end{array}C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\left(t-s\right)}_{}^{-\alpha -\frac{3}{2}}}\left(t-\lfloor \frac{t}{\delta }\rfloor \delta \right){\left|{\displaystyle {\int }_{t\left(\delta \right)}^t\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2{1}_{l^c}\text{d}s\right]\\ +C{\delta }^{-1}\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\left(t-t\left(\delta \right)\right)}^{\frac{1}{2}-\alpha }{\displaystyle {\sum }_{k=\lfloor \frac{s}{\delta }\rfloor +1}^{\lfloor \frac{t}{\delta }\rfloor -1}{\left|{\displaystyle {\int }_{k\delta }^{\left(k+1\right)\delta }\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2}{1}_{l^c}\text{d}}s\right]\\ \le C\delta +C\frac{{\epsilon }^2}{{\delta }^2}\underset{0\le k\le \lfloor \frac{T}{\delta }\rfloor -1}{\max }{\displaystyle {\int }_0^{\frac{\delta }{\epsilon }}{\displaystyle {\int }_{\zeta }^{\frac{\delta }{\epsilon }}{J}_k\left(s,\zeta \right)\text{d}s\text{d}\zeta }}.\end{array}$

对于 $C_{122}$ ，设 $j:=\left\{\lfloor \frac{t}{\delta }\rfloor >1\right\}$ 和 $j^c:=\left\{\lfloor \frac{t}{\delta }\rfloor \le 1\right\}$ ，由条件(H1)和 $t-s<\lfloor \frac{s}{\delta }\rfloor \delta -s+2\delta \le 2\delta$ ，有

$\begin{array}{c}{C}_{122}\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^{\left(\lfloor \frac{t}{\delta }\rfloor -1\right)\delta }{\left(t-s\right)}^{-\alpha -\frac{3}{2}}{\left|{\displaystyle {\int }_s^t\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2{1}_{j\cap l}\text{d}s}\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_{\left(\lfloor \frac{t}{\delta }\rfloor -1\right)\delta }^t{\left(t-s\right)}^{-\alpha -\frac{3}{2}}{\left|{\displaystyle {\int }_s^t\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2{1}_{j\cap l}\text{d}s}\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha -\frac{3}{2}}{\left|{\displaystyle {\int }_s^t\left(f\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon },{\hat{Y}}_r^{\epsilon }\right)-\bar{f}\left(r\left(\delta \right),X_{r\left(\delta \right)}^{\epsilon }\right)\right)\text{d}r}\right|}^2{1}_{j^c\cap l}\text{d}s}\right]\\ \le C{\delta }^2\underset{t\in \left[0,T\right]}{\sup }\left({\displaystyle {\int }_0^{\left(\lfloor \frac{t}{\delta }\rfloor -1\right)\delta }{\left(t-s\right)}^{-\alpha +\frac{1}{2}}}{1}_{j\cap l}\text{d}s\right)\begin{array}{c}+\end{array}C\underset{t\in \left[0,T\right]}{\sup }\left({\displaystyle {\int }_{\left(\lfloor \frac{t}{\delta }\rfloor -1\right)\delta }^t{\left(t-s\right)}^{-\alpha +\frac{1}{2}}}{1}_{j\cap l}\text{d}s\right)\\ +C\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha +\frac{1}{2}}{1}_{j^c\cap l}\text{d}s}\right]\\ \le C{\delta }^{\frac{3}{2}-\alpha }.\end{array}$

因此， $C_1\le C_{\alpha ,T}\frac{{\epsilon }^2}{{\delta }^2}{\max }_{0\le k\le \lfloor \frac{T}{\delta }\rfloor -1}{\displaystyle {\int }_0^{\frac{\delta }{\epsilon }}{\displaystyle {\int }_{\zeta }^{\frac{\delta }{\epsilon }}{J}_k\left(s,\zeta \right)\text{d}s\text{d}\zeta }}$ 。

其中 $0\le \zeta \le s\le \frac{\delta }{\epsilon }$ ， $J_k\left(s,\zeta \right)=\mathbb{E}\left[\langle f\left(k\delta ,X_{k\delta }^{\epsilon },{\hat{Y}}_{s\epsilon +k\delta }^{\epsilon }\right)-\bar{f}\left(k\delta ,X_{k\delta }^{\epsilon }\right),f\left(k\delta ,X_{k\delta }^{\epsilon },{\hat{Y}}_{\zeta \epsilon +k\delta }^{\epsilon }\right)-\bar{f}\left(k\delta ,X_{k\delta }^{\epsilon }\right)\rangle \right]$ 。(24)

下面，我们来估计 $J_k\left(s,\zeta \right)$ ：对 ${\hat{Y}}^{\epsilon }\left(t\right)$ 作时间尺度变换，即对固定的k和 $s\in \left[0,\delta \right]$ ，

$\begin{array}{c}{\hat{Y}}^{\epsilon }\left(s+k\delta \right)={\hat{Y}}^{\epsilon }\left(k\delta \right)+{\displaystyle {\int }_{k\delta }^{s+k\delta }g\left(X_{k\delta }^{\epsilon },{\hat{Y}}^{\epsilon }\left(r\right),{\hat{Y}}^{\epsilon }\left(r-\tau \right)\right)\text{d}r}+{\displaystyle {\int }_0^{s/\epsilon }{\sigma }_2\left(X_{k\delta }^{\epsilon },{\hat{Y}}^{\epsilon }\left(r\right),{\hat{Y}}^{\epsilon }\left(r-\tau \right)\right)\text{d}{W}_r}\\ ={\hat{Y}}^{\epsilon }\left(k\delta \right)+\frac{1}{\epsilon }{\displaystyle {\int }_0^{s}g\left(X_{k\delta }^{\epsilon },{\hat{Y}}^{\epsilon }\left(r+k\delta \right),{\hat{Y}}^{\epsilon }\left(r+k\delta -\tau \right)\right)\text{d}r}\\ +\frac{1}{\sqrt{\epsilon }}{\displaystyle {\int }_0^s{\sigma }_2\left(X_{k\delta }^{\epsilon },{\hat{Y}}^{\epsilon }\left(r+k\delta \right),{\hat{Y}}^{\epsilon }\left(r+k\delta -\tau \right)\right)\text{d}{W}_r^{\ast }},\end{array}$ (25)

其中 $W_t^{*}=W_{t+k\delta }-W_{k\delta }$ 。对于固定的 $\epsilon >0$ 和 $r\ge 0$ ，设 $Y^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\left(r/\epsilon +\theta \right)={\hat{Y}}^{\epsilon }\left(r+k\delta +\theta \right)$ ， $\theta \in \left[-\tau ,0\right]$ 。令 ${\bar{W}}_t$ 是布朗运动且与 $W_t$ 独立。构建过程 $Y^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}$

$\begin{array}{c}{Y}^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\left(s/\epsilon \right)={\hat{Y}}^{\epsilon }\left(k\delta \right)+{\displaystyle {\int }_0^{s/\epsilon }g\left(X_{k\delta }^{\epsilon },Y^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\left(r\right),Y^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\left(r-\tau \right)\right)\text{d}r}\\ +{\displaystyle {\int }_0^{s/\epsilon }{\sigma }_2(X_{k\delta }^{\epsilon },Y^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\left(r\right),}{Y}^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\left(r-\tau \right))\text{d}{\bar{W}}_r\\ ={\hat{Y}}^{\epsilon }\left(k\delta \right)+\frac{1}{\epsilon }{\displaystyle {\int }_0^{s}g\left(X_{k\delta }^{\epsilon },Y^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\left(r/\epsilon \right),Y^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\left(r/\epsilon -\tau \right)\right)\text{d}r}\\ +\frac{1}{\sqrt{\epsilon }}{\displaystyle {\int }_0^s{\sigma }_2\left(X_{k\delta }^{\epsilon },Y^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\left(r/\epsilon \right),Y^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\left(r/\epsilon -\tau \right)\right)\text{d}{\tilde{W}}_r^{\epsilon }},\end{array}$ (26)

其中 ${\tilde{W}}_t^{\epsilon }=\sqrt{\epsilon }{\bar{W}}_{t/\epsilon }$ 。

结合(25)和(26)，可得 $\left(X_{k\delta }^{\epsilon },{\hat{Y}}_{s+k\delta }^{\epsilon }\right)~\left(X_{k\delta }^{\epsilon },Y_{s/\epsilon }^{{}_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\right)$ ， $s\in \left[0,\delta \right)$ ，这里∼表示同分布。

对于当 $s\in \left[0,\delta \right)$ 时，由不等式(24)，有

$J_k\left(s,\zeta \right)=\mathbb{E}\left[\langle f\left(k\delta ,X_{k\delta }^{\epsilon },Y_s^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\right)-\bar{f}\left(k\delta ,X_{k\delta }^{\epsilon }\right),f\left(k\delta ,X_{k\delta }^{\epsilon },Y_{\zeta }^{X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }}\right)-\bar{f}\left(k\delta ,X_{k\delta }^{\epsilon }\right)\rangle \right].$

设 ${\mathcal{M}}_{k\delta }^{\epsilon }$ 是由 $X_{k\delta }^{\epsilon },{\hat{Y}}_{k\delta }^{\epsilon }$ 生成的σ-代数，它与 $\left\{Y_r^{\chi ,\eta }:r\ge 0\right\}$ 独立。由文献 [28] 中的定理7.1.2，再结合引理2.7，可以得到 $J_k\left(s,\zeta \right)\le C\left(1+\mathbb{E}\left[{\Vert X_{k\delta }^{\epsilon }\Vert }_{\infty }^2\right]+\mathbb{E}\left[{\Vert {\hat{Y}}_{k\delta }^{\epsilon }\Vert }_{\infty }^2\right]\right){\text{e}}^{-\frac{\rho }{2}\left(s-\zeta \right)}$ 。

因此，对于 $C_1$ ， $C_1\le C\left(\epsilon {\delta }^{-1}+\delta \right)$ 。

通过定理3.6，估计 ${{\displaystyle \sum }}_{i=2}^5{C}_i$ ，可以得到

$\begin{array}{c}{\displaystyle \underset{i=2}{\overset{5}{\sum }}{C}_i}\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha }{\left|\bar{f}\left(s\left(\delta \right),X_{s\left(\delta \right)}^{\epsilon }\right)-\bar{f}\left(s,X_{s\left(\delta \right)}^{\epsilon }\right)\right|}^2}\text{d}s{1}_D\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha }{\left|\bar{f}\left(s,X_{s\left(\delta \right)}^{\epsilon }\right)-\bar{f}\left(s,X_s^{\epsilon }\right)\right|}^2}\text{d}s{1}_D\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha }{\left|\bar{f}\left(s,X_s^{\epsilon }\right)-\bar{f}\left(s,{\hat{X}}_s^{\epsilon }\right)\right|}^2\text{d}s}{1}_D\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\text{e}}^{-\lambda t}{\displaystyle {\int }_0^t{\left(t-s\right)}^{-2\alpha }{\left|\bar{f}\left(s,{\hat{X}}_s\right)-\bar{f}\left(s,{\bar{X}}_s^{\epsilon }\right)\right|}^2\text{d}}s{1}_D\right]\\ \le C{\delta }^2\left(1+{\delta }^{-1-2\alpha }\left(1+{\epsilon }^{-1}{\text{e}}^{\beta \frac{\delta }{\epsilon }}\right)\right)+C{\lambda }^{\alpha -1}\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_D\right].\end{array}$

对于 $C_6$ ，有

$\begin{array}{c}{C}_6\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\text{e}}^{-\lambda t}\left[{\left(t-r\right)}^{-2\alpha }+r^{-\alpha }\right]{\Vert {\hat{\sigma }}_1\left(r,{\hat{X}}^{\epsilon }\left(r-\tau \right)\right)-{\sigma }_1\left(r,\bar{X}\left(r-\tau \right)\right)\Vert }_{\alpha }^2{1}_D\text{d}r}\right]\\ \le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\text{e}}^{-\lambda t}\left[{\left(t-r\right)}^{-2\alpha }+r^{-\alpha }\right]{\left|{\hat{\sigma }}_1\left(r,{\hat{X}}^{\epsilon }\left(r-\tau \right)\right)-{\sigma }_1\left(r,\bar{X}\left(r-\tau \right)\right)\right|}^2{1}_D\text{d}r}\right]\\ +C\mathbb{E}[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_0^t{\text{e}}^{-\lambda t}\left[{\left(t-r\right)}^{-2\alpha }+r^{-\alpha }\right]}({\displaystyle {\int }_0^r{\left(t-u\right)}^{-\alpha -1}}\\ \times {\left|{\sigma }_1\left(r,{\hat{X}}^{\epsilon }\left(r-\tau \right)\right)-{\sigma }_1\left(r,{\bar{X}}^{\epsilon }\left(r-\tau \right)\right)-{\sigma }_1\left(u,{\hat{X}}^{\epsilon }\left(u-\tau \right)\right)+{\sigma }_1\left(u,\bar{X}\left(u-\tau \right)\right)\right|\text{d}u)}^2{1}_D\text{d}r]\\ =:G_{61}+G_{62}.\end{array}$

对于 $C_{61}$ ，通过Hölder不等式，

$\begin{array}{c}{G}_{61}\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle \underset{0}{\overset{t}{\int }}{\text{e}}^{-\lambda \left(t-r\right)}\left[{\left(t-r\right)}^{-2\alpha }+r^{-\alpha }\right]\underset{q\in \left[-\tau ,r\right]}{\sup }{\text{e}}^{-\lambda r}{\left|{\hat{X}}^{\epsilon }\left(u\right)-\bar{X}\left(u\right)\right|}^2{1}_D\text{d}r}\right]\\ \le C{\lambda }^{2\alpha -1}\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_D\right].\end{array}$

根据(H2)和文献 [25] 的引理7.1，存在正常数C，使得

$\begin{array}{l}\left|\sigma \left(t_1,x_1\right)-\sigma \left(t_2,x_2\right)-\sigma \left(t_1,x_3\right)+\sigma \left(t_2,x_4\right)\right|\\ \le C\left|x_1-x_2-x_3+x_4\right|+C\left|x_1-x_3\right|{\left|t_2-t_1\right|}^{\beta }+C\left|x_1-x_3\right|\left({\left|x_1-x_2\right|}^{\gamma }+{\left|x_3-x_4\right|}^{\gamma }\right).\end{array}$

对于 $C_{62}$ ，有

$\begin{array}{c}{C}_{62}\le C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_{-\tau }^{t-\tau }{\text{e}}^{-\lambda t}\left[{\left(t-s-\tau \right)}^{-2\alpha }+{\left(s+\tau \right)}^{-\alpha }\right]}\times {\left({\displaystyle {\int }_{-\tau }^s\frac{\left|{\hat{X}}^{\epsilon }\left(s\right)-\bar{X}\left(s\right)-{\hat{X}}^{\epsilon }\left(u\right)+\bar{X}\left(u\right)\right|}{{\left(s-u\right)}^{\alpha +1}}\text{d}u}\right)}^2{1}_D\text{d}s\right]\\ +C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_{-\tau }^{t-\tau }{\text{e}}^{-\lambda t}\left[{\left(t-s-\tau \right)}^{-2\alpha }+{\left(s+\tau \right)}^{-\alpha }\right]}\times {\left(\left|{\hat{X}}^{\epsilon }\left(s\right)-\bar{X}\left(s\right)\right|{\displaystyle {\int }_{-\tau }^s\frac{{\left|{\hat{X}}^{\epsilon }\left(s\right)-{\hat{X}}^{\epsilon }\left(u\right)\right|}^{\gamma }}{{\left(s-u\right)}^{1+\alpha }}\text{d}u}\right)}^2{1}_D\text{d}s\right]\end{array}$

$\begin{array}{c}+C\mathbb{E}\left[\underset{t\in \left[0,T\right]}{\sup }{\displaystyle {\int }_{-\tau }^{t-\tau }{\text{e}}^{-\lambda t}\left[{\left(t-s-\tau \right)}^{-2\alpha }+{\left(s+\tau \right)}^{-\alpha }\right]}\times {\left(\left|{\hat{X}}^{\epsilon }\left(s\right)-\bar{X}\left(s\right)\right|{\displaystyle {\int }_{-\tau }^s\frac{{\left|\bar{X}\left(s\right)-\bar{X}\left(u\right)\right|}^{\gamma }}{{\left(s-u\right)}^{1+\alpha }}\text{d}u}\right)}^2{1}_D\text{d}s\right]\\ \le C\mathbb{E}{\displaystyle {\int }_{-\tau }^{t-\tau }{\text{e}}^{-\lambda t}\left[{\left(t-s-\tau \right)}^{-2\alpha }+{\left(s+\tau \right)}^{-\alpha }\right]}\times {\left(1+\Delta \left({\hat{X}}^{\epsilon }\right)+\Delta \left(\bar{X}\right)\right)}^2{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_D\text{d}s],\end{array}$

其中 $\Delta \left({\hat{X}}^{\epsilon }\right)=\underset{s\in \left[-\tau ,T\right]}{\sup }{\displaystyle {\int }_{-\tau }^s\frac{{\left|{\hat{X}}^{\epsilon }\left(s\right)-\hat{X}\left(u\right)\right|}^{\gamma }}{{\left(s-u\right)}^{\alpha +1}}\text{d}u}$ 和 $\Delta \left(\bar{X}\right)=\underset{s\in \left[-\tau ,T\right]}{\sup }{\displaystyle {\int }_{-\tau }^s\frac{{\left|\bar{X}\left(s\right)-\bar{X}\left(u\right)\right|}^{\gamma }}{{\left(s-u\right)}^{\alpha +1}}\text{d}u}$ 。

由引理3.4，在 ${\Vert B^H\Vert }_{\alpha ,0,T}\le R$ 的条件下，存在正常数C，使得

$\begin{array}{c}\Delta \left({\hat{X}}^{\epsilon }\right)+\Delta \left(\bar{X}\right)\le C{\Lambda }_{\alpha }^{\gamma }\left(B^H\right)\left[{\left(1+{\Vert {\hat{X}}^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}\right)}^{\gamma }+{\left(1+{\Vert \bar{X}\Vert }_{\alpha ,\infty \left(\tau \right)}\right)}^{\gamma }\right]\underset{s\in \left[-\tau ,T\right]}{\sup }{\displaystyle {\int }_{-\tau }^s{\left(s-r\right)}^{\left(1-\alpha \right)\gamma -1-\alpha }\text{d}r}\\ \le C{\Lambda }_{\alpha }^{\gamma }\left(B^H\right)\left(1+{\Vert {\hat{X}}^{\epsilon }\Vert }_{\alpha ,\infty \left(\tau \right)}^{\gamma }+{\Vert \bar{X}\Vert }_{\alpha ,\infty \left(\tau \right)}^{\gamma }\right)\le C.\end{array}$

其中 $\left(1-\alpha \right)\gamma -\alpha >0$ 且 $\alpha \in \left(0,\gamma /2\right)$ 。

因此，可得 $C_6\le C_{61}+C_{62}\le C{\lambda }^{2\alpha -1}\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_D\right]$ 。

同理， $C_7$ 的估计如下 $C_7\le C{\lambda }^{2\alpha -1}\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-X\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_D\right]$ 。

同理， $B_3$ 的估计为

$B_3\le C\left(\epsilon {\delta }^{-1}+\delta \right)+C{\lambda }^{2\alpha -1}\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2{1}_D\right]+C{\delta }^2\left(1+{\delta }^{-1-2\alpha }\left(1+{\epsilon }^{-1}{\text{e}}^{\beta \frac{\delta }{\epsilon }}\right)\right).$

综上所述，

$\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2\right]\le C{\delta }^2\left(1+{\delta }^{-1-2\alpha }\left(1+{\epsilon }^{-1}{\text{e}}^{\beta \frac{\delta }{\epsilon }}\right)\right)+C\left(\delta +\epsilon {\delta }^{-1}\right)+C\sqrt{R^{-1}\mathbb{E}\left[{\Vert B^H\Vert }_{\alpha ,0,T}^2\right]}.$

引理3.7得证。

Step4：对 $X^{\epsilon }-\bar{X}$ 进行估计。

应用基本不等式，由式(8)和(12)可以得到

$\mathbb{E}\left[{\Vert X^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2\right]\le 2\mathbb{E}\left[{\Vert X^{\epsilon }-{\hat{X}}^{\epsilon }\Vert }_{\alpha ,\lambda \left(\tau \right)}^2\right]+2\mathbb{E}\left[{\Vert {\hat{X}}^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2\right].$

根据定理3.6和定理3.7，有

$\mathbb{E}\left[{\Vert X^{\epsilon }-\bar{X}\Vert }_{\alpha ,\infty \left(\tau \right)}^2\right]\le C{\delta }^2\left(1+{\delta }^{-1-2\alpha }\left(1+{\epsilon }^{-1}{\text{e}}^{\beta \frac{\delta }{\epsilon }}\right)\right)+C\left(\delta +\epsilon {\delta }^{-1}\right)+C\sqrt{R^{-1}\mathbb{E}\left[{\Vert B^H\Vert }_{\alpha ,0,T}^2\right]}.$

综上所述，

$\begin{array}{c}\mathbb{E}\left[{\Vert X^{\epsilon }-\bar{X}\Vert }_{\alpha ,\infty \left(\tau \right)}^2\right]\le {\text{e}}^{\lambda T}\mathbb{E}\left[{\Vert X^{\epsilon }-\bar{X}\Vert }_{\alpha ,\lambda \left(\tau \right)}^2\right]\\ \le C{\delta }^2\left(1+{\delta }^{-1-2\alpha }\left(1+{\epsilon }^{-1}{\text{e}}^{\beta \frac{\delta }{\epsilon }}\right)\right)+C\left(\delta +\epsilon {\delta }^{-1}\right)+C\sqrt{R^{-1}\mathbb{E}\left[{\Vert B^H\Vert }_{\alpha ,0,T}^2\right]}.\end{array}$

取 $\delta =\epsilon {\left(-\ln \epsilon \right)}^{1/2}$ ，当 $R\to \infty$ 时，有

$\underset{\epsilon \to 0}{\lim }\mathbb{E}\left[{\Vert X^{\epsilon }-\bar{X}\Vert }_{\alpha ,\infty \left(\tau \right)}^2\right]=0.$

综上所述，即证明了定理3.1。

4. 结束语

在本篇论文中，我们研究了由分数布朗运动驱动的非自治双尺度随机时滞微分方程的平均原理，其中该方程的慢分量由Hurst参数 $H\in \left(1/2,1\right)$ 的分数布朗运动驱动，快分量由布朗运动驱动。首先，我们使用广义Stieltjes积分对快慢过程进行估计，得到方程(2.1)的解 $\left(X^{\epsilon },Y^{\epsilon }\right)$ 的矩估计。其次，我们利用Khasminskii技巧，构造了辅助过程 $\left({\hat{X}}_t^{\epsilon },{\hat{Y}}_t^{\epsilon }\right)$ ，估计了 $X_t^{\epsilon }-{\hat{X}}_t^{\epsilon }$ 和 $Y_t^{\epsilon }-{\hat{Y}}_t^{\epsilon }$ 这两项。然后，基于快方程的指数遍历性，我们引入停时序列来控制分数布朗运动的大小进而估计辅助过程 ${\hat{X}}_t^{\epsilon }$ 与平均方程解 ${\bar{X}}_t$ 之间的误差。最后，证明了慢分量 $X_t^{\epsilon }$ 在均方意义下收敛于平均方程的解 ${\bar{X}}_t$ 。

基金项目

长安大学中央高校基本科研业务费专项资金资助(CHD300102122113)，陕西省自然科学基础研究计划项目(2023-JC-QN-0009)。