一类带共同噪声和Lévy过程驱动的McKean-Vlasov随机微分方程的数值分析

Numerical analysis for a class of McKean-Vlasov stochastic differential equations driven by common noise and Lévy processes

摘要

研究了一类带共同噪声和Lévy过程驱动的McKean-Vlasov随机微分方程的数值方法，其中方程系数满足超线性增长条件. 对相应的交互粒子系统构造了自适应Euler-Maruyama算法，并给出了该数值算法的收敛速率.

Abstract

The numerical scheme for a class of McKean-Vlasov stochastic differential equations driven by common noise and Lévy processes is studied， whose coefficients satisfy the superlinear growth condition. The adaptive Euler-Maruyama scheme is constructed for the corresponding interacting particle system， and its convergence rate is shown.

关键词

McKean-Vlasov随机微分方程 / 共同噪声 / 自适应Euler-Maruyama算法 / Lévy过程

Key words

McKean-Vlasov stochastic differential equations / common noise / adaptive Euler-Maruyama scheme / Lévy process

1 相关知识

本文主要研究一类带共同噪声和Lévy过程驱动的McKean-Vlasov随机微分方程的数值算法，方程的形式如下

\mathrm{d}{X}_t=b\left(X_t,{\mathrm{ℒ}}_{X_t}\right)\mathrm{d}t+\sigma \left(X_t,{\mathrm{ℒ}}_{X_t}\right)\mathrm{d}{W}_t^{\mathrm{0}}+\kappa \left(X_t,{\mathrm{ℒ}}_{X_t}\right)\mathrm{d}{P}_t，（1）

其中b:{\mathbb{R}}^d\times {\mathrm{𝒫}}_{\mathrm{2}}\left({\mathbb{R}}^d\right)\to {\mathbb{R}}^d，\sigma :{\mathbb{R}}^d\times {\mathrm{𝒫}}_{\mathrm{2}}\left({\mathbb{R}}^d\right)\to {\mathbb{R}}^{d\times l}，\kappa :{\mathbb{R}}^d\times{\mathrm{𝒫}}_{\mathrm{2}}\left({\mathbb{R}}^d\right)\to {\mathbb{R}}^{d\times l},其初值为X_{\mathrm{0}}=\xi，W_t^{\mathrm{0}}表示共同噪声且是一个l-维布朗运动，P_t={\int }_{\mathrm{0}}^t{\int }_{Z}z\left(N(\mathrm{d}s,\mathrm{d}z)-\nu (\mathrm{d}z)\mathrm{d}s\right)是一个纯跳的Lévy过程，并且Lévy测度\nu满足{\int }_{{\mathit{ℝ}}^d}{\left|z\right|}^{\mathrm{2}}\nu \left(d\right.z)\infty，Z={\mathbb{R}}^d/\{\mathrm{0}\}，N(\mathrm{d}t,\mathrm{d}z)表示Poisson测度.另外，{\mathrm{ℒ}}_{X_t}表示随机变量X_t的条件边缘分布流.

文献［1］提出了一种逼近McKean-Vlasov随机微分方程的方法，其分为两步.第1步是用经验测度{\mu }_t^{X,N}\left(\mathrm{d}x\right):=\frac{\mathrm{1}}{N}{\displaystyle \sum _{i=\mathrm{1}}^N}{\delta }_{X_t^{i,N}}\left(\mathrm{d}x\right)对真实分布{\mathrm{ℒ}}_{X_t}近似，其中{\delta }_{X_t^{i,N}}表示X_t^{i,N}的Dirac测度，且{\left(X^{i,N}\right)}_{i\in \left\{\mathrm{1},...,N\right\}}是如下的{\mathbb{R}}^{d\times N}维交互粒子系统的解

\mathrm{d}{X}_t^{i,N}=b\left(X_t^{i,N},{\mu }_t^{X,N}\right)\mathrm{d}t+\sigma \left(X_t^{i,N},{\mu }_t^{X,N}\right)\mathrm{d}{W}_t^{\mathrm{0}}+\kappa \left(X_t,{\mu }_t^{X,N}\right)\mathrm{d}{P}_t^i,（2）

第2步构造适当的数值算法来近似上述交互粒子系统，从而实现对原方程的数值逼近.

1956年，McKean在文献［2］中首次展开了对McKean-Vlasov随机微分方程的研究. 当带有共同噪声的McKean-Vlasov随机微分方程的系数满足Lipschitz条件时，文献［3］讨论了其强解的存在唯一性.当系数的状态变量满足单调条件时，文献［4-5］研究了McKean-Vlasov随机微分方程强解的适定性.文献［6-7］研究了McKean-Vlasov随机微分方程弱解的存在唯一性. 文献［8］用自适应Euler-Maruyama算法对一类随机微分方程进行数值逼近.随后，文献［9］研究了不带共同噪声且漂移项和扩散项系数满足超线性条件的McKean-Vlasov随机微分方程的自适应算法的收敛性.文献［10-11］考虑了驯服Milstein算法对满足单边Lipschitz条件的McKean-Vlasov随机微分方程的收敛性.文献［12］研究了驯服Euler-Maruyama算法对Lévy过程驱动的McKean-Vlasov随机微分方程的收敛性.文献［13］研究了驯服的自适应Euler-Maruyama算法对Lévy过程驱动的McKean-Vlasov随机微分方程在无限时间尺度下的数值逼近.

令\left|\cdot \right|表示Euclidean范数，且‖\cdot ‖表示Hilbert-Schmidt范数.令\left(\mathrm{\Omega },\mathrm{ℱ},{\left({\mathrm{ℱ}}_t\right)}_{t\ge \mathrm{0}},\mathbb{P}\right)表示完备的概率空间，其\sigma代数流{\left({\mathrm{ℱ}}_t\right)}_{t\ge \mathrm{0}}满足一般条件（即单调递增和右连续的，且{\mathrm{ℱ}}_{\mathrm{0}}包含所有零测集）.用〈a,b〉:=a_{\mathrm{1}}{b}_{\mathrm{1}}+\dots +a_d{b}_d表示两个{\mathbb{R}}^d向量a和b的内积.用\mathrm{𝒫}\left({\mathbb{R}}^d\right)表示在\left({\mathbb{R}}^d,\mathrm{ℬ}\left({\mathbb{R}}^d\right)\right)上的所有概率测度集，其中\mathrm{ℬ}\left({\mathbb{R}}^d\right)是一个Borel\sigma-域.对于p\ge \mathrm{1}，定义：

，

对于任意的\mu ,\nu \in {\mathrm{𝒫}}_p\left({\mathbb{R}}^d\right)，其Wasserstein距离定义为

{\mathrm{𝒲}}_p\left(\mu ,\nu \right):={\left(\underset{\pi \in \mathrm{\Pi }\left(\mu ,\nu \right)}{\mathrm{i}\mathrm{n}\mathrm{f}}{\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}{\left|x-y\right|}^p\pi \left(\mathrm{d}x,\mathrm{d}y\right)\right)}^{\frac{\mathrm{1}}{p}}，

其中\mathrm{\Pi }\left(\mu ,\nu \right)是μ和ν全体耦合.

对于任意i\in \left\{\mathrm{1},\dots ,N\right\}，给定自适应的时间步长h_n^i=h\left({\widehat{X}}_{t_n}^{i,N},{\mu }_{t_n}^{\widehat{X},N}\right).将交互粒子系统（2）的自适应Euler-Maruyama算法定义为

{\widehat{X}}_{t_{n+\mathrm{1}}}^{i,N}={\widehat{X}}_{t_n}^{i,N}+b\left({\widehat{X}}_{t_n}^{i,N},{\mu }_{t_n}^{\widehat{X},N}\right)h_n^{\mathrm{m}\mathrm{i}\mathrm{n}}+\sigma \left({\widehat{X}}_{t_n}^{i,N},{\mu }_{t_n}^{\widehat{X},N}\right)\mathrm{\Delta }{W}_{t_n}^{\mathrm{0}}+\kappa \left({\widehat{X}}_{t_n}^{i,N},{\mu }_{t_n}^{\widehat{X},N}\right)\mathrm{\Delta }{P}_{t_n}^i,（3）

其中\mathrm{\Delta }{W}_{t_n}^i=W_{t_{n+\mathrm{1}}}^i-W_{t_n}^i,\mathrm{\Delta }{P}_{t_n}^i=P_{t_{n+\mathrm{1}}}^i-P_{t_n}^i.且时间步长h_n^{\mathrm{m}\mathrm{i}\mathrm{n}}:=\mathrm{m}\mathrm{i}\mathrm{n}\left\{h_n^{\mathrm{1}},\dots ,h_n^N\right\},t_{n+\mathrm{1}}=t_n+h_n^{\mathrm{m}\mathrm{i}\mathrm{n}}，t_n随着n的增大而增大，故当n=M时，有t_M\ge T.定义符号\underline{t}:=\mathrm{m}\mathrm{a}\mathrm{x}\left\{t_n:t_n\le t\right\}是时间t前最靠近t的时间点，以及n_t:=\mathrm{m}\mathrm{a}\mathrm{x}\left\{n:t_n\le t\right\}表示直到时间t近似步长数量.因此，定义连续插值过程如下：

{\widehat{X}}_t^{i,N}={\widehat{X}}_{\underline{t}}^{i,N}+b\left({\widehat{X}}_{\underline{t}}^{i,N},{\mu }_{\underline{t}}^{\widehat{X},N}\right)\left(t-\underline{t}\right)+\sigma \left({\widehat{X}}_{\underline{t}}^{i,N},{\mu }_{\underline{t}}^{\widehat{X},N}\right)\left(W_t^{\mathrm{0}}-W_{\underline{t}}^{\mathrm{0}}\right)+\kappa \left({\widehat{X}}_{\underline{t}}^{i,N},{\mu }_{\underline{t}}^{\widehat{X},N}\right)\left(P_t^i-P_{\underline{t}}^i\right),（4）

因此{\left({\widehat{X}}_t^{i,N}\right)}_{t\in \left[\mathrm{0},T\right]}满足

\mathrm{d}{\widehat{X}}_t^{i,N}=b\left({\widehat{X}}_{\underline{t}}^{i,N},{\mu }_{\underline{t}}^{\widehat{X},N}\right)\mathrm{d}t+\sigma \left({\widehat{X}}_{\underline{t}}^{i,N},{\mu }_{\underline{t}}^{\widehat{X},N}\right)\mathrm{d}{W}_t^{\mathrm{0}}+\kappa \left({\widehat{X}}_{\underline{t}}^{i,N},{\mu }_{\underline{t}}^{\widehat{X},N}\right)\mathrm{d}{P}_t^i.（5）

令方程（1）的系数满足如下的假设：

（H1）对于任意的p\ge \mathrm{2}，有.

（H2）存在常数L_{\mathrm{1}}>\mathrm{0},q>\mathrm{0}，以及{\mathrm{ℱ}}_{\mathrm{0}}可测的非负随机变量序列{\left\{{\tilde{A}}_t\right\}}_{t\in [\mathrm{0},T]}满足，使得对任意的p\ge \mathrm{2},x,x\mathrm{\text{'}}\in {\mathbb{R}}^d,\mu ,\mu \mathrm{\text{'}}\in {\mathrm{𝒫}}_{\mathrm{2}}\left({\mathbb{R}}^d\right)，有：

\begin{array}{c}\mathrm{2}{\left|x-x\mathrm{\text{'}}\right|}^{p-\mathrm{2}}〈x-x\mathrm{\text{'}},b\left(x,\mu \right)-b\left(x\mathrm{\text{'}},\mu \mathrm{\text{'}}\right)〉+\mathrm{2}\left(p-\mathrm{1}\right){\left|x-x\mathrm{\text{'}}\right|}^{p-\mathrm{2}}{‖\sigma \left(x,\mu \right)-\sigma \left(x\mathrm{\text{'}},\mu \mathrm{\text{'}}\right)‖}^{\mathrm{2}}+\\ {\mathrm{2}}^{p-\mathrm{2}}\left(p-\mathrm{1}\right){‖\kappa \left(x,\mu \right)-\kappa \left(x\mathrm{\text{'}},\mu \mathrm{\text{'}}\right)‖}^{\mathrm{2}}{\int }_Z{\int }_{\mathrm{0}}^{\mathrm{1}}\left(\mathrm{1}-\theta \right)\times \\ {\left|x-x\mathrm{\text{'}}+\theta z\left(\kappa \left(x,\mu \right)-\kappa \left(x\mathrm{\text{'}},\mu \mathrm{\text{'}}\right)\right)\right|}^{p-\mathrm{2}}{\left|z\right|}^{\mathrm{2}}\mathrm{d}\theta \nu (\mathrm{d}z)\le L_{\mathrm{1}}\left({\left|x-x\mathrm{\text{'}}\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left(\mu ,\mu \mathrm{\text{'}}\right)\right).\end{array}（6）

\left|b\left(x,\mu \right)-b\left(x\mathrm{\text{'}},\mu \mathrm{\text{'}}\right)\right|\le L_{\mathrm{1}}\left(\mathrm{1}+{\left|x\right|}^q+{\left|x\mathrm{\text{'}}\right|}^q\right)\left|x-x\mathrm{\text{'}}\right|+{\mathrm{𝒲}}_{\mathrm{2}}\left(\mu ,\mu \mathrm{\text{'}}\right).（7）

\begin{array}{l}{‖\sigma \left(x,\mu \right)-\sigma \left(x\mathrm{\text{'}},\mu \mathrm{\text{'}}\right)‖}^{\mathrm{2}}+{‖\kappa \left(x,\mu \right)-\kappa \left(x\mathrm{\text{'}},\mu \mathrm{\text{'}}\right)‖}^{\mathrm{2}}{\int }_Z{\left|z\right|}^{\mathrm{2}}\nu \left(d\right.z)\le \\ L_{\mathrm{1}}\left(\mathrm{1}+{\left|x\right|}^q+{\left|x\mathrm{\text{'}}\right|}^q\right){\left|x-x\mathrm{\text{'}}\right|}^{\mathrm{2}}+{\mathrm{𝒲}}_{\mathrm{2}}^{\mathrm{2}}\left(\mu ,\mu \mathrm{\text{'}}\right).\end{array}（8）

\begin{array}{c}\mathrm{2}{\left|x\right|}^{p-\mathrm{2}}〈x,b(x,\mu )〉+\left(p-\mathrm{1}\right){\left|x\right|}^{p-\mathrm{2}}{‖\sigma (x,\mu )‖}^{\mathrm{2}}+\mathrm{2}\left(p-\mathrm{1}\right){‖\kappa (x,\mu )‖}^{\mathrm{2}}\times \\ {\int }_Z{\int }_{\mathrm{0}}^{\mathrm{1}}\left(\mathrm{1}-\theta \right){\left|x+\theta z\kappa (x,\mu )\right|}^{p-\mathrm{2}}{\left|z\right|}^{\mathrm{2}}\mathrm{d}\theta \nu (\mathrm{d}z)\le L_{\mathrm{1}}\left({\tilde{A}}_t+{\left|x\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left(\mu ,{\delta }_{\mathrm{0}}\right)\right).\end{array}（9）

由于步长h依赖于状态变量{X_t}^{i,N}，因此用h^{\epsilon }(x,\mu )表示修正后的步长函数.

（H3）对于步长函数h,h^{\epsilon }:{\mathbb{R}}^d\times {\mathrm{𝒫}}_{\mathrm{2}}\left({\mathbb{R}}^d\right)\to {\mathbb{R}}^{+}，存在常数L_{\mathrm{2}},q,{\eta }_{\mathrm{1}},{\eta }_{\mathrm{2}},{\eta }_{\mathrm{3}},\varpi >\mathrm{0}，使得对任意的x\in {\mathbb{R}}^d,\mu \in {\mathrm{𝒫}}_{\mathrm{2}}\left({\mathbb{R}}^d\right)，有：

\begin{array}{c}{\left({\eta }_{\mathrm{1}}{\left|x\right|}^{\varpi }+{\eta }_{\mathrm{2}}{\mathrm{𝒲}}_{\mathrm{2}}^{\varpi }\left(\mu ,{\delta }_{\mathrm{0}}\right)+{\eta }_{\mathrm{3}}\right)}^{-\mathrm{1}}\le h\left(x,\mu \right)\le \\ L_{\mathrm{2}}{\left(\mathrm{1}+{\left|x\right|}^{\mathrm{3}q}+{\mathrm{𝒲}}_{\mathrm{2}}^{\mathrm{3}q}\left(\mu ,{\delta }_{\mathrm{0}}\right)\right)}^{-\mathrm{1}}.\end{array}（10）

\epsilon \mathrm{m}\mathrm{i}\mathrm{n}\left(T,h\left(x,\mu \right)\right)\le h^{\epsilon }\left(x,\mu \right)\le \mathrm{m}\mathrm{i}\mathrm{n}\left(\epsilon T,h\left(x,\mu \right)\right).（11）

由假设（H1），（H2）可推出方程（1）存在唯一的强解并且E\left[{\left|X_t\right|}^r\right]\le C(r\ge \mathrm{0})，其证明可以参考文献［12-13］及其参考文献.同理，由假设（H1），（H2）还可推出交互粒子系统（2）存在唯一的强解{X_t}^{i,N}及其p阶矩有界性.

2 主要结论及证明

定理1 对于任意的p\ge \mathrm{2}，在假设（H1），（H2），（H3）下，存在依赖于时间T和p的常数C>\mathrm{0}，使得自适应Euler-Maruyama算法（3）的解满足：

\underset{i\in \{\mathrm{1},\cdot \cdot \cdot ,N\}}{\mathrm{m}\mathrm{a}\mathrm{x}}\underset{t\in [\mathrm{0},T]}{\mathrm{s}\mathrm{u}\mathrm{p}}E\left[{\left|{\widehat{X}}_t^{i,N}\right|}^p\right]\le C.（12）

证明 定义停时{\rho }_{K>\mathrm{0}}:=\mathrm{i}\mathrm{n}\mathrm{f}\left\{t\ge \mathrm{0}:\underset{{}_{i\in \{\mathrm{1},...,N]}}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|{\widehat{X}}_t^{i,N}\right|\ge K\right\}，并使用Itô公式可得：

\begin{array}{c}{\left|{\widehat{X}}_{t\wedge {\rho }_K}^{i,N}\right|}^p\le {\left|{\widehat{X}}_{\mathrm{0}}^{i,N}\right|}^p+p{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|{\widehat{X}}_s^{i,N}\right|}^{p-\mathrm{2}}〈{\widehat{X}}_s^{i,N},b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)〉\mathrm{d}s+\\ p{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|{\widehat{X}}_s^{i,N}\right|}^{p-\mathrm{2}}〈{\widehat{X}}_s^{i,N},\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\mathrm{d}{W}_t^{\mathrm{0}}〉+\frac{p\left(p-\mathrm{1}\right)}{\mathrm{2}}{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|{\widehat{X}}_s^{i,N}\right|}^{p-\mathrm{2}}{‖\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)‖}^{\mathrm{2}}\mathrm{d}s+\\ p{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\int }_Z{\left|{\widehat{X}}_s^{i,N}\right|}^{p-\mathrm{2}}〈{\widehat{X}}_s^{i,N},z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)〉{\tilde{N}}^i(\mathrm{d}s,\mathrm{d}z)+\\ {\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\int }_Z\left({\left|{\widehat{X}}_s^{i,N}+z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^p-{\left|{\widehat{X}}_s^{i,N}\right|}^p-p{\left|{\widehat{X}}_s^{i,N}\right|}^{p-\mathrm{2}}〈{\widehat{X}}_s^{i,N},z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)〉\right)N^i(\mathrm{d}s,\mathrm{d}z),\end{array}

两边同时取期望并由（10）式和等式

{\left|y\right|}^p={\left|a\right|}^p+p{\left|a\right|}^{p-\mathrm{2}}〈a,y-a〉+p(p-\mathrm{1}){\int }_{\mathrm{0}}^{\mathrm{1}}(\mathrm{1}-\theta ){\left|y-a\right|}^{\mathrm{2}}{\left|a+\theta (y-a)\right|}^{p-\mathrm{2}}\mathrm{d}\theta可得到

\begin{array}{c}E\left[{\left|{\widehat{X}}_{t\wedge {\rho }_K}^{i,N}\right|}^p\right]\le E\left[{\left|{\widehat{X}}_{\mathrm{0}}^{i,N}\right|}^p\right]+\frac{p}{\mathrm{2}}E\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|{\widehat{X}}_s^{i,N}\right|}^{p-\mathrm{2}}\left(\mathrm{2}〈{\widehat{X}}_{\underline{s}}^{i,N},b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)〉+\right.\right.\\ \left(p-\mathrm{1}\right){‖\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)‖}^{\mathrm{2}}+\mathrm{2}\left(p-\mathrm{1}\right){\int }_Z{‖\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)‖}^{\mathrm{2}}\times \\ \left.\left.{\int }_{\mathrm{0}}^{\mathrm{1}}(\mathrm{1}-\theta ){\left|{\widehat{X}}_{\underline{s}}^{i,N}+\theta z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{p-\mathrm{2}}{\left|z\right|}^{\mathrm{2}}\mathrm{d}\theta \nu (\mathrm{d}z)\right)\mathrm{d}s\right]+\\ pE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|{\widehat{X}}_s^{i,N}\right|}^{p-\mathrm{2}}\right.\left.〈{\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N},b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)〉\mathrm{d}s\right]+\\ p\left(p-\mathrm{1}\right)E\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\int }_Z{‖\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)‖}^{\mathrm{2}}{\int }_{\mathrm{0}}^{\mathrm{1}}(\mathrm{1}-\theta )\times \right.\\ \left.\left({\left|{\widehat{X}}_s^{i,N}+\theta z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{p-\mathrm{2}}-{\left|{\widehat{X}}_{\underline{s}}^{i,N}+\theta z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{p-\mathrm{2}}\right){\left|z\right|}^{\mathrm{2}}\mathrm{d}\theta \nu (\mathrm{d}z)\mathrm{d}s\right]\le \\ E\left[{\left|{\widehat{X}}_{\mathrm{0}}^{i,N}\right|}^p\right]+CE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\tilde{A}}_{\underline{s}}+{\left|{\widehat{X}}_{\underline{s}}^{i,N}\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_{\underline{s}}^{\widehat{X},N},{\delta }_{\mathrm{0}}\right)\mathrm{d}s\right]+U_{\mathrm{1}}+U_{\mathrm{2}},\end{array}（13）

其中

\begin{array}{c}{U}_{\mathrm{1}}=pE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|{\widehat{X}}_s^{i,N}\right|}^{p-\mathrm{2}}\right.\left.〈{\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N},b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)〉\mathrm{d}s\right],\\ U_{\mathrm{2}}=p\left(p-\mathrm{1}\right)E\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\int }_Z{\left|\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\mathrm{2}}{\int }_{\mathrm{0}}^{\mathrm{1}}(\mathrm{1}-\theta )\times \right.\\ \left.\left\{{\left|{\widehat{X}}_s^{i,N}+\theta z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{p-\mathrm{2}}-{\left|{\widehat{X}}_{\underline{s}}^{i,N}+\theta z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{p-\mathrm{2}}\right\}{\left|z\right|}^{\mathrm{2}}\mathrm{d}\theta \nu (\mathrm{d}z)\mathrm{d}s\right].\end{array}

由假设（H3）以及Hölder不等式可以得到

\begin{array}{c}{U}_{\mathrm{1}}=pE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|{\widehat{X}}_s^{i,N}\right|}^{p-\mathrm{2}}\right.\left.〈{\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N},b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)〉\mathrm{d}s\right]\le \\ pE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|{\widehat{X}}_s^{i,N}\right|}^p\mathrm{d}s\right]+E\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|{\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N}\right|}^{\frac{p}{\mathrm{2}}}{\left|b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\frac{p}{\mathrm{2}}}\mathrm{d}s\right]\le \\ pE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|{\widehat{X}}_s^{i,N}\right|}^p\mathrm{d}s\right]+CE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^p{\left(s-\underline{s}\right)}^{\frac{p}{\mathrm{2}}}\mathrm{d}s\right]+\\ CE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\left(W_s^{\mathrm{0}}-W_{\underline{s}}^{\mathrm{0}}\right)\right|}^{\frac{p}{\mathrm{2}}}{\left|b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\frac{p}{\mathrm{2}}}\mathrm{d}s\right]+\\ CE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\left|\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\left(P_s^i-P_{\underline{s}}^i\right)\right|}^{\frac{p}{\mathrm{2}}}{\left|b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\frac{p}{\mathrm{2}}}\mathrm{d}s\right]\le \\ CE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}\left(\mathrm{1}+{\left|{\widehat{X}}_{\underline{s}}^{i,N}\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_{\underline{s}}^{\widehat{X},N},{\delta }_{\mathrm{0}}\right)\right)\mathrm{d}s\right]+\\ C{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}E\left[E\left[{\left|\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\left(W_s^{\mathrm{0}}-W_{\underline{s}}^{\mathrm{0}}\right)\right|}^{\frac{p}{\mathrm{2}}}{\left|b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\frac{p}{\mathrm{2}}}\left|\right.{\mathrm{ℱ}}_{\underline{s}}\right]\right]\mathrm{d}s+\\ {\int }_{\mathrm{0}}^{t\wedge {\rho }_K}E\left[E\left[{\left|\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\left(P_s^i-P_{\underline{s}}^i\right)\right|}^{\frac{p}{\mathrm{2}}}{\left|b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\frac{p}{\mathrm{2}}}\left|\right.{\mathrm{ℱ}}_{\underline{s}}\right]\right]\mathrm{d}s,\end{array}

其中

\begin{array}{c}E\left[{‖W_s^{\mathrm{0}}-W_{\underline{s}}^{\mathrm{0}}‖}^p\left|{\mathrm{ℱ}}_{\underline{s}}\right.\right]\le C{\left(s-\underline{s}\right)}^{\frac{p}{\mathrm{2}}},E\left[{‖P_s^i-P_{\underline{s}}^i‖}^p\left|{\mathrm{ℱ}}_{\underline{s}}\right.\right]\le C{\left({\int }_Z{\left|z\right|}^{\mathrm{2}}\nu \left(d\right.z)\right)}^{\frac{p}{\mathrm{2}}}{\left(s-\underline{s}\right)}^{\frac{p}{\mathrm{2}}},\\ E\left[E\left[{\left|\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\left(W_s^{\mathrm{0}}-W_{\underline{s}}^{\mathrm{0}}\right)\right|}^{\frac{p}{\mathrm{2}}}{\left|b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\frac{p}{\mathrm{2}}}\left|{\mathrm{ℱ}}_{\underline{s}}\right.\right]\right]\le E\left[\mathrm{1}+{\left|{\widehat{X}}_{\underline{s}}^{i,N}\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_{\underline{s}}^{\widehat{X},N},{\delta }_{\mathrm{0}}\right)\right],\\ E\left[E\left[{\left|\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\left(P_s^i-P_{\underline{s}}^i\right)\right|}^{\frac{p}{\mathrm{2}}}{\left|b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\frac{p}{\mathrm{2}}}\left|{\mathrm{ℱ}}_{\underline{s}}\right.\right]\right]\le E\left[\mathrm{1}+{\left|{\widehat{X}}_{\underline{s}}^{i,N}\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_{\underline{s}}^{\widehat{X},N},{\delta }_{\mathrm{0}}\right)\right],\\ E\left[{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_{\underline{s}}^{\widehat{X},N},{\delta }_{\mathrm{0}}\right)\right]=E\left[{\left|{\widehat{X}}_{\underline{s}}^{i,N}\right|}^p\right],\end{array}

故代入上式得到

U_{\mathrm{1}}\le C{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}E\left[\mathrm{1}+{\left|{\widehat{X}}_{\underline{s}}^{i,N}\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_{\underline{s}}^{\widehat{X},N},{\delta }_{\mathrm{0}}\right)\right]\mathrm{d}s,

使用等式{\left|y\right|}^{p-\mathrm{2}}={\left|a\right|}^{p-\mathrm{2}}+\left(p-\mathrm{2}\right){\int }_{\mathrm{0}}^{\mathrm{1}}{\left|a+\theta (y-a)\right|}^{p-\mathrm{4}}\left(y-a\right)\left(a+\theta (y-a)\right)\mathrm{d}\theta得到

\begin{array}{c}{U}_{\mathrm{2}}=p\left(p-\mathrm{1}\right)E\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\int }_Z{\left|\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\mathrm{2}}{\int }_{\mathrm{0}}^{\mathrm{1}}(\mathrm{1}-\theta )\times \right.\\ \left.\left\{{\left|{\widehat{X}}_s^{i,N}+\theta z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{p-\mathrm{2}}-{\left|{\widehat{X}}_{\underline{s}}^{i,N}+\theta z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{p-\mathrm{2}}\right\}{\left|z\right|}^{\mathrm{2}}\mathrm{d}\theta \nu (\mathrm{d}z)\mathrm{d}s\right]=\\ p\left(p-\mathrm{1}\right)\left(p-\mathrm{2}\right)E\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\int }_Z{‖\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)‖}^{\mathrm{2}}{\int }_{\mathrm{0}}^{\mathrm{1}}(\mathrm{1}-\theta )\times \right.\\ {\int }_{\mathrm{0}}^{\mathrm{1}}\left\{{\left|{\widehat{X}}_{\underline{s}}^{i,N}+\theta z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)+\overline{\theta }\left({\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N}\right)\right|}^{p-\mathrm{4}}\left({\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N}\right)\times \right.\\ \left.\left.\left({\widehat{X}}_{\underline{s}}^{i,N}+\theta z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)+\overline{\theta }\left({\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N}\right)\right)\right\}{\left|z\right|}^{\mathrm{2}}\mathrm{d}\overline{\theta }\mathrm{d}\theta \nu (\mathrm{d}z)\mathrm{d}s\right]\le \\ E\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}{\int }_Z{‖\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)‖}^{\mathrm{2}}{\left(\left|{\widehat{X}}_{\underline{s}}^{i,N}\right|+\left|z\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|+\left|{\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N}\right|\right)}^{p-\mathrm{3}}\times \right.\\ \left.\left|{\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N}\right|{\left|z\right|}^{\mathrm{2}}\nu (\mathrm{d}z)\mathrm{d}s\right]\le CE\left[{\int }_{\mathrm{0}}^{t\wedge {\rho }_K}\mathrm{1}+{\left|{\widehat{X}}_{\underline{s}}^{i,N}\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_{\underline{s}}^{\widehat{X},N},{\delta }_{\mathrm{0}}\right)\mathrm{d}s\right].\end{array}

将上式代入（13）式并使用Gronwall不等式得到

\underset{s\in [\mathrm{0},t]}{\mathrm{s}\mathrm{u}\mathrm{p}}E\left[{\left|{\widehat{X}}_{s\wedge {\rho }_K}^{i,N}\right|}^p\right]\le C+E\left[{\left|{\widehat{X}}_{\mathrm{0}}^{i,N}\right|}^p\right]+C{\int }_{\mathrm{0}}^t\underset{r\in [\mathrm{0},s]}{\mathrm{s}\mathrm{u}\mathrm{p}}E\left[{\left|{\widehat{X}}_{s\wedge {\rho }_K}^{i,N}\right|}^p\right]\mathrm{d}s\le C.

为了确定T是可以达到的，使用Markov不等式得到

P\left({\rho }_K\le T\right)\le {\displaystyle \sum _{i=\mathrm{1}}^N}P\left(\left|{\widehat{X}}_{{\rho }_K\wedge T}^{i,N}\right|>K\right)\le NP\left(\underset{i\in \{\mathrm{1},...,N\}}{\mathrm{m}\mathrm{a}\mathrm{x}}\left|{\widehat{X}}_{{\rho }_K\wedge T}^{i,N}\right|>K\right)\le \frac{N}{K^{\mathrm{2}}}E\left[\underset{i\in \{\mathrm{1},...,N\}}{\mathrm{m}\mathrm{a}\mathrm{x}}{\left|{\widehat{X}}_{{\rho }_K\wedge T}^{i,N}\right|}^{\mathrm{2}}\right]\le \frac{CN}{K^{\mathrm{2}}}.

因此当K\to \mathrm{\infty }时，有

当K\to \mathrm{\infty },{\rho }_K\to \mathrm{\infty }时，\underset{i\in \{\mathrm{1},...,N\}}{\mathrm{m}\mathrm{a}\mathrm{x}}\underset{t\in [\mathrm{0},T]}{\mathrm{s}\mathrm{u}\mathrm{p}}\left|{\widehat{X}}_t^{i,N}\right|\mathrm{\infty }几乎必然成立，则T可以达到.

定理2 对于任意的p\ge \mathrm{2}，假设方程（1）满足定理1的条件，则存在依赖于时间T和p的常数C>\mathrm{0}，有：

（14）

其中\epsilon来自于假设（H3）.

证明 定义S_t^i={\widehat{X}}_t^{i,N}-X_t^{i,N}，则有：

\mathrm{d}{S}_t^i=\left(b\left({\widehat{X}}_{\underline{t}}^{i,N},{\mu }_{\underline{t}}^{\widehat{X},N}\right)-b\left(X_t^{i,N},{\mu }_t^{X,N}\right)\right)\mathrm{d}t+\left(\sigma \left({\widehat{X}}_{\underline{t}}^{i,N},{\mu }_{\underline{t}}^{\widehat{X},N}\right)-\sigma \left(X_t^{i,N},{\mu }_t^{X,N}\right)\right)\mathrm{d}{W}_t^{\mathrm{0}}+\left(\kappa \left({\widehat{X}}_{\underline{t}}^{i,N},{\mu }_{\underline{t}}^{\widehat{X},N}\right)-\kappa \left(X_t^{i,N},{\mu }_t^{X,N}\right)\right)\mathrm{d}{P}_t^i.

对{\left|S_t^i\right|}^p用Itô公式可以得到

\begin{array}{l}{\left|S_t^i\right|}^p\le \frac{p}{\mathrm{2}}{\int }_{\mathrm{0}}^t{\left|S_s^i\right|}^{p-\mathrm{2}}\mathrm{2}〈S_s^i,b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-b\left(X_s^{i,N},{\mu }_s^{X,N}\right)〉\mathrm{d}s+\\ p{\int }_{\mathrm{0}}^t{\left|S_s^i\right|}^{p-\mathrm{2}}〈S_s^i,\left(\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\sigma \left(X_s^{i,N},{\mu }_s^{X,N}\right)\right)\mathrm{d}{W}_s^{\mathrm{0}}〉+\\ {\int }_{\mathrm{0}}^t{\int }_Z\left({\left|S_s^i+z\left(\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)\right)\right|}^p-{\left|S_s^i\right|}^p\right){\tilde{N}}^i(\mathrm{d}s,\mathrm{d}z)+\\ \frac{p\left(p-\mathrm{1}\right)}{\mathrm{2}}{{\int }_{\mathrm{0}}^t{\left|S_s^i\right|}^{p-\mathrm{2}}‖\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\sigma \left(X_s^{i,N},{\mu }_s^{X,N}\right)‖}^{\mathrm{2}}\mathrm{d}s+\\ {\int }_{\mathrm{0}}^t{\int }_Z\left({\left|S_s^i+z\left(\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)\right)\right|}^p-{\left|S_s^i\right|}^p-\right.\\ \left.pz{\left|S_s^i\right|}^{p-\mathrm{2}}〈S_s^i,\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)〉\right)\nu (\mathrm{d}z)\mathrm{d}s=R_t+M_t,\end{array}（15）

其中

\begin{array}{c}{R}_t=\frac{p}{\mathrm{2}}{\int }_{\mathrm{0}}^t{\left|S_s^i\right|}^{p-\mathrm{2}}\mathrm{2}〈S_s^i,b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-b\left(X_s^{i,N},{\mu }_s^{X,N}\right)〉\mathrm{d}s+\\ \frac{p\left(p-\mathrm{1}\right)}{\mathrm{2}}{{\int }_{\mathrm{0}}^t{\left|S_s^i\right|}^{p-\mathrm{2}}‖\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\sigma \left(X_s^{i,N},{\mu }_s^{X,N}\right)‖}^{\mathrm{2}}\mathrm{d}s+\\ {\int }_{\mathrm{0}}^t{\int }_Z\left({\left|S_s^i+z\left(\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)\right)\right|}^p-{\left|S_s^i\right|}^p-\right.\\ \left.pz{\left|S_s^i\right|}^{p-\mathrm{2}}〈S_s^i,\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)〉\right)\nu (\mathrm{d}z)\mathrm{d}s,\\ M_t=p{\int }_{\mathrm{0}}^t{\left|S_s^i\right|}^{p-\mathrm{2}}〈S_s^i,\left(\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\sigma \left(X_s^{i,N},{\mu }_s^{X,N}\right)\right)\mathrm{d}{W}_s^{\mathrm{0}}〉+\\ {\int }_{\mathrm{0}}^t{\int }_Z\left({\left|S_s^i+z\left(\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)\right)\right|}^p-{\left|S_s^i\right|}^p\right){\tilde{N}}^i(\mathrm{d}s,\mathrm{d}z).\end{array}

由等式{\left|y\right|}^p={\left|a\right|}^p+p{\left|a\right|}^{p-\mathrm{2}}〈a,y-a〉+p(p-\mathrm{1}){\int }_{\mathrm{0}}^{\mathrm{1}}(\mathrm{1}-\theta ){\left|y-a\right|}^{\mathrm{2}}{\left|a+\theta (y-a)\right|}^{p-\mathrm{2}}\mathrm{d}\theta得到

\begin{array}{c}{R}_t=p{\left|S_s^i\right|}^{p-\mathrm{2}}〈S_s^i,b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-b\left(X_s^{i,N},{\mu }_s^{X,N}\right)〉+\frac{p\left(p-\mathrm{1}\right)}{\mathrm{2}}{\left|S_s^i\right|}^{p-\mathrm{2}}{‖\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\sigma \left(X_s^{i,N},{\mu }_s^{X,N}\right)‖}^{\mathrm{2}}+\\ {\int }_Z\left\{{\left|S_s^i+z\left(\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)\right)\right|}^p-{\left|S_s^i\right|}^p-\right.\left.pz{\left|S_s^i\right|}^{p-\mathrm{2}}〈S_s^i,\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)〉\right\}\nu (\mathrm{d}z)=\end{array}

\begin{array}{c}p{\left|S_s^i\right|}^{p-\mathrm{2}}〈S_s^i,b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-b\left(X_s^{i,N},{\mu }_s^{X,N}\right)〉+\frac{p\left(p-\mathrm{1}\right)}{\mathrm{2}}{\left|S_s^i\right|}^{p-\mathrm{2}}{‖\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\sigma \left(X_s^{i,N},{\mu }_s^{X,N}\right)‖}^{\mathrm{2}}+\\ {\int }_Z{\int }_{\mathrm{0}}^{\mathrm{1}}\left\{\left(\mathrm{1}-\theta \right){\left|S_s^i+\theta z\left(\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)\right)\right|}^{p-\mathrm{2}}\times \right.\left.{‖\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)‖}^{\mathrm{2}}{\left|z\right|}^{\mathrm{2}}\mathrm{d}\theta \nu (\mathrm{d}z)\right\},\end{array}

由假设条件（H2）、Young不等式得到

\begin{array}{c}{R}_t\le p{\left|S_s^i\right|}^{p-\mathrm{2}}\left(〈S_s^i,b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-b\left({\widehat{X}}_s^{i,N},{\mu }_s^{\widehat{X},N}\right)〉+〈S_s^i,b\left({\widehat{X}}_s^{i,N},{\mu }_s^{\widehat{X},N}\right)-b\left(X_s^{i,N},{\mu }_s^{X,N}\right)〉\right)+\\ p\left(p-\mathrm{1}\right){\left|S_s^i\right|}^{p-\mathrm{2}}\left({‖\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\sigma \left({\widehat{X}}_s^{i,N},{\mu }_s^{\widehat{X},N}\right)‖}^{\mathrm{2}}+{‖\sigma \left({\widehat{X}}_s^{i,N},{\mu }_s^{\widehat{X},N}\right)-\sigma \left(X_s^{i,N},{\mu }_s^{X,N}\right)‖}^{\mathrm{2}}\right)+\\ {\mathrm{2}}^{p-\mathrm{2}}{\int }_Z{\int }_{\mathrm{0}}^{\mathrm{1}}\left\{\left(\mathrm{1}-\theta \right){\left|z\right|}^{\mathrm{2}}\right.\left({‖\theta z\left(\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left({\widehat{X}}_s^{i,N},{\mu }_s^{\widehat{X},N}\right)\right)‖}^{p-\mathrm{2}}+\right.\\ \left.{\left|S_s^i+\theta z\left(\kappa \left({\widehat{X}}_s^{i,N},{\mu }_s^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)\right)\right|}^{p-\mathrm{2}}\right)\times \\ \left.\left({‖\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)-\kappa \left({\widehat{X}}_s^{i,N},{\mu }_s^{\widehat{X},N}\right)‖}^{\mathrm{2}}+{‖\kappa \left({\widehat{X}}_s^{i,N},{\mu }_s^{\widehat{X},N}\right)-\kappa \left(X_s^{i,N},{\mu }_s^{X,N}\right)‖}^{\mathrm{2}}\right)\right\}\mathrm{d}\theta \nu (\mathrm{d}z)\le \\ C{Q}^p\left({\widehat{X}}_{\underline{s}}^{i,N},{\widehat{X}}_s^{i,N}\right){\left|{\widehat{X}}_{\underline{s}}^{i,N}-{\widehat{X}}_s^{i,N}\right|}^p+C{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_{\underline{s}}^{\widehat{X},N},{\mu }_s^{\widehat{X},N}\right)+\\ C\left(\left({Q^{\text{'}}}^p\left(X_s^{i,N},{\widehat{X}}_s^{i,N}\right)+\mathrm{1}\right){\left|S_s^i\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_s^{\widehat{X},N},{\mu }_s^{X,N}\right)\right),\end{array}

其中Q\left({\widehat{X}}_{\underline{s}}^{i,N},{\widehat{X}}_s^{i,N}\right)=L_{\mathrm{1}}\left(\mathrm{1}+{\left|{\widehat{X}}_{\underline{s}}^{i,N}\right|}^q+{\left|{\widehat{X}}_s^{i,N}\right|}^q\right),Q^{\text{'}}\left(X_s^{i,N},{\widehat{X}}_s^{i,N}\right)=L_{\mathrm{1}}\left(\mathrm{1}+{\left|X_s^{i,N}\right|}^q+{\left|{\widehat{X}}_s^{i,N}\right|}^q\right).

将上述估计式代入（15）式并两边同时取期望得

\begin{array}{c}E\left[{\left|S_t^i\right|}^p\right]\le C{\int }_{\mathrm{0}}^{t}E\left[\left({Q^{\text{'}}}^p\left(X_s^{i,N},{\widehat{X}}_s^{i,N}\right)+\mathrm{1}\right){\left|S_s^i\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_s^{\widehat{X},N},{\mu }_s^{X,N}\right)\right]\mathrm{d}s+\\ C{\int }_{\mathrm{0}}^{t}E\left[Q^p\left({\widehat{X}}_{\underline{s}}^{i,N},{\widehat{X}}_s^{i,N}\right){\left|{\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N}\right|}^p+{\mathrm{𝒲}}_{\mathrm{2}}^p\left({\mu }_s^{\widehat{X},N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right]\mathrm{d}s.\end{array}

由Hölder不等式知

\begin{array}{c}E\left[Q^p\left({\widehat{X}}_{\underline{s}}^{i,N},{\widehat{X}}_s^{i,N}\right){\left|{\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N}\right|}^p\right]\le {\left(E\left[Q^{\mathrm{2}p}\left({\widehat{X}}_{\underline{s}}^{i,N},{\widehat{X}}_s^{i,N}\right)\right]E\left[{\left|{\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N}\right|}^{\mathrm{2}p}\right]\right)}^{\frac{\mathrm{1}}{\mathrm{2}}},\\ E\left[{Q^{\text{'}}}^p\left(X_s^{i,N},{\widehat{X}}_s^{i,N}\right){\left|S_s^i\right|}^p\right]\le {\left(E\left[{Q^{\text{'}}}^{\mathrm{2}p}\left(X_s^{i,N},{\widehat{X}}_s^{i,N}\right)\right]E\left[{\left|S_s^i\right|}^{\mathrm{2}p}\right]\right)}^{\frac{\mathrm{1}}{\mathrm{2}}},\end{array}

由定理1知E\left[Q^{\mathrm{2}p}\left({\widehat{X}}_{\underline{s}}^{i,N},{\widehat{X}}_s^{i,N}\right)\right]\le C,E\left[{Q^{\text{'}}}^{\mathrm{2}p}\left(X_s^{i,N},{\widehat{X}}_s^{i,N}\right)\right]\le C，并通过（4）式得到

\begin{array}{c}E\left[{\left|{\widehat{X}}_s^{i,N}-{\widehat{X}}_{\underline{s}}^{i,N}\right|}^{\mathrm{2}p}\right]\le C{\left(E\left[{\left|b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\mathrm{4}p}\right]E\left[{\left(s-\underline{s}\right)}^{\mathrm{4}p}\right]\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}+C{\left(E\left[{‖\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)‖}^{\mathrm{4}p}\right]E\left[{‖W_s^i-W_{\underline{s}}^i‖}^{\mathrm{4}p}\right]\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}+\\ C{\left(E\left[{‖\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)‖}^{\mathrm{4}p}\right]E\left[{‖P_s^i-P_{\underline{s}}^i‖}^{\mathrm{4}p}\right]\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}.\end{array}

由定理1得到

E\left[{\left|b\left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)\right|}^{\mathrm{4}p}\right]\le C,E\left[{‖\sigma \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)‖}^{\mathrm{4}p}\right]\le C,E\left[{‖\kappa \left({\widehat{X}}_{\underline{s}}^{i,N},{\mu }_{\underline{s}}^{\widehat{X},N}\right)‖}^{\mathrm{4}p}\right]\le C,

注意到

\begin{array}{c}E\left[{\left(s-\underline{s}\right)}^{\mathrm{4}p}\right]\le {\left(\epsilon T\right)}^{\mathrm{4}p}\le C{\epsilon }^{\mathrm{2}p},\\ E\left[{‖W_s^{\mathrm{0}}-W_{\underline{s}}^{\mathrm{0}}‖}^{\mathrm{4}p}\right]\le E\left[E\left[{‖W_s^{\mathrm{0}}-W_{\underline{s}}^{\mathrm{0}}‖}^{\mathrm{4}p}\right]\left|{\mathrm{ℱ}}_s\right.\right]\le C{\left(s-\underline{s}\right)}^{\mathrm{2}p}\le C{\epsilon }^{\mathrm{2}p},\\ E\left[{‖P_s^i-P_{\underline{s}}^i‖}^{\mathrm{4}p}\right]\le E\left[E\left[{‖P_s^i-P_{\underline{s}}^i‖}^{\mathrm{4}p}\right]\left|\right.{\mathrm{ℱ}}_s\right]\le C{\left({\int }_Z{\left|z\right|}^{\mathrm{2}}\nu \left(d\right.z)\right)}^{\mathrm{2}p}{\left(s-\underline{s}\right)}^{\mathrm{2}p}\le C{\epsilon }^{\mathrm{2}p}.\end{array}

因此结合上面所有的估计得到

\underset{i\in \{\mathrm{1},...,N\}}{\mathrm{m}\mathrm{a}\mathrm{x}}\underset{\mathrm{0}\le t\le T}{\mathrm{s}\mathrm{u}\mathrm{p}}E\left[{\left|S_t^i\right|}^p\right]\le C{\int }_{\mathrm{0}}^t\underset{i\in \{\mathrm{1},...,N\}}{\mathrm{m}\mathrm{a}\mathrm{x}}\underset{\mathrm{0}\le u\le s}{\mathrm{s}\mathrm{u}\mathrm{p}}E\left[{\left|S_u^i\right|}^p\right]\mathrm{d}s+C{\epsilon }^{\frac{p}{\mathrm{2}}},

最后使用Gronwall不等式即可得到结果.

推论1 对于任意的\mathrm{2}\le p\le r，假设（H1），（H2），（H3）成立，则存在常数\mathrm{C}>\mathrm{0}，有：

\underset{i\in \{\mathrm{1},\dots ,N\}}{\mathrm{s}\mathrm{u}\mathrm{p}}\underset{\mathrm{0}\le t\le T}{\mathrm{s}\mathrm{u}\mathrm{p}}E\left[{\left|{X_t}^i-{\widehat{X}}_t^{i,N}\right|}^p\right]\le \left\{\begin{array}{l}C\left(N^{-\frac{\mathrm{1}}{\mathrm{2}}}+N^{-\frac{r-p}{r}}+{\epsilon }^{\frac{p}{\mathrm{2}}}\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}p>\frac{d}{\mathrm{2}}\mathrm{且}r\ne \mathrm{2}p,\\ C\left(N^{-\frac{\mathrm{1}}{\mathrm{2}}}\mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{1}+N)+N^{-\frac{r-p}{r}}+{\epsilon }^{\frac{p}{\mathrm{2}}}\right),p=\frac{d}{\mathrm{2}}\mathrm{且}r\ne \mathrm{2}p,\\ C\left(N^{-\frac{\mathrm{1}}{\mathrm{2}}}+N^{-\frac{r-p}{r}}+{\epsilon }^{\frac{p}{\mathrm{2}}}\right),\hspace{1em}\hspace{1em}p\in [\mathrm{2},\frac{d}{\mathrm{2}})\mathrm{且}\mathrm{r}\ne \frac{d}{d-p}.\end{array}\right.

结合文献［13-14］所推出的混沌传播结论及本文定理2即可得到上述推论1.

3 实例

考虑如下的一维McKean-Vlasov随机微分方程

\mathrm{d}{X}_t=\left(-X_t^{\mathrm{3}}+E{X}_t\right)\mathrm{d}t+X_t\mathrm{d}{W}_t^{\mathrm{0}}+X_t\mathrm{d}{P}_t,（16）

其初值X_{\mathrm{0}}=\mathrm{1}.

首先对于任意p\ge \mathrm{2}，存在\overline{L}\mathrm{0}使得

\begin{array}{c}\mathrm{2}{\left|x-y\right|}^{p-\mathrm{2}}〈x-y,\left(-x^{\mathrm{3}}+Ex\right)-\left(-y^{\mathrm{3}}+Ey\right)〉+\\ \mathrm{3}\left(p-\mathrm{1}\right){\left|x-y\right|}^p+\mathrm{2}\times {\mathrm{3}}^{p-\mathrm{1}}\left(p-\mathrm{1}\right){\left|x-y\right|}^{\mathrm{2}}{\int }_Z{\int }_{\mathrm{0}}^{\mathrm{1}}\left(\mathrm{1}-\theta \right){\left|x-y+\theta z\left(x-y\right)\right|}^{p-\mathrm{2}}{\left|z\right|}^{\mathrm{2}}\mathrm{d}\theta \nu (\mathrm{d}z)\le \\ \mathrm{2}{\left|x-y\right|}^{p-\mathrm{1}}\left(-\left(x^{\mathrm{3}}-y^{\mathrm{3}}\right)+E\left(x-y\right)\right)+\\ \mathrm{3}\left(p-\mathrm{1}\right){\left|x-y\right|}^p+\mathrm{2}\times {\mathrm{3}}^{p-\mathrm{1}}\left(p-\mathrm{1}\right){\left|x-y\right|}^{\mathrm{2}}{\int }_Z{\left(\left|x-y\right|+\left|z\left(x-y\right)\right|\right)}^{p-\mathrm{2}}{\left|z\right|}^{\mathrm{2}}\nu \left(d\right.z)\le \\ \overline{L}\left({\left|x-y\right|}^p+E^p\left(x-y\right)\right),\end{array}

故易知假设（H2）中（6）式成立，

\begin{array}{c}\left|\left(-x^{\mathrm{3}}+Ex\right)-\left(-y^{\mathrm{3}}+Ey\right)\right|\le \\ \mathrm{2}\left(\mathrm{1}+{\left|x\right|}^{\mathrm{2}}+{\left|y\right|}^{\mathrm{2}}\right)\left|x-y\right|+E\left(x-y\right),\end{array}

即可验证（7）式成立.选取h\left(x,\mu \right)=\frac{\mathrm{1}}{\mathrm{1}+{\left|x\right|}^{\mathrm{6}}+Ex}，那么在假设（H3）选取\varpi =\mathrm{7},q=\mathrm{2}，对于适当的L_{\mathrm{2}}即可得到假设（H3）也成立.因此，根据定理1和定理2可以知道自适应Euler-Maruyama算法的数值解的有界性及其收敛速率.

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基金资助

国家自然科学基金资助项目(62373383)