数学物理学报, 2022, 42(2): 520-556 doi:

论文

非Lipschitz条件下超前带跳倒向耦合随机微分方程的Wong-Zakai逼近

徐杰,1, 孙艳华2

1 河南师范大学数学与信息科学学院 河南 新乡 453002

2 河南科技学院数学科学学院 河南 新乡 453001

Wong-Zakai Approximations of Anticipated Backward Doubly Stochastic Differential Equations with Jumps in Non-Lipschitz Conditions

Xu Jie,1, Sun Yanhua2

1 College of Mathematics and Information Science, Henan Normal University, Henan Xinxiang 453002

2 School of Mathematical Sciences, Henan Institute of Science and Technology, Henan Xinxiang 453001

通讯作者: 徐杰, E-mail: xujiescu@163.com

收稿日期: 2020-08-20  

基金资助: 河南省高等学校重点科研项目计划.  21A110011

Received: 2020-08-20  

Fund supported: the Key Scientific Research Project Plans of Henan Province.  21A110011

Abstract

In this paper we will prove the Wong-Zakai approximation of anticipated backward doubly stochastic differential equations with Poisson jumps under the non-Lipschitz conditions.

Keywords: Anticipated backward doubly stochastic differential equations ; Possion jumps ; Wong-Zakai approximations ; Non-Lipschitz.

PDF (419KB) 元数据 多维度评价 相关文章 导出 EndNote| Ris| Bibtex  收藏本文

本文引用格式

徐杰, 孙艳华. 非Lipschitz条件下超前带跳倒向耦合随机微分方程的Wong-Zakai逼近. 数学物理学报[J], 2022, 42(2): 520-556 doi:

Xu Jie, Sun Yanhua. Wong-Zakai Approximations of Anticipated Backward Doubly Stochastic Differential Equations with Jumps in Non-Lipschitz Conditions. Acta Mathematica Scientia[J], 2022, 42(2): 520-556 doi:

1 引言

假设$ \{W_t, 0\leq t\leq T\} $$ \{B_t, 0\leq t\leq T\} $是分别取值在$ {\mathbb R}^l $$ {\mathbb R}^m $上两个独立的标准布朗运动, $ \{N_t, 0\leq t\leq T\} $是独立于$ \{W_t, 0\leq t\leq T\} $$ \{B_t, 0\leq t\leq T\} $的一个泊松过程, 且它们定义在同一个概率空间$ (\Omega, {\cal F}, P) $.$ {\cal N} $是一个$ P $零测集, 对任意$ t\in[0, T] $, 定义$ {\cal F}_t={\cal F}_t^{W, N}\vee{\cal F}_{t, T}^B. $对任给过程$ \{\alpha_t\} $, 我们定义$ {\cal F}_{s, t}^\alpha=\sigma\{\alpha_r-\alpha_s; s\leq r\leq t\}\vee{\cal N} $, $ {\cal F}_t^\alpha={\cal F}_{0, t}^\alpha. $注意$ {\cal F}_t, t\geq 0 $不是一个流.

$ \xi\in L^2(\Omega, {\mathbb R}^d) $且关于$ {\cal F}_T $可测, 考虑超前带跳倒向耦合随机微分方程(ABDSDEs):

$ \begin{eqnarray} \left\{ \begin{array}{rl} Y_t=&{ } \xi+\int_t^T f(s, Y_s, Z_s, U_s, Y_{s+\mu(s)}, Z_{s+\nu(s)}, U_{s+\delta(s)}){{\mathord{{{\rm{d}}}}}} s+\int_t^T g(Y_s){{\mathord{{{\rm{d}}}}}} B_s\\ &{ } +\frac{1}{2} \int_t^T gg'(Y_s){{\mathord{{{\rm{d}}}}}} s -\int_t^T Z_s{{\mathord{{{\rm{d}}}}}} W_s-\int_t^T U_s{{\mathord{{{\rm{d}}}}}}\hat{N}_s, \\ Y_t^n=&{ } \xi+\int_t^T f(s, Y_s^n, Z_s^n, U_s^n, Y_{s+\mu(s)}^n, Z_{s+\nu(s)}^n, U_{s+\delta(s)}^n){{\mathord{{{\rm{d}}}}}} s+\int_t^T g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &{ } -\int_t^T Z_s^n{{\mathord{{{\rm{d}}}}}} W_s-\int_t^T U_s^n{{\mathord{{{\rm{d}}}}}} \hat{N}_s, \\ Y_t=&\xi(t), Y_t^n=\xi(t), t\in[T, T+K], \\ Z_t=&\eta(t), Z_t^n=\eta(t), t\in[T, T+K], \\ U_t=&\gamma(t), U_t^n=\gamma(t), t\in[T, T+K], \end{array} \right. \end{eqnarray} $

其中$ {{\mathord{{{\rm{d}}}}}} B_s $为倒向Itô积分, $ {{\mathord{{{\rm{d}}}}}} W_s $$ {{\mathord{{{\rm{d}}}}}}\hat{N}_s $为正向Itô积分, $ \hat{N}(t)=N(t)-\lambda t $是一个均值为零的泊松过程. $ {{\mathord{{{\rm{d}}}}}} B_s^n $表示$ \dot{B}_s^n\overrightarrow{{{\mathord{{{\rm{d}}}}}} s} $, 其中$ \dot{B}_s^n $表示$ B^n $的分段导数, $ \overrightarrow{{{\mathord{{{\rm{d}}}}}} s} $表示关于Lebesgue测度的倒向积分, 即

假设:

(B1) $ f: {\mathbb R}_+\times {\mathbb R}^d\times {\mathbb R}^{d\times l}\times{\mathbb R}^d \times {\mathbb R}^d\times {\mathbb R}^{d\times l}\times{\mathbb R}^d\rightarrow {\mathbb R}^d $是一个有界可测函数.

(B2) 对$ \forall $$ (s, Y_s^1, Z_s^1, U_s^1, Y_{s+\mu(s)}^1, Z_{s+\nu(s)}^1, U_{s+\delta(s)}^1), (s, Y_s^2, Z_s^2, U_s^2, Y_{s+\mu(s)}^2, Z_{s+\nu(s)}^2, U_{s+\delta(s)}^2) $$ \in{\mathbb R}_+\times {\mathbb R}^d\times {\mathbb R}^{d\times l}\times{\mathbb R}^d \times {\mathbb R}^d\times {\mathbb R}^{d\times l}\times{\mathbb R}^d $$ T>0 $, $ \exists $$ C>0 $满足

其中$ \rho $为定义在从$ {\mathbb R}_+ $$ {\mathbb R}_+ $上的一个非降连续凹函数, 并满足$ \rho(0)=0 $

$ \mu(\cdot):[0, T]\rightarrow {\mathbb R}^+\backslash\{0\} $, $ \nu(\cdot):[0, T]\rightarrow {\mathbb R}^+\backslash\{0\} $$ \delta(\cdot):[0, T]\rightarrow {\mathbb R}^+\backslash\{0\} $都是连续函数, 且使得:

(i) $ \forall $$ t\in[0, T] $, $ \exists $$ K\geq 0 $满足

(ii) $ \forall $$ t\in[0, T] $和非负可积函数$ {\cal J}(\cdot) $, $ \exists $$ L\geq 0 $满足

$ g(\cdot)=(g_{i, j}(\cdot))\in C_b^2({\mathbb R}^d, {\mathbb R}^{d\times m}) $, 并定义$ gg': {\mathbb R}^d\rightarrow {\mathbb R}^d $

$ \begin{eqnarray} (gg'(y))_i=\sum\limits_{j=1}^m\sum\limits_{k=1}^d\frac{\partial g_{i, j}(y)}{\partial y_k}g_{k, j}(y), \quad i=1, \ldots, d. \end{eqnarray} $

(B3) 假设$ yg(y), y^3g(y)\in C_b^1({\mathbb R}^d) $.

$ n\geq 1 $, 定义$ B $的线性插值$ B^n $

$ \begin{eqnarray} B_t^n=B_{\frac{k+2}{2^n}}+2^n\big(t-\frac{k+1}{2^n}\big)\big(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\big), \quad\forall t\in\bigg[\frac{k}{2^n}, \frac{k+1}{2^n}\bigg]. \end{eqnarray} $

$ \forall $$ p\in {\mathbb N} $, $ L_{{\cal F}}^2([0, T];{\mathbb R}^p) $表示$ p $维联合可测随机过程$ \{\phi_t;t\in[0, T]\} $的集合, 且满足

(a1) $ {\mathbb E}[\int_0^T|\phi_t|^2{{\mathord{{{\rm{d}}}}}} t]<\infty $,

(a2) 对$ t\in[0, T] $, $ \phi_t $关于$ {\cal F}_t $几乎处处可测.

类似地, 我们用$ S_{{\cal F}}^2([0, T];{\mathbb R}^p) $表示$ p $维连续随机过程的集合, 且满足

(b1) $ {\mathbb E}[\sup\limits_{0\leq t\leq T}|\phi_t|^2]<\infty $,

(b2) $ \forall $$ t\in[0, T] $, $ \phi_t $关于$ {\cal F}_t $可测.

在上述假设条件下, 结合文献[15], 方程(1.1)解的存在唯一性可以直接得到.假设$ (Y, Z, U) $$ (Y^n, Z^n, U^n) $是方程(1.1)的解, 由文献[14, 15]可得$ (Y, Z, U), (Y^n, Z^n, U^n)\in S_{{\cal F}}^2([0, T];{\mathbb R}^d)\times L_{{\cal F}}^2([0, T];{\mathbb R}^{d\times l})\times L_{{\cal F}}^2([0, T];{\mathbb R}^d) $.

线性倒向随机微分方程由Bismut在1973年第一次提出, 非线性倒向随机微分方程由Pardoux和Peng在1990年研究控制问题时创立[6].近年来, 倒向随机微分方程被广泛关注, Pardoux和Peng在1994年研究了倒向耦合随机微分方程[7]. Ma和Zhang在2002年首次研究倒向耦合随机微分方程的欧拉逼近[5]. Hu, Anis和Zhang在2015年给出倒向耦合随机微分方程(BDSDE)的Wong-Zakai逼近[3].在2009年, Peng和Yang研究超前倒向耦合随机微分方程, 在这个方程中, 生成子不仅包括现在时刻的状态还包括未来时刻的状态.之后, 超前倒向耦合随机微分方程被很多专家学者从各个方面进行研究[2, 4, 9-11, 15, 16].现在超前带跳倒向耦合随机微分方程已经被广泛应用于金融和控制问题.一个自然的问题是能否将Hu, Anis和Zhang的结果推广到非Lipschitz系数超前带跳倒向耦合随机微分方程?这是本文的研究动机.因此我们的目的是研究非Lipschitz条件下超前带跳倒向耦合随机微分方程的Wong-Zakai逼近.简单地说, 证明在$ L^2 $意义下$ (Y^n, Z^n, U^n) $收敛于$ (Y, Z, U) $.

与文献[3]相比, 本文的主要创新点如下:一是将文献[3]的结果推广到带跳的情形, 即方程(1.1)中$ U $不等于0;二是将文献[3]中的生成子$ f $关于$ y $由Lipschitz条件推广到非Lipschitz条件, 即$ f $满足$ \rm (H2) $; 三是将文献[3]中倒向耦合随机微分方程推广到超前倒向耦合随机微分方程, 即方程(1.1)含有超前项.

本文的结构如下:在第二部分, 我们通过引理2.1–2.3给出解的一些先验估计, 接着通过引理2.1–2.3和引理2.4–2.5给出定理2.1的证明.由于引理2.4和2.5的证明非常复杂, 因此将在定理2.1之后分别进行证明.

在整篇论文中, $ C $表示与$ n $无关的正常数.

2 主要结果

为了证明简单, 我们假设$ d=m=l=1 $, 即均是一维随机变量, 多维证明过程类似.

引理2.1  假设(B1)成立, 则存在一个常数$ C $满足

$ \begin{eqnarray} \sup\limits_n\sup\limits_{0\leq t\leq T}\bigg\{{\mathbb E}[(Y_t^n)^2]+{\mathbb E}\int_t^T(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s +\lambda {\mathbb E}\int_t^T(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg\}\leq C, \end{eqnarray} $

其中$ C $$ n $无关.

  利用Itô公式, 可得

$ \begin{eqnarray} &&(Y_t^n)^{2}+\int_t^T(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s+\lambda\int_t^T(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\\ &=&\xi^2+2\int_t^TY_s^nf(s, Y_s^n, Z_s^n, U_s^n, Y_{s+\mu(s)}^n, Z_{s+\nu(s)}^n, U_{s+\delta(s)}^n){{\mathord{{{\rm{d}}}}}} s+ 2\int_t^TY_s^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^TY_s^nZ_s^n{{\mathord{{{\rm{d}}}}}} W_s-2\int_t^TY_s^nZ_s^n{{\mathord{{{\rm{d}}}}}} \hat{N}_s. \end{eqnarray} $

$ \forall $$ s\in[\frac{k}{2^n}, \frac{k+1}{2^n}] $, 令$ s^+=\frac{k+2}{2^n} $, $ s^-=\frac{k-1}{2^n} $.$ f $的有界性和Young不等式可知, $ \exists $$ C>0 $使得

$ \begin{equation} 2\int_t^TY_s^nf(s, Y_s^n, Z_s^n, U_s^n, Y_{s+\mu(s)}^n, Z_{s+\nu(s)}^n, U_{s+\delta(s)}^n){{\mathord{{{\rm{d}}}}}} s\leq C\int_t^T|Y_s^n|{{\mathord{{{\rm{d}}}}}} s \leq C\int_t^T|Y_s^n|^2{{\mathord{{{\rm{d}}}}}} s+C. \end{equation} $

注意到$ 2\int_t^TY_s^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n $可以分解为

$ \begin{eqnarray} 2\int_t^TY_s^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n&=&2\int_t^TY_{s^+}^ng(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n+ 2\int_t^T(Y_s^n-Y_{s^+}^n)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^TY_{s^+}^n(g(Y_s^n)-g(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&A_1+A_2+A_3. \end{eqnarray} $

由于$ A_1 $是随机积分, 因此$ {\mathbb E}[A_1]=0 $.由(1.1)式可得

$ \begin{eqnarray} A_2&=&2\int_t^T\bigg(\int_s^{s^+}f(u, Y_u^n, Z_u^n, U_u^n, Y_{u+\mu(u)}^n, Z_{u+\nu(u)}^n, U_{u+\delta(u)}^n){{\mathord{{{\rm{d}}}}}} u\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^T\bigg(\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n -2\int_t^T\bigg(\int_s^{s^+}Z_u^n{{\mathord{{{\rm{d}}}}}} W_u\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^T\bigg(\int_s^{s^+}U_u^n{{\mathord{{{\rm{d}}}}}} \hat{N}_u\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&A_{2, 1}+A_{2, 2}+A_{2, 3}+A_{2, 4}. \end{eqnarray} $

$ f $的有界性, 得

$ \begin{equation} {\mathbb E}[A_{2, 1}]\leq C\int_t^T\bigg(\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u\bigg){\mathbb E}[|\dot{B}_s^n|]{{\mathord{{{\rm{d}}}}}} s \leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}, \end{equation} $

其中

$ g $的有界性和Cauchy-Schwarz不等式, 得

$ \begin{eqnarray} {\mathbb E}[A_{2, 2}]&\leq& C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}|\dot{B}_u^n|{{\mathord{{{\rm{d}}}}}} u\bigg)|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\leq C\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u{\mathbb E}[|\dot{B}_u^n||\dot{B}_s^n|]\\ &\leq &C\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u({\mathbb E}[|\dot{B}_u^n|^2])^{\frac{1}{2}}({\mathbb E}[|\dot{B}_s^n|^2])^{\frac{1}{2}}\\ &\leq &C\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u(2^n)^{\frac{1}{2}}(2^n)^{\frac{1}{2}}\leq C, \end{eqnarray} $

其中$ C $是与$ n $无关的常数.

$ A_{2, 3} $, 由全期望公式和鞅的性质, 可得

$ \begin{eqnarray} {\mathbb E}[A_{2, 3}]&=&-2\int_t^T{\mathbb E}\bigg[\bigg(\int_s^{s^+}Z_u^n{{\mathord{{{\rm{d}}}}}} W_u\bigg)g(Y_s^n)\dot{B}_s^n\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &=&-2\int_t^T{\mathbb E}\bigg[g(Y_s^n)\dot{B}_s^n{\mathbb E}\bigg[\bigg(\int_s^{s^+}Z_u^n{{\mathord{{{\rm{d}}}}}} W_u\bigg)\Big| {\cal F}_s\bigg]\bigg]{{\mathord{{{\rm{d}}}}}} s=0. \end{eqnarray} $

类似地

$ \begin{eqnarray} {\mathbb E}[A_{2, 4}]&=&-2\int_t^T{\mathbb E}\bigg[\bigg(\int_s^{s^+}U_u^n{{\mathord{{{\rm{d}}}}}} \hat{N}_u\bigg)g(Y_s^n)\dot{B}_s^n\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &=&-2\int_t^T{\mathbb E}\bigg[g(Y_s^n)\dot{B}_s^n{\mathbb E}\bigg[\bigg(\int_s^{s^+}U_u^n{{\mathord{{{\rm{d}}}}}}\hat{N}_u\bigg)\Big| {\cal F}_s\bigg]\bigg]{{\mathord{{{\rm{d}}}}}} s=0. \end{eqnarray} $

由(2.6)–(2.9)式得

$ \begin{equation} \sup\limits_n{\mathbb E}[A_2]\leq C, \end{equation} $

其中$ C $$ n $无关.

接下来我们处理$ A_3 $, 由$ g' $的有界性和(1.1)式可得

$ \begin{eqnarray} A_3&=&2\int_t^TY_{s^+}^ng'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))(Y_s^n-Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &\leq&2\int_t^TY_{s^+}^n|g'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))|(Y_s^n-Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &\leq&C\int_t^TY_{s^+}^n(Y_s^n-Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &=&C\int_t^TY_{s^+}^n\bigg(\int_s^{s^+}f(u, Y_u^n, Z_u^n, U_u^n, Y_{u+\mu(u)}^n, Z_{u+\nu(u)}^n, U_{u+\delta(u)}^n)\bigg){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+C\int_t^TY_{s^+}^n\bigg(\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg){{\mathord{{{\rm{d}}}}}} B_s^n -C\int_t^TY_{s^+}^n\bigg(\int_s^{s^+}Z_u^n{{\mathord{{{\rm{d}}}}}} W_u\bigg){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-C\int_t^TY_{s^+}^n\bigg(\int_s^{s^+}U_u^n{{\mathord{{{\rm{d}}}}}} \hat{N}_u\bigg){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&A_{3, 1}+A_{3, 2}+A_{3, 3}+A_{3, 4}, \end{eqnarray} $

其中$ \eta\in[0, 1] $.

$ A_{3, 1} $, 由$ f $的有界性和Young不等式, 得

$ \begin{eqnarray} {\mathbb E}[A_{3, 1}]&\leq& C\int_t^T{\mathbb E}\bigg[|Y_{s^+}^n|\frac{1}{2^n}|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s+C\int_t^T{\mathbb E}\bigg[\bigg(\frac{1}{2^n}\bigg)^2|\dot{B}_s^n|^2\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq &C\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s+C. \end{eqnarray} $

类似地, 由$ g $的有界性, 得

$ \begin{eqnarray} {\mathbb E}[A_{3, 2}]&\leq &C\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u{\mathbb E}[|Y_{s^+}^n||\dot{B}_u^n||\dot{B}_s^n|]\\ &\leq& C\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u({\mathbb E}[(Y_{s^+}^n)^2])^{\frac{1}{2}}({\mathbb E}[|\dot{B}_u^n|^2|\dot{B}_s^n|^2]) ^{\frac{1}{2}}\\ &\leq& C\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u({\mathbb E}[(Y_{s^+}^n)^2])^{\frac{1}{2}}2^n\\ &\leq& C\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s+C. \end{eqnarray} $

$ Y_{s^+}^n $$ \dot{B}_s^n $的独立性, Cauchy-Schwarz不等式, Young不等式和Itô等距, 可得

$ \begin{eqnarray} {\mathbb E}[A_{3, 3}]&\leq &C{\mathbb E}\bigg[\int_t^T|Y_{s^+}^n||\dot{B}_s^n|\Big|\int_s^{s^+}Z_u^n{{\mathord{{{\rm{d}}}}}} W_u\Big|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C\int_t^T({\mathbb E}[(Y_{s^+}^n)^2|\dot{B}_s^n|^2])^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}Z_u^n{{\mathord{{{\rm{d}}}}}} W_u\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &=&C\int_t^T({\mathbb E}[(Y_{s^+}^n)^2])^{\frac{1}{2}}({\mathbb E}[|\dot{B}_s^n|^2])^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\int_s^{s^+}(Z_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &=&C\int_t^T({\mathbb E}[(Y_{s^+}^n)^2])^{\frac{1}{2}}(2^n)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\int_s^{s^+}(Z_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq&\frac{1}{4}\int_t^T2^n{\mathbb E}\bigg[\int_s^{s^+}(Z_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]{{\mathord{{{\rm{d}}}}}} s+C_2\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s\\ &=&\frac{1}{4}{\mathbb E}\bigg[\int_t^T(Z_u^n)^2{{\mathord{{{\rm{d}}}}}} u2^n\bigg(\int_{u^-}^u{{\mathord{{{\rm{d}}}}}} s\bigg)\bigg]+C_2\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s\\ &\leq&\frac{1}{4}{\mathbb E}\bigg[\int_t^T(Z_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]+C_2\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s, \end{eqnarray} $

$ \begin{eqnarray} {\mathbb E}[A_{3, 4}]&\leq& C{\mathbb E}\bigg[\int_t^T|Y_{s^+}^n||\dot{B}_s^n|\Big|\int_s^{s^+}U_u^n{{\mathord{{{\rm{d}}}}}}\hat{N}_u\Big|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C\int_t^T({\mathbb E}[(Y_{s^+}^n)^2|\dot{B}_s^n|^2])^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}U_u^n{{\mathord{{{\rm{d}}}}}}\hat{N}_u\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &=&C\lambda\int_t^T({\mathbb E}[(Y_{s^+}^n)^2])^{\frac{1}{2}}({\mathbb E}[|\dot{B}_s^n|^2])^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\int_s^{s^+}(U_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &=&C\lambda\int_t^T({\mathbb E}[(Y_{s^+}^n)^2])^{\frac{1}{2}}(2^n)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\int_s^{s^+}(U_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq&\frac{1}{4}\lambda\int_t^T2^n{\mathbb E}\bigg[\int_s^{s^+}(U_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]{{\mathord{{{\rm{d}}}}}} s+C_3\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s\\ &=&\frac{1}{4}\lambda{\mathbb E}\bigg[\int_t^T(U_u^n)^2{{\mathord{{{\rm{d}}}}}} u2^n\bigg(\int_{u^-}^u{{\mathord{{{\rm{d}}}}}} s\bigg)\bigg]+C_3\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s\\ &\leq&\frac{1}{4}\lambda{\mathbb E}\bigg[\int_t^T(U_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]+C_3\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s. \end{eqnarray} $

由(2.12)–(2.15)式, 可得

$ \begin{eqnarray} {\mathbb E}[A_3]\leq\frac{1}{4}{\mathbb E}\bigg[\int_t^T(Z_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]+\frac{1}{4}\lambda{\mathbb E}\bigg[\int_t^T(U_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]+ C\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s+C. \end{eqnarray} $

通过(2.2), (2.3), (2.10)和(2.16)式, 得

$ \begin{eqnarray} &&{\mathbb E}[(Y_t^n)^2]+\frac{3}{4}{\mathbb E}\bigg[\int_t^T(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg]+\frac{3}{4}\lambda {\mathbb E}\bigg[\int_t^T(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq &{\mathbb E}[(\xi)^2]+C\int_t^T{\mathbb E}[(Y_{s^+}^n)^2]{{\mathord{{{\rm{d}}}}}} s+C\int_t^T{\mathbb E}[(Y_s^n)^2]{{\mathord{{{\rm{d}}}}}} s+C. \end{eqnarray} $

由Gronwall不等式可知结论成立.引理2.1得证.

引理2.2  存在一个常数$ C $满足

$ \begin{equation} {\mathbb E}\bigg[\int_0^T|Y_{s^+}-Y_s|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\leq C\frac{1}{2^n}, \end{equation} $

$ \begin{equation} {\mathbb E}\bigg[\int_0^T|Y_{s^+}^n-Y_s^n|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\leq C\frac{1}{2^n}. \end{equation} $

  我们证明(2.18)式, (2.19)式的证明类似.

$ \begin{eqnarray} {\mathbb E}\bigg[\int_0^T|Y_{s^+}-Y_s|^2{{\mathord{{{\rm{d}}}}}} s\bigg] &\leq &C{\mathbb E}\bigg[\int_0^T\bigg|\int_s^{s^+}f(u, Y_u, Z_u, U_u, Y_{u+\mu(u)}, Z_{u+\nu(u)}, U_{u+\delta(u)}){{\mathord{{{\rm{d}}}}}} u\bigg|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&+C{\mathbb E}\bigg[\int_0^T\bigg|\int_s^{s^+}g(Y_u){{\mathord{{{\rm{d}}}}}} B_u\bigg|^2{{\mathord{{{\rm{d}}}}}} s\bigg] +C{\mathbb E}\bigg[\int_0^T\bigg|\int_s^{s^+}gg'(Y_u){{\mathord{{{\rm{d}}}}}} u\bigg|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&+C{\mathbb E}\bigg[\int_0^T\bigg|\int_s^{s^+}Z_u{{\mathord{{{\rm{d}}}}}} W_u\bigg|^2{{\mathord{{{\rm{d}}}}}} s\bigg] +C{\mathbb E}\bigg[\int_0^T\bigg|\int_s^{s^+}U_u{{\mathord{{{\rm{d}}}}}} \hat{N}_u\bigg|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\frac{1}{2^n} +C{\mathbb E}\bigg[\int_0^T\int_s^{s^+}(Z_u)^2{{\mathord{{{\rm{d}}}}}} u{{\mathord{{{\rm{d}}}}}} s\bigg] +C\lambda {\mathbb E}\bigg[\int_0^T\int_s^{s^+}(U_u)^2{{\mathord{{{\rm{d}}}}}} u{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\frac{1}{2^n}+C\bigg({\mathbb E}\bigg[\int_0^T(Z_u)^2{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)\frac{1}{2^n}+ C\bigg({\mathbb E}\bigg[\int_0^T(U_u)^2{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)\frac{1}{2^n}\\ &\leq&C\frac{1}{2^n}. \end{eqnarray} $

由Cauchy-Schwarz不等式, (2.18)和(2.19)式可得

$ \begin{equation} {\mathbb E}\bigg[\int_0^T|Y_{s^+}-Y_s|{{\mathord{{{\rm{d}}}}}} s\bigg]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}, \end{equation} $

$ \begin{equation} {\mathbb E}\bigg[\int_0^T|Y_{s^+}^n-Y_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{equation} $

引理2.2得证.

引理2.3  令$ \xi\in L^4(\Omega;{\mathbb R}^d) $, 在$ \rm (B1) $, $ \rm (B2) $$ \rm (B3) $假设下, 则存在一个常数$ C $满足

$ \begin{eqnarray} \sup\limits_n\bigg\{{\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}|Y_t^n|^4\bigg]+{\mathbb E}\bigg[\bigg(\int_t^T(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]+ \lambda {\mathbb E}\bigg[\bigg(\int_t^T(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg\}\leq C, \end{eqnarray} $

其中$ C $$ n $无关.

  由Itô公式, 得

$ \begin{eqnarray} |Y_t^n|^4&=&(\xi)^4+4\int_t^T(Y_s^n)^3f(s, Y_s^n, Z_s^n, U_s^n, Y_{s+\mu(s)}^n, Z_{s+\nu(s)}^n, U_{s+\delta(u)}^n){{\mathord{{{\rm{d}}}}}} s \\ &&+4\int_t^T(Y_s^n)^3g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n-4\int_t^T(Y_s^n)^3Z_s^n{{\mathord{{{\rm{d}}}}}} W_s-4\int_t^T(Y_s^n)^3U_s^n{{\mathord{{{\rm{d}}}}}} \hat{N}_s\\ &&-6\int_t^T(Y_s^n)^2(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s-6\lambda\int_t^T(Y_s^n)^2(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\\ :&=&(\xi)^4+B_1^n+B_2^n+B_3^n+B_4^n+B_5^n+B_6^n. \end{eqnarray} $

因为$ B_5^n $$ B_6^n $都是负的, 所以

$ \begin{eqnarray} |Y_t^n|^4\leq(\xi)^4+B_1^n+B_2^n+B_3^n+B_4^n. \end{eqnarray} $

$ f $的有界性和Young不等式, 得

$ \begin{eqnarray} B_1^n&=&4\int_t^T(Y_s^n)^3f(s, Y_s^n, Z_s^n, U_s^n, Y_{s+\mu(s)}^n, Z_{s+\nu(s)}^n, U_{s+\delta(s)}^n){{\mathord{{{\rm{d}}}}}} s\\ &\leq &C\int_t^T|Y_s^n|^3{{\mathord{{{\rm{d}}}}}} s\leq C\int_t^T|Y_s^n|^4{{\mathord{{{\rm{d}}}}}} s+C. \end{eqnarray} $

$ F(y)=4y^3g(y) $.由(2.19)式和假设$ \rm (B3) $, 得

$ \begin{eqnarray} {\mathbb E}[B_2^n]&=&{\mathbb E}\bigg[\int_t^TF(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\bigg]\\ &=&{\mathbb E}\bigg[\int_t^TF(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\bigg]+{\mathbb E}\bigg[\int_t^T(F(Y_s^n)-F(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_s^n\bigg]\\ &\leq& C+C{\mathbb E}\bigg[\int_t^T|Y_s^n-Y_{s^+}^n||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq &C+C\bigg({\mathbb E}\bigg[\int_t^T|Y_s^n-Y_{s^+}^n|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\int_t^T|\dot{B}_s^n|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg)^{\frac{1}{2}}\\ &\leq &C+C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}(2^n)^{\frac{1}{2}}\leq C. \end{eqnarray} $

对(2.25)式两端取期望, 可得

$ \begin{eqnarray} {\mathbb E}[|Y_t^n|^4]\leq C+{\mathbb E}[(\xi)^4]+C\int_t^T{\mathbb E}[|Y_s^n|^4]{{\mathord{{{\rm{d}}}}}} s, \end{eqnarray} $

其中$ C $是与$ n $无关的常数.通过Gronwall不等式, 可得

上式和(2.24)式可以进一步推出

$ \begin{eqnarray} \sup\limits_n\bigg\{\sup\limits_{0\leq t\leq T}{\mathbb E}[|Y_t^n|^4]+6{\mathbb E}\bigg[\int_0^T|Y_s^n|^2(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg]+ 6\lambda{\mathbb E}\bigg[\int_0^T|Y_s^n|^2(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg\}\leq C. \end{eqnarray} $

由Burkholder不等式, 假设$ \rm (B3) $和(2.19)式, 得

$ \begin{eqnarray} {\mathbb E}[\sup\limits_{0\leq t\leq T}|B_2^n(t)|]&\leq&{\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}\bigg|\int_t^TF(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\bigg|\bigg] +{\mathbb E}\bigg[\bigg|\int_0^T(F(Y_s^n)-F(Y_{s^+}^n))\dot{B}_s^n{{\mathord{{{\rm{d}}}}}} s\bigg|\bigg]\\ &\leq& C+C{\mathbb E}\bigg[\bigg(\int_0^T(Y_{s^+}^n)^6g^2(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg)\bigg]^{\frac{1}{2}}\leq C, \end{eqnarray} $

$ \begin{eqnarray} {\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}4\bigg|\int_t^T(Y_s^n)^3Z_s^n{{\mathord{{{\rm{d}}}}}} W_s\bigg|\bigg] &\leq &C{\mathbb E}\bigg[\bigg(\int_0^T(Y_s^n)^6(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\bigg]\\ &\leq &C{\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}(Y_t^n)^2\bigg(\int_0^T(Y_s^n)^2(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\bigg]\\ &\leq&\frac{1}{4}{\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}(Y_t^n)^4\bigg]+C{\mathbb E}\bigg[\int_0^T(Y_s^n)^2(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg] \end{eqnarray} $

$ \begin{eqnarray} {\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}4\bigg|\int_t^T(Y_s^n) ^3U_s^n{{\mathord{{{\rm{d}}}}}} \hat{N}_s\bigg|\bigg] &\leq &C{\mathbb E}\bigg[\bigg(\int_0^T(Y_s^n)^6(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\bigg]\\ &\leq &C{\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}(Y_t^n)^2\bigg(\int_0^T(Y_s^n)^2(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\bigg]\\ &\leq&\frac{1}{4}{\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}(Y_t^n)^4\bigg]+C{\mathbb E}\bigg[\int_0^T(Y_s^n)^2(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg].{\qquad} \end{eqnarray} $

对(2.25)式两端取上确界可得

$ \begin{eqnarray} \sup\limits_n{\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}(Y_t^n)^4\bigg]<\infty. \end{eqnarray} $

接下来, 我们证明

$ \begin{equation} \sup\limits_n{\mathbb E}\bigg[\bigg(\int_0^T(Z_t^n)^2{{\mathord{{{\rm{d}}}}}} t\bigg)^2\bigg]<\infty. \end{equation} $

由Itô公式, 可得

$ \begin{eqnarray} \int_0^T(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s&=&(\xi)^2-(Y_0^n)^2+\int_0^T(2Y_s^nf(s, Y_s^n, Z_s^n, U_s^n, Y_{s+\mu(s)}^n, Z_{s+\nu(s)}^n, U_{s+\delta(s)}^n) \\ &&-\lambda(U_s^n)^2){{\mathord{{{\rm{d}}}}}} s+2\int_0^TY_s^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n-2\int_0^TY_s^nZ_s^n{{\mathord{{{\rm{d}}}}}} W_s -2\int_0^TY_s^nU_s^n{{\mathord{{{\rm{d}}}}}} \hat{N}_s.{}\\ \end{eqnarray} $

于是

$ \begin{eqnarray} &&{\mathbb E}\bigg[\bigg(\int_0^T(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]+\lambda {\mathbb E}\bigg[\bigg(\int_0^T(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^2 \bigg]\\ &\leq&C{\mathbb E}[(\xi)^4]+C{\mathbb E}[(Y_0^n)^4]+C{\mathbb E}\bigg[\bigg(\int_0^T2Y_s^nf(s, Y_s^n, Z_s^n, U_s^n, Y_{s+\mu(s)}^n, Z_{s+\nu(s)}^n, U_{s+\delta(s)}^n){{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\\ &&+C{\mathbb E}\bigg[\bigg(\int_0^TY_s^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\bigg)^2\bigg] +C{\mathbb E}\bigg[\bigg(\int_0^TY_s^nZ_s^n{{\mathord{{{\rm{d}}}}}} W_s\bigg)^2\bigg] +C{\mathbb E}\bigg[\bigg(\int_0^TY_s^nU_s^n{{\mathord{{{\rm{d}}}}}} \hat{N}_s\bigg)^2\bigg].{}\\ \end{eqnarray} $

$ f $的有界性, 得

$ \begin{eqnarray} &&{\mathbb E}\bigg[\bigg(\int_0^T2Y_s^nf(s, Y_s^n, Z_s^n, U_s^n, Y_{s+\mu(s)}^n, Z_{s+\nu(s)}^n, U_{s+\delta(s)}^n){{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\\ &\leq &C{\mathbb E}\bigg[\bigg(\int_0^T(Y_s^n){{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg] \leq C{\mathbb E}\bigg(\int_0^T(Y_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^{1/2} \leq C{\mathbb E}\bigg[\int_0^T(Y_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg] \leq C. \end{eqnarray} $

由Itô等距和Young不等式可得

$ \begin{eqnarray} C{\mathbb E}\bigg[\bigg(\int_0^TY_s^nZ_s^n{{\mathord{{{\rm{d}}}}}} W_s\bigg)^2\bigg] &=&C{\mathbb E}\bigg[\int_0^T(Y_s^n)^2(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg] \leq C{\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}(Y_t^n)^2\int_0^T(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq &C{\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}(Y_t^n)^4\bigg]+\frac{1}{4}{\mathbb E}\bigg[\bigg(\int_0^T(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg] \end{eqnarray} $

$ \begin{eqnarray} C{\mathbb E}\bigg[\bigg(\int_0^TY_s^nU_s^n{{\mathord{{{\rm{d}}}}}} \hat{N}_s\bigg)^2\bigg]&=&C\lambda {\mathbb E}\bigg[\int_0^T(Y_s^n)^2(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg] \leq C\lambda {\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}(Y_t^n)^2\int_0^T(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C{\mathbb E}\bigg[\sup\limits_{0\leq t\leq T}(Y_t^n)^4\bigg]+\frac{1}{4}{\mathbb E}\bigg[\bigg(\int_0^T(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]. \end{eqnarray} $

为了估计$ {\mathbb E}[(\int_0^TY_s^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n)^2] $, 令$ h_t^n=\int_t^TY_s^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n $, 由链式法则可得

$ \begin{eqnarray} (h_t^n)^2&=&2\int_t^Th_s^nY_s^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &=&2\int_t^Th_{s^+}^nY_{s^+}^ng(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n+2\int_t^T(h_s^n-h_{s^+}^n)Y_{s^+}^ng(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^Th_s^n(Y_s^ng(Y_s^n)-Y_{s^+}^ng(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_s^n. \end{eqnarray} $

注意到$ \int_t^Th_{s^+}^nY_{s^+}^ng(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n $是随机积分, 于是$ {\mathbb E}[\int_t^Th_{s^+}^nY_{s^+}^ng(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n]=0 $.进一步, 由Cauchy-Schwarz不等式, 得

$ \begin{eqnarray} 2{\mathbb E}\bigg[\bigg|\int_t^T(h_s^n-h_{s^+}^n)Y_{s^+}^ng(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\bigg|\bigg] &\leq& 2{\mathbb E}\bigg[\int_t^T\bigg|\int_s^{s^+}Y_u^ng(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg||Y_{s^+}^ng(Y_{s^+}^n)||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq &C{\mathbb E}\bigg[\int_t^T\bigg|\int_s^{s^+}|\dot{B}_u^n|{{\mathord{{{\rm{d}}}}}} u\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\leq C. \end{eqnarray} $

现在我们处理(2.40)左边最后一项.由Cauchy-Schwarz不等式, 得

$ \begin{eqnarray} &&2{\mathbb E}\bigg[\bigg|\int_t^Th_s^n(Y_s^ng(Y_s^n)-Y_{s^+}^ng(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_s^n\bigg|\bigg]\\ &\leq &C{\mathbb E}\bigg[\int_t^T|h_s^n||\dot{B}_s^n||Y_s^n-Y_{s^+}^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq &C\bigg({\mathbb E}\bigg[\int_t^T|Y_s^n-Y_{s^+}^n|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg)^{\frac{1}{2}} \bigg(\int_t^T{\mathbb E}[(|h_s^n|^2\dot|\dot{B}_s^n|)^2]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq &C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}\bigg(\int_t^T{\mathbb E}[|h_s^n|^2|\dot{B}_s^n|^2]{{\mathord{{{\rm{d}}}}}} s\bigg)^ {\frac{1}{2}}\\ &\leq &C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}\bigg(\int_t^T{\mathbb E}\bigg[\bigg(\int_s^{s^+}Y_u^ng(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n +\int_{s^+}^TY_u^ng(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg)^2|\dot{B}_s^n|^2\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq &C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}\bigg(\int_t^T\bigg[C{\mathbb E}\bigg(\int_s^{s^+}Y_u^ng(Y_u^n) {{\mathord{{{\rm{d}}}}}} B_u^n\bigg)^2|\dot{B}_s^n|^2 \\ && +C{\mathbb E}\bigg(\int_{s^+}^TY_u^ng(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg)^2 |\dot{B}_s^n|^2\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}\bigg(\int_t^TC\bigg(\frac{1}{2^n}\bigg)^2{\mathbb E} [|\dot{B}_u^n|^2|\dot{B}_s^n|^2]{{\mathord{{{\rm{d}}}}}} s+\int_t^TC2^n{\mathbb E}[|h_{s^+}^n|^2]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}\bigg(\int_t^TC\bigg(\frac{1}{2^n}\bigg)^2(2^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg) ^{\frac{1}{2}}+C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}\bigg(\int_t^TC2^n{\mathbb E}[|h_{s^+}^n|^2] {{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}+C\bigg(\int_t^T{\mathbb E}[|h_{s^+}^n|^2]{{\mathord{{{\rm{d}}}}}} s\bigg)+C. \end{eqnarray} $

对(2.40)式两端取期望, 由(2.41)和(2.42)式, 可得

于是

$ \begin{equation} \sup\limits_n{\mathbb E}[(h_0^n)^2]<\infty. \end{equation} $

由(2.36)–(2.38)和(2.43)式, 可得

$ \begin{eqnarray} \sup\limits_n\bigg\{{\mathbb E}\bigg[\bigg(\int_0^T(Z_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]+\lambda {\mathbb E}\bigg[\bigg(\int_0^T(U_s^n)^2{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg\}<\infty. \end{eqnarray} $

引理2.3证毕.

接下来给出本文的主要定理.

定理2.1  令$ {\mathbb E}[|\xi|^4]<\infty $, 在$ \rm (B1) $, $ \rm (B2) $$ \rm (B3) $假设下, 则

$ \begin{equation} \lim\limits_{n\rightarrow \infty}\sup\limits_{0\leq t\leq T}\bigg\{{\mathbb E}[(Y_t^n-Y_t)^2]+ {\mathbb E}\bigg[\int_t^T(Z_s^n-Z_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]+{\mathbb E}\bigg[\int_t^T(U_s^n-U_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg\}=0. \end{equation} $

  由Itô公式可得

$ \begin{eqnarray} &&(Y_t^n-Y_t)^2+\int_t^T(Z_s^n-Z_s)^2{{\mathord{{{\rm{d}}}}}} s+\lambda\int_t^T(U_s^n-U_s)^2{{\mathord{{{\rm{d}}}}}} s\\ &=&2\int_t^T(Y_s^n-Y_s)(f(s, Y_s^n, Z_s^n, U_s^n, Y_{s+\mu(s)}^n, Z_{s+\nu(s)}^n, U_{s+\delta(s)}^n)\\ &&-f(s, Y_s, Z_s, U_s, Y_{s+\mu(s)}, Z_{s+\nu(s)}, U_{s+\delta(s)})){{\mathord{{{\rm{d}}}}}} s+2\int_t^T(Y_s^n-Y_s)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+\int_s^Tg^2(Y_s){{\mathord{{{\rm{d}}}}}} s-2\int_t^T(Y_s^n-Y_s)g(Y_s){{\mathord{{{\rm{d}}}}}} B_s-\int_t^T(Y_s^n-Y_s)gg'(Y_s){{\mathord{{{\rm{d}}}}}} s\\ &&-2\int_t^T(Y_s^n-Y_s)(Z_s^n-Z_s){{\mathord{{{\rm{d}}}}}} W_s-2\int_t^T(Y_s^n-Y_s)(U_s^n-U_s){{\mathord{{{\rm{d}}}}}} \hat{N}_s\\ :&=&D_1^n+D_2^n+D_3^n+D_4^n+D_5^n+D_6^n+D_7^n. \end{eqnarray} $

$ D_2^n $, 可写为

$ \begin{eqnarray} D_2^n&=&2\int_t^T(Y_s^n-Y_s)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &=&2\int_t^T\bigg[(Y_s^n-Y_s)-(Y_{s^+}^n-Y_{s^+})\bigg]g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^T(Y_{s^+}^n-Y_{s^+})(g(Y_s^n)-g(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_s^n +2\int_t^T(Y_{s^+}^n-Y_{s^+})g(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&F+G+K. \end{eqnarray} $

由随机积分定义可知$ {\mathbb E}[K]=0 $.$ {\mathbb E}[F] $$ {\mathbb E}[G] $, 我们仍想得到它们的估计.然而, 这两个引理的证明非常复杂, 因此我们先给出主要定理的证明, 接下来再给出引理2.4和引理2.5的证明.

对(2.46)式两端取期望, 可得

$ \begin{eqnarray} &&{\mathbb E}[(Y_t^n-Y_t)^2]+{\mathbb E}\bigg[\int_t^T(Z_s^n-Z_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]+\lambda {\mathbb E}\bigg[\int_t^T(U_s^n-U_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &=&2{\mathbb E}\bigg[\int_t^T(f(s, Y_s^n, Z_s^n, U_s^n, Y_{s+\mu(s)}^n, Z_{s+\nu(s)}^n, U_{s+\delta(s)}^n)\\ &&-f(s, Y_s, Z_s, U_s, Y_{s+\mu(s)}, Z_{s+\nu(s)}, U_{s+\delta(s)})) (Y_s^n-Y_s){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&+2{\mathbb E}\bigg[\int_t^T(Y_s^n-Y_s)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\bigg]+{\mathbb E}\bigg[\int_t^Tg^2(Y_s){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&-{\mathbb E}\bigg[\int_t^T(Y_s^n-Y_s)gg'(Y_s){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{eqnarray} $

我们将对$ \lambda $取值进行分类讨论:当$ \lambda\geq1 $时, 我们按照下述步骤进行定理2.1的证明; 当$ 0<\lambda<1 $时, 令$ \lambda=\frac{1}{\theta}, \theta>1 $, 对(2.48)式左右两侧同乘以$ \theta $, 此时与$ \lambda>1 $情况类似.

对上式中的第一项, 由Young不等式, 得

由假设$ \rm (B2) $, 得

于是

把上式带入(2.48)式, 结合引理2.4和引理2.5, 对$ \forall $$ \delta<\frac{1}{2} $, 得

由假设$ \rm (B2) $中(ⅱ)可得

$ \begin{eqnarray} &&{\mathbb E}[(Y_t^n-Y_t)^2]+{\mathbb E}\bigg[\int_t^T(Z_s^n-Z_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]+\lambda {\mathbb E}\bigg[\int_t^T(U_s^n-U_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ & \leq&4C{\mathbb E}\bigg[\int_t^T|Y_s^n-Y_s|^2{{\mathord{{{\rm{d}}}}}} s\bigg]+\frac{1}{2}{\mathbb E}\bigg[\int_t^T\rho(|Y_s^n-Y_s|^2){{\mathord{{{\rm{d}}}}}} s\bigg]+\frac{1}{2}{\mathbb E}\bigg[\int_t^T|Z_s^n-Z_s|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&+\frac{1}{2}{\mathbb E}\bigg[\int_t^T|U_s^n-U_s|^2{{\mathord{{{\rm{d}}}}}} s\bigg]+ C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta} -2{\mathbb E}\bigg[\int_t^Tg(Y_{s^+})g(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&+{\mathbb E}\bigg[\int_t^Tg^2(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg] +{\mathbb E}\bigg[\int_t^T(Y_{s^+}^n-Y_{s^+})gg'(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg] +{\mathbb E}\bigg[\int_t^Tg^2(Y_s){{\mathord{{{\rm{d}}}}}} s\bigg]\\ & &-{\mathbb E}\bigg[\int_t^T(Y_s^n-Y_s)gg'(Y_s){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{eqnarray} $

利用(2.18)和(2.19)式, 在上式中用$ s $来代替$ s^+ $, 有

$ \begin{eqnarray} & &{\mathbb E}[(Y_t^n-Y_t)^2]+\frac{1}{2}{\mathbb E}\bigg[\int_t^T(Z_s^n-Z_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]+(\lambda-\frac{1}{2}) {\mathbb E}\bigg[\int_t^T(U_s^n-U_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}+4C{\mathbb E}\bigg[\int_t^T|Y_s^n-Y_s|^2{{\mathord{{{\rm{d}}}}}} s\bigg]+ \frac{1}{2}{\mathbb E}\bigg[\int_t^T\rho(|Y_s^n-Y_s|^2){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&+{\mathbb E}\bigg[\int_t^Tg^2(Y_s){{\mathord{{{\rm{d}}}}}} s\bigg] -{\mathbb E}\bigg[\int_t^T(Y_s^n-Y_s)gg'(Y_s){{\mathord{{{\rm{d}}}}}} s\bigg]+{\mathbb E}\bigg[\int_t^T(Y_s^n-Y_s)gg'(Y_s^n){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&-2{\mathbb E}\bigg[\int_t^Tg(Y_s)g(Y_s^n){{\mathord{{{\rm{d}}}}}} s\bigg]+{\mathbb E}\bigg[\int_t^Tg^2(Y_s^n){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{eqnarray} $

整理(2.50)式可得

$ \begin{eqnarray} & &{\mathbb E}[(Y_t^n-Y_t)^2]+\frac{1}{2}{\mathbb E}\bigg[\int_t^T(Z_s^n-Z_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]+(\lambda-\frac{1}{2}) {\mathbb E}\bigg[\int_t^T(U_s^n-U_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}+4C{\mathbb E}\bigg[\int_t^T|Y_s^n-Y_s|^2{{\mathord{{{\rm{d}}}}}} s\bigg]+ \frac{1}{2}{\mathbb E}\bigg[\int_t^T\rho(|Y_s^n-Y_s|^2){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&+{\mathbb E}\bigg[\int_t^T(g(Y_s)-g(Y_s^n))^2{{\mathord{{{\rm{d}}}}}} s\bigg] +{\mathbb E}\bigg[\int_t^T(Y_s^n-Y_s)(gg'(Y_s^n)-gg'(Y_s)){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{eqnarray} $

利用$ gg' $$ g $的Lipschitz连续性, 由(2.51)式可得

$ \begin{eqnarray} &&{\mathbb E}[(Y_t^n-Y_t)^2]+\frac{1}{2}{\mathbb E}\bigg[\int_t^T(Z_s^n-Z_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]+(\lambda-\frac{1}{2}) {\mathbb E}\bigg[\int_t^T(U_s^n-U_s)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}+4C{\mathbb E}\bigg[\int_t^T|Y_s^n-Y_s|^2{{\mathord{{{\rm{d}}}}}} s\bigg]+ \frac{1}{2}{\mathbb E}\bigg[\int_t^T\rho(|Y_s^n-Y_s|^2){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{eqnarray} $

注意到(2.52)式第二项和第三项均非负, 则

首先, 对上式左右两边取上确界

再对上式左右两边取上极限

因此

利用Fubini定理, Fatou引理和$ \rho $的性质可得

$ \begin{eqnarray} &&\overline{\lim\limits_{n\rightarrow \infty}}\sup\limits_{0\leq t\leq T}{\mathbb E}[(Y_t^n-Y_t)^2]\\ &\leq&4C\bigg[\int_0^T\overline{\lim\limits_{n\rightarrow \infty}}\sup\limits_{0\leq r\leq s}{\mathbb E}|Y_r^n-Y_r|^2{{\mathord{{{\rm{d}}}}}} s\bigg] +\frac{1}{2}\bigg[\int_0^T\rho(\overline{\lim\limits_{n\rightarrow \infty}}\sup\limits_{0\leq r\leq s}{\mathbb E}|Y_r^n-Y_r|^2){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&+\overline{\lim\limits_{n\rightarrow \infty}}C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}\\ &\leq&C\bigg[\int_0^T\overline{\lim\limits_{n\rightarrow \infty}}\sup\limits_{0\leq r\leq s}{\mathbb E}|Y_r^n-Y_r|^2+\rho(\overline{\lim\limits_{n\rightarrow \infty}}\sup\limits_{0\leq r\leq s}{\mathbb E} |Y_r^n-Y_r|^2)\bigg]. \end{eqnarray} $

若定义$ \psi(T)=\overline{\lim\limits_{n\rightarrow \infty}}\sup\limits_{0\leq t\leq T}{\mathbb E}[(Y_t^n-Y_t)^2] $, 则(2.53)式可改写为

进一步, 通过常微分方程的比较定理可得

定理2.1证明完成.

下面将给出定理证明过程中用到的两个引理及证明.

引理2.4  对任意$ \delta<\frac{1}{2} $, 在定理$ 2.1 $的假设下, 则

$ \begin{eqnarray} {\mathbb E}[F]&=&2{\mathbb E}\bigg[\int_t^T[(Y_s^n-Y_s)-(Y_{s^+}^n-Y_{s^+})]g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\bigg]\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}-2{\mathbb E}\bigg[\int_t^Tg(Y_{s^+})g(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg] +{\mathbb E}\bigg[\int_t^Tg^2(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{eqnarray} $

  由(1.1)式中$ Y^n $$ Y $的定义, 得

$ \begin{eqnarray} F&=&2\int_t^T[(Y_s^n-Y_s)-(Y_{s^+}^n-Y_{s^+})]g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ & =&2\int_t^T\bigg(\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n- 2\int_t^T\bigg(\int_s^{s^+}g(Y_u){{\mathord{{{\rm{d}}}}}} B_u\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^T\bigg(\int_s^{s^+}(Z_u^n-Z_u){{\mathord{{{\rm{d}}}}}} W_u\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n- \int_t^T\bigg(\int_s^{s^+}gg'(Y_u){{\mathord{{{\rm{d}}}}}} u\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^T\bigg(\int_s^{s^+}\bigg[f(u, Y_u^n, Z_u^n, U_u^n, Y_{u+\mu(u)}^n, Z_{u+\nu(u)}^n, U_{u+\delta(u)}^n) \\ && -f(u, Y_u, Z_u, U_u, Y_{u+\mu(u)}, Z_{u+\nu(u)}, U_{u+\delta(u)})\bigg]{{\mathord{{{\rm{d}}}}}} u\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^T\bigg(\int_s^{s^+}(U_u^n-U_u){{\mathord{{{\rm{d}}}}}}\hat{N}_u\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&F_1+F_2+F_3+F_4+F_5+F_6. \end{eqnarray} $

显然

$ \begin{equation} {\mathbb E}[|F_3|]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}, \end{equation} $

$ \begin{equation} {\mathbb E}[|F_5|]\leq C\frac{1}{2^n}\int_t^T{\mathbb E}[|\dot{B}_s^n|]{{\mathord{{{\rm{d}}}}}} s\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{equation} $

由全期望公式和鞅的性质可得

$ \begin{eqnarray} {\mathbb E}[F_4]&=&-2\int_t^T{\mathbb E}\bigg[\bigg(\int_s^{s^+}(Z_u^n-Z_u){{\mathord{{{\rm{d}}}}}} W_u\bigg)g(Y_s^n)\dot{B}_s^n\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &=&-2\int_t^T{\mathbb E}\bigg[{\mathbb E}\bigg[\bigg(\int_s^{s^+}(Z_u^n-Z_u){{\mathord{{{\rm{d}}}}}} W_u\bigg)\Big|{\cal F}_s\bigg] g(Y_s^n)\dot{B}_s^n\bigg]{{\mathord{{{\rm{d}}}}}} s=0, \end{eqnarray} $

$ \begin{eqnarray} {\mathbb E}[F_6]&=&-2\int_t^T{\mathbb E}\bigg[\bigg(\int_s^{s^+}(U_u^n-U_u){{\mathord{{{\rm{d}}}}}} \hat{N}_u\bigg)g(Y_s^n)\dot{B}_s^n\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &=&-2\int_t^T{\mathbb E}\bigg[{\mathbb E}\bigg[\bigg(\int_s^{s^+}(Z_u^n-Z_u){{\mathord{{{\rm{d}}}}}} \hat{N}_u\bigg)\Big|{\cal F}_s\bigg] g(Y_s^n)\dot{B}_s^n\bigg]{{\mathord{{{\rm{d}}}}}} s=0. \end{eqnarray} $

接下来估计$ F_1 $.

$ \begin{eqnarray} F_1&=&2\int_t^T\bigg(\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &=&2\int_t^T\bigg[\int_s^{s^+}(g(Y_u^n)-g(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_u^n\bigg]g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^Tg(Y_{s^+}^n)(B_s^n-B_{s^+}^n)(g(Y_s^n)-g(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^Tg(Y_{s^+}^n)(B_s^n-B_{s^+}^n)g(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&F_{11}+F_{12}+F_{13}. \end{eqnarray} $

$ F_{13} $, 由$ B_s^n $的定义可得

$ \begin{eqnarray} F_{13}&=&2\sum\limits_k\int_{\frac{k}{2^n}}^{\frac{k+1}{2^n}}g^2\bigg(Y_{\frac{k+2}{2^n}}^n\bigg)2^n \bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+3}{2^n}}\bigg)\bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg) \overrightarrow{{{\mathord{{{\rm{d}}}}}} s}\\ &&+2\sum\limits_k\int_{\frac{k}{2^n}}^{\frac{k+1}{2^n}}g^2\bigg(Y_{\frac{k+2}{2^n}}^n\bigg)(2^n)^2 \bigg(s-\frac{k+1}{2^n}\bigg)\bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg)^2\overrightarrow{{{\mathord{{{\rm{d}}}}}} s}\\ :&=&F_{13, 1}+F_{13, 2}. \end{eqnarray} $

由全期望公式可得

$ \begin{eqnarray} {\mathbb E}[F_{13, 1}]=-2\sum\limits_k{\mathbb E}\bigg[g^2\bigg(Y_{\frac{k+2}{2^n}}^n\bigg)\bigg(B_{\frac{k+2}{2^n}}- B_{\frac{k+3}{2^n}}\bigg){\mathbb E}\bigg[\bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg)\Big| {\cal F}_{\frac{k+2}{2^n}}\bigg]\bigg]=0. \end{eqnarray} $

对(2.61)式两端取期望可得

$ \begin{eqnarray} {\mathbb E}[F_{13}]&=&{\mathbb E}[F_{13, 2}]\\ &=&{\mathbb E}\bigg[\sum\limits_kg^2\bigg(Y_{\frac{k+2}{2^n}}^n\bigg)\bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}} \bigg)^2\bigg]\\ &=&{\mathbb E}\bigg[\sum\limits_kg^2\bigg(Y_{\frac{k+2}{2^n}}^n\bigg)\bigg\{\bigg(B_{\frac{k+2}{2^n}}-B_{\frac{K+1}{2^n}} \bigg)^2-\frac{1}{2^n}\bigg\}\bigg]+{\mathbb E}\bigg[\int_t^Tg^2(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &=&{\mathbb E}\bigg[\int_t^Tg^2(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg], \end{eqnarray} $

其中序列$ \{(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}})^2-\frac{1}{2^n}, k\geq 0\} $是一个鞅.

对(2.60)式中$ F_{11} $, 对$ \forall $$ \eta\in[0, 1] $

$ \begin{eqnarray} F_{11}&=&2\int_t^T\int_s^{s^+}g'(Y_{s^+}^n+\eta(Y_u^n-Y_{s^+}^n))(Y_u^n-Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_u^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &=&2\int_t^T\int_s^{s^+}g'(Y_{s^+}^n+\eta(Y_u^n-Y_{s^+}^n))\\ &&\times\bigg[\int_u^{s^+}f(v, Y_v^n, Z_v^n, U_v^n, Y_{v+\mu(v)}^n, Z_{v+\nu(v)}^n, U_{v+\delta(v)}^n){{\mathord{{{\rm{d}}}}}} v\bigg] {{\mathord{{{\rm{d}}}}}} B_u^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^T\int_s^{s^+}g'(Y_{s^+}^n+\eta(Y_u^n-Y_{s^+}^n))\bigg[\int_u^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg] {{\mathord{{{\rm{d}}}}}} B_u^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^T\int_s^{s^+}g'(Y_{s^+}^n+\eta(Y_u^n-Y_{s^+}^n))\bigg[\int_u^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg] {{\mathord{{{\rm{d}}}}}} B_u^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^T\int_s^{s^+}g'(Y_{s^+}^n+\eta(Y_u^n-Y_{s^+}^n))\bigg[\int_u^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg] {{\mathord{{{\rm{d}}}}}} B_u^ng(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&F_{11, 1}+F_{11, 2}+F_{11, 3}+F_{11, 4}. \end{eqnarray} $

$ F_{11, 1} $$ F_{11, 2} $, 由$ g, g', f $的有界性可得

$ \begin{equation} {\mathbb E}[F_{11, 1}]\leq C\frac{1}{2^n}\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{\mathbb E}[|\dot{B}_u^n||\dot{B}_s^n|]{{\mathord{{{\rm{d}}}}}} u \leq C\frac{1}{2^n}(2^n)^{\frac{1}{2}}(2^n)^{\frac{1}{2}}\frac{1}{2^n}\leq C\frac{1}{2^n}, \end{equation} $

$ \begin{equation} {\mathbb E}[F_{11, 2}]\leq C\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u\int_u^{s^+}{{\mathord{{{\rm{d}}}}}} v{\mathbb E}[|\dot{B}_u^n||\dot{B}_v^n||\dot{B}_s^n|] \leq C(2^n)^{\frac{3}{2}}\bigg(\frac{1}{2^n}\bigg)^2\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{equation} $

$ F_{11, 3} $, 由$ g, g' $的有界性和Itô等距可得

$ \begin{eqnarray} {\mathbb E}[F_{11, 3}]&\leq &C{\mathbb E}\bigg[\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}|\dot{B}_u^n||\dot{B}_s^n|\bigg|\int_u^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v \bigg|{{\mathord{{{\rm{d}}}}}} u\bigg]\\ &\leq &C\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u({\mathbb E}[|\dot{B}_u^n|^2|\dot{B}_s^n|^2])^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg|\int_u^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq & C2^n\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u\bigg({\mathbb E}\bigg[\int_u^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}\\ &\leq &C\int_t^T{{\mathord{{{\rm{d}}}}}} s\bigg({\mathbb E}\bigg[\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}\\ &\leq& C\bigg(\int_t^T{{\mathord{{{\rm{d}}}}}} s{\mathbb E}\bigg[\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}\\ &\leq &C\bigg({\mathbb E}\bigg[\int_t^T(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\int_{v^-}^v{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg)^{\frac{1}{2}}\\ &\leq & C\bigg(\sup\limits_n\bigg({\mathbb E}\bigg[\int_t^T(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}\bigg) \bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{eqnarray} $

类似地

$ \begin{eqnarray} {\mathbb E}[F_{11, 4}]&\leq& C{\mathbb E}\bigg[\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}|\dot{B}_u^n||\dot{B}_s^n|\bigg|\int_u^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg|{{\mathord{{{\rm{d}}}}}} u\bigg]\\ &\leq &C\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u({\mathbb E}[|\dot{B}_u^n|^2|\dot{B}_s^n|^2])^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg|\int_u^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg|^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq & C\lambda2^n\int_t^T{{\mathord{{{\rm{d}}}}}} s\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u\bigg({\mathbb E}\bigg[\int_u^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}\\ &\leq& C\lambda\int_t^T{{\mathord{{{\rm{d}}}}}} s\bigg({\mathbb E}\bigg[\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}\\ &\leq & C\lambda\bigg(\int_t^T{{\mathord{{{\rm{d}}}}}} s{\mathbb E}\bigg[\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}\\ &\leq& C\lambda\bigg({\mathbb E}\bigg[\int_t^T(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\int_{v^-}^v{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg)^{\frac{1}{2}}\\ &\leq & C\lambda\bigg(\sup\limits_n\bigg({\mathbb E}\bigg[\int_t^T(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}\bigg) \left(\frac{1}{2^n}\right)^{\frac{1}{2}}. \end{eqnarray} $

由(2.65)–(2.68)式可得

$ \begin{equation} {\mathbb E}[F_{11}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{equation} $

类似地, 对$ F_{12} $, 有

$ \begin{eqnarray} F_{12}&=&2\int_t^Tg(Y_{s^+}^n)\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} B_u^ng'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))(Y_s^n-Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ & =&2\int_t^Tg(Y_{s^+}^n)\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} B_u^ng'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))\\ &&\times\bigg[\int_s^{s^+}f(v, Y_v^n, Z_v^n, U_v^n, Y_{v+\mu(v)}^n, Z_{v+\nu(v)}^n, U_{v+\delta(v)}^n){{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} B_s^n\\ & &+2\int_t^Tg(Y_{s^+}^n)\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} B_u^ng'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n)) \bigg[\int_s^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg]{{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^Tg(Y_{s^+}^n)\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} B_u^ng'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n)) \bigg[\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg]{{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^Tg(Y_{s^+}^n)\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} B_u^ng'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n)) \bigg[\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg]{{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&F_{12, 1}+F_{12, 2}+F_{12, 3}+F_{12, 4}. \end{eqnarray} $

与(2.65)–(2.68)式类似, 可得

$ \begin{equation} {\mathbb E}[F_{12, j}]\leq C\frac{1}{2^n}, \quad j=1, 2, 3, 4. \end{equation} $

因此

$ \begin{equation} {\mathbb E}[F_{12}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{equation} $

从(2.63), (2.69)和(2.72)式可得

$ \begin{equation} {\mathbb E}[F_1]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}+{\mathbb E}\bigg[\int_t^Tg^2(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{equation} $

现在估计$ F_2 $.

$ \begin{eqnarray} F_2&=&-2\int_t^T\bigg(\int_s^{s^+}g(Y_u){{\mathord{{{\rm{d}}}}}} B_u\bigg)g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &=&-2\int_t^T\bigg[\int_s^{s^+}(g(Y_u)-g(Y_{s^+})){{\mathord{{{\rm{d}}}}}} B_u\bigg]g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^Tg(Y_{s^+})(B_s-B_{s^+})(g(Y_s^n)-g(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^Tg(Y_{s^+})(B_s-B_{s^+})g(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&F_{21}+F_{22}+F_{23}. \end{eqnarray} $

$ F_{23} $, 我们将区间$ [t, T] $分割为一些小区间$ [\frac{k}{2^n}, \frac{k+1}{2^n}] $的和, 得

$ \begin{eqnarray} F_{23}&=&-2\sum\limits_k\int_{\frac{k}{2^n}}^{\frac{k+1}{2^n}}g\bigg(Y_{\frac{k+2}{2^n}}\bigg) g\bigg(Y_{\frac{k+2}{2^n}}^n\bigg)\bigg(B_s-B_{\frac{k+2}{2^n}}\bigg)2^n \bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg)\overrightarrow{{{\mathord{{{\rm{d}}}}}} s}\\ &=&-2\sum\limits_k\int_{\frac{k}{2^n}}^{\frac{k+1}{2^n}}g\bigg(Y_{\frac{k+2}{2^n}}\bigg) g\bigg(Y_{\frac{k+2}{2^n}}^n\bigg)\bigg(B_s-B_{\frac{k+1}{2^n}}\bigg)2^n \bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg)\overrightarrow{{{\mathord{{{\rm{d}}}}}} s}\\ &&-2\sum\limits_kg\bigg(Y_{\frac{k+2}{2^n}}\bigg)g\bigg(Y_{\frac{k+2}{2^n}}^n\bigg) \bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg)^2\\ :&=&F_{23, 1}+F_{23, 2}. \end{eqnarray} $

由全期望公式和鞅的性质可得

$ \begin{eqnarray} {\mathbb E}[F_{23, 1}]&=&-2\sum\limits_k\int_{\frac{k}{2^n}}^{\frac{k+1}{2^n}}2^n\overrightarrow{{{\mathord{{{\rm{d}}}}}} s} {\mathbb E}\bigg[g\bigg(Y_{\frac{k+2}{2^n}}\bigg)g\bigg(Y_{\frac{k+2}{2^n}}^n\bigg) \\ && \times {\mathbb E}\big[\bigg(B_s-B_{\frac{k+1}{2^n}}\bigg)\bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg) \Big|{\cal F}_{\frac{k+2}{2^n}}\big]\bigg]\\ &=&-2\sum\limits_k\int_{\frac{k}{2^n}}^{\frac{k+1}{2^n}}2^n\overrightarrow{{{\mathord{{{\rm{d}}}}}} s}{\mathbb E}\bigg[ g\bigg(Y_{\frac{k+2}{2^n}}\bigg)g\bigg(Y_{\frac{k+2}{2^n}}^n\bigg) \\ && \times {\mathbb E}\big[\bigg( B_s-B_{\frac{k+1}{2^n}}\bigg)\bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg)\big]\bigg] =0. \end{eqnarray} $

$ F_{23, 2} $, 有

$ \begin{equation} F_{23, 2}=-2\sum\limits_kg\bigg(Y_{\frac{k+2}{2^n}}\bigg)g\bigg(Y_{\frac{k+2}{2^n}}^n\bigg)\bigg\{ \bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg)^2-\frac{1}{2^n}\bigg\} -2\int_t^Tg(Y_{s^+})g(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s. \end{equation} $

$ \begin{equation} {\mathbb E}[F_{23}]={\mathbb E}[F_{23, 2}]=-2{\mathbb E}\bigg[\int_t^Tg(Y_{s^+})g(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{equation} $

$ F_{21} $可改写为

$ \begin{eqnarray} F_{21}&=&-2\int_t^T\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))(Y_u-Y_{s^+}){{\mathord{{{\rm{d}}}}}} B_ug(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &=&-2\int_t^T\int_s^{s^+}\bigg[\int_u^{s^+}f(v, Y_v, Z_v, U_v, Y_{v+\mu(v)}, Z_{v+\nu(v)}, U_{v+\delta(v)}){{\mathord{{{\rm{d}}}}}} v\bigg]\\ &&\times g'(Y_{s^+}+\eta(Y_u-Y_{s^+})){{\mathord{{{\rm{d}}}}}} B_ug(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^T\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))\bigg[\int_u^{s^+}g(Y_v){{\mathord{{{\rm{d}}}}}} B_v\bigg]{{\mathord{{{\rm{d}}}}}} B_u g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-\int_t^T\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))\bigg[\int_u^{s^+}gg'(Y_v){{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} B_u g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^T\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))\bigg[\int_u^{s^+}Z_v{{\mathord{{{\rm{d}}}}}} W_v\bigg]{{\mathord{{{\rm{d}}}}}} B_u g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^T\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))\bigg[\int_u^{s^+}U_v{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg]{{\mathord{{{\rm{d}}}}}} B_u g(Y_s^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&F_{21, 1}+F_{21, 2}+F_{21, 3}+F_{21, 4}+F_{21, 5}. \end{eqnarray} $

$ F_{21, 1} $, 由$ g, g', f $的有界性可得

$ \begin{eqnarray} {\mathbb E}[F_{21, 1}]&\leq& C\int_t^T{\mathbb E}\bigg[\bigg|\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+})) \\ && \times\bigg[\int_u^{s^+}f(v, Y_v, Z_v, U_v, Y_{v+\mu(v)}, Z_{v+\nu(v)}, U_{v+\delta(v)}){{\mathord{{{\rm{d}}}}}} v\bigg] {{\mathord{{{\rm{d}}}}}} B_u\bigg||\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\int_t^T\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+})) \\ && \times\bigg[\int_u^{s^+}f(v, Y_v, Z_v, U_v, Y_{v+\mu(v)}, Z_{v+\nu(v)}, U_{v+\delta(v)}){{\mathord{{{\rm{d}}}}}} v\bigg] {{\mathord{{{\rm{d}}}}}} B_u\bigg|^2\bigg]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}[|\dot{B}_s^n|^2]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}C{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s \leq C(2^n)^{\frac{1}{2}}\int_t^T[(s^+-s)(s^+-s)^2]^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\bigg(\frac{1}{2^n}\bigg). \end{eqnarray} $

类似, 可得

$ \begin{eqnarray} {\mathbb E}[F_{21, 2}] {}&\leq& C\int_t^T{\mathbb E}\bigg[\bigg|\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))\times \big[\int_u^{s^+}g(Y_v){{\mathord{{{\rm{d}}}}}} B_v\big]{{\mathord{{{\rm{d}}}}}} B_u\bigg||\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\int_t^T\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))\times\big[\int_u^{s^+}g(Y_v){{\mathord{{{\rm{d}}}}}} B_v\big] {{\mathord{{{\rm{d}}}}}} B_u\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{}\\ &&\times\Big({\mathbb E}\big[|\dot{B}_s^n|^2\big]\Big)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{S^+}))^2\times \bigg(\int_u^{s^+}g(Y_v){{\mathord{{{\rm{d}}}}}} B_v\bigg)^2{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T\bigg(\int_s^{s^+}{\mathbb E}\bigg[\bigg(\int_u^{s^+}g(Y_v){{\mathord{{{\rm{d}}}}}} B_v \bigg)^2\bigg]{{\mathord{{{\rm{d}}}}}} u\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T\bigg(\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u\bigg)^{\frac{1}{2}}\bigg({\mathbb E}\bigg[\bigg( \int_s^{s^+}g^2(Y_v){{\mathord{{{\rm{d}}}}}} v\bigg)\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s \leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}, \end{eqnarray} $

$ \begin{eqnarray} {\mathbb E}[F_{21, 3}]&\leq& C\int_t^T{\mathbb E}\bigg[\bigg|\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+})) \bigg[\int_u^{s^+}gg'(Y_v){{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} B_u\bigg||\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\int_t^T\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+})) \bigg[\int_u^{s^+}gg'(Y_v){{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} B_u\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{}\\ && \Big({\mathbb E}[|\dot{B}_s^n|^2]\Big)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))^2 \bigg(\int_u^{s^+}gg'(Y_v){{\mathord{{{\rm{d}}}}}} v\bigg)^2{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T[(s^+-s)(s^+-s)^2]^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s \leq C\bigg(\frac{1}{2^n}\bigg). \end{eqnarray} $

由Hölder不等式和Itô等距, 可得

$ \begin{eqnarray} {} {\mathbb E}[F_{21, 4}] &\leq& C\int_t^T{\mathbb E}\bigg[\bigg|\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))\times\bigg[ \int_u^{s^+}Z_v{{\mathord{{{\rm{d}}}}}} W_v\bigg]{{\mathord{{{\rm{d}}}}}} B_u\bigg||\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\int_t^T\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))\times\bigg[\int_u^{s^+} Z_v{{\mathord{{{\rm{d}}}}}} W_v\bigg]{{\mathord{{{\rm{d}}}}}} B_u\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{}\\ &&\times\Big({\mathbb E}[|\dot{B}_s^n|^2]\Big)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))^2\times \bigg(\int_u^{s^+}Z_v{{\mathord{{{\rm{d}}}}}} W_v\bigg)^2{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T\bigg(\int_s^{s^+}{\mathbb E}\bigg[\bigg(\int_u^{s^+}Z_v{{\mathord{{{\rm{d}}}}}} W_v\bigg)^2\bigg] {{\mathord{{{\rm{d}}}}}} u\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T\bigg(\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u\bigg)^{\frac{1}{2}}\bigg({\mathbb E}\bigg[\int_s^{s^+}Z_v^2 {{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\bigg(\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}Z_v^2{{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)\bigg)^{\frac{1}{2}} {}\\ & \leq& C\bigg({\mathbb E}\bigg[\int_t^TZ_v^2{{\mathord{{{\rm{d}}}}}} v\int_{v^-}^v{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg)^{\frac{1}{2}} \leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}, \end{eqnarray} $

$ \begin{eqnarray} {\mathbb E}[F_{21, 5}]&\leq& C\int_t^T{\mathbb E}\bigg[\bigg|\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))\times\bigg[ \int_u^{s^+}U_v{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg]{{\mathord{{{\rm{d}}}}}} B_u\bigg||\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\int_t^T\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))\times\bigg[\int_u^{s^+} U_v{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg]{{\mathord{{{\rm{d}}}}}} B_u\bigg|^2\bigg]\bigg)^{\frac{1}{2}}\\ &&\times\Big({\mathbb E}[|\dot{B}_s^n|^2]\Big)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}g'(Y_{s^+}+\eta(Y_u-Y_{s^+}))^2\times \bigg(\int_u^{s^+}U_v{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg)^2{{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(2^n)^{\frac{1}{2}}\int_t^T\bigg(\int_s^{s^+}{\mathbb E}\bigg[\bigg(\int_u^{s^+}U_v{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg)^2\bigg] {{\mathord{{{\rm{d}}}}}} u\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\lambda(2^n)^{\frac{1}{2}}\int_t^T\bigg(\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u\bigg)^{\frac{1}{2}}\bigg({\mathbb E}\bigg[\int_s^{s^+}U_v^2 {{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\lambda\bigg(\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}U_v^2{{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)\bigg)^{\frac{1}{2}}\\ &\leq& C\lambda\bigg({\mathbb E}\bigg[\int_t^TU_v^2{{\mathord{{{\rm{d}}}}}} v\int_{v^-}^v{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg)^{\frac{1}{2}} \leq C\lambda\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{eqnarray} $

由(2.79)–(2.84)式可得

$ \begin{equation} {\mathbb E}[F_{21}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{equation} $

现在估计$ F_{22} $.

$ \begin{eqnarray} F_{22}&\leq& C\int_t^T|B_s-B_{s^+}||Y_s^n-Y_{s^+}^n||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\int_t^T|B_s-B_{s^+}|\bigg|\int_s^{s^+}f(u, Y_u^n, Z_u^n, U_u^n, Y_{u+\mu(u)}^n, Z_{u+\nu(u)}^n, U_{u+\delta(u)}^n){{\mathord{{{\rm{d}}}}}} u\bigg| |\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|B_s-B_{s^+}|\bigg|\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s +C\int_t^T|B_s-B_{s^+}|\\ &&\times\bigg|\int_s^{s^+}Z_u^n{{\mathord{{{\rm{d}}}}}} W_u\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s +C\int_t^T|B_s-B_{s^+}|\bigg|\int_s^{s^+}U_u^n{{\mathord{{{\rm{d}}}}}}\hat{N}_u\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ :&=&F_{22, 1}+F_{22, 2}+F_{22, 3}+F_{22, 4}. \end{eqnarray} $

$ F_{22, 1} $, 由$ f $的有界性可得

$ \begin{eqnarray} {\mathbb E}[F_{22, 1}]\leq C\frac{1}{2^n}\int_t^T\bigg({\mathbb E}[|B_s-B_{s^+}|^2]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}[|\dot{B}_s^n|^2]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s \leq C\frac{1}{2^n}. \end{eqnarray} $

下面这两个不等式将会在下文中经常用到

$ \begin{eqnarray} \sup\limits_u|\dot{B}_u^n|\leq2^n\sup\limits_{|r-s|\leq\frac{1}{2^n}}|B_r-B_s|. \end{eqnarray} $

$ \forall $$ \delta>0 $$ p\geq 1 $, $ \exists $$ C_{p, \delta} $满足

$ \begin{eqnarray} {\mathbb E}\bigg[\sup\limits_{|r-s|\leq\frac{1}{2^n}}|B_r-B_s|^p\bigg]\leq C_{p, \delta}\bigg(\frac{1}{2^n}\bigg) ^{\frac{p}{2}-\delta}. \end{eqnarray} $

于是

$ \begin{eqnarray} {\mathbb E}[F_{22, 2}]&\leq&C\frac{1}{2^n}\int_t^T\bigg({\mathbb E}[|B_s-B_{s^+}|^2]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\sup\limits_n|\dot{B}_u^n|^4\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\frac{1}{2^n}\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}(2^n)^2\int_t^T \bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^4\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

由Hölder不等式, Fubini定理和Itô等距可得

$ \begin{eqnarray} {\mathbb E}[F_{22, 3}]&\leq&C\int_t^T\bigg({\mathbb E}[|B_s-B_{s^+}|^4]\bigg)^{\frac{1}{4}}\bigg({\mathbb E} [|\dot{B}_s^n|^4]\bigg)^{\frac{1}{4}}\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}Z_u^n{{\mathord{{{\rm{d}}}}}} W_u\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\bigg(\int_t^T{\mathbb E}\bigg[\int_s^{s^+}(Z_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}} \leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}, \end{eqnarray} $

$ \begin{eqnarray} {\mathbb E}[F_{22, 4}]&\leq&C\int_t^T\bigg({\mathbb E}[|B_s-B_{s^+}|^4]\bigg)^{\frac{1}{4}}\bigg({\mathbb E} [|\dot{B}_s^n|^4]\bigg)^{\frac{1}{4}}\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}U_u^n{{\mathord{{{\rm{d}}}}}} \hat{N}_u\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\lambda\bigg(\int_t^T{\mathbb E}\bigg[\int_s^{s^+}(U_u^n)^2{{\mathord{{{\rm{d}}}}}} u\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}} \leq C\lambda\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{eqnarray} $

由(2.86)–(2.91)式可得

$ \begin{equation} {\mathbb E}[F_{22}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

因此

$ \begin{equation} {\mathbb E}[F_2]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}-2{\mathbb E}\bigg[\int_t^Tg(Y_{s^+})g(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{equation} $

由(2.57)–(2.59), (2.73)和(2.94)式完成了引理2.4的证明.

引理2.5  对任意$ \delta<\frac{1}{2} $, 在定理$ 2.1 $的假设下, 得

$ \begin{eqnarray} {\mathbb E}[G]&=&2{\mathbb E}\bigg[\int_t^T(Y_{s^+}^n-Y_{s^+})(g(Y_s^n)-g(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_s^n\bigg]\\ &\leq & C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}+{\mathbb E}\bigg[\int_t^T(Y_{s^+}^n-Y_{s^+}) gg'(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{eqnarray} $

  注意到$ G $可改写为

$ \begin{eqnarray} G&=&2\int_t^T(Y_{s^+}^n-Y_{s^+})(g(Y_s^n)-g(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_s^n\\ &=&2\int_t^T(Y_{s^+}^n-Y_{s^+})g'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))(Y_s^n-Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ &=&2\int_t^T(Y_{s^+}^n-Y_{s^+})g'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))\\ &&\times\bigg[ \int_s^{s^+}f(v, Y_v^n, Z_v^n, U_v^n, Y_{v+\mu(v)}^n, Z_{v+\nu(v)}^n, U_{v+\delta(v)}^n){{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^T(Y_{s^+}^n-Y_{s^+})g'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))\bigg[ \int_s^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg]{{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^T(Y_{s^+}^n-Y_{s^+})g'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))\bigg[ \int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg]{{\mathord{{{\rm{d}}}}}} B_s^n\\ &&-2\int_t^T(Y_{s^+}^n-Y_{s^+})g'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))\bigg[ \int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg]{{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&G_1+G_2+G_3+G_4. \end{eqnarray} $

$ g', f $的有界性和引理2.2, 可得

$ \begin{eqnarray} {\mathbb E}[G_1]&\leq &C\frac{1}{2^n}{\mathbb E}\bigg[\int_t^T|Y_{s^+}-Y_{s^+}||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ {}&\leq &C\frac{1}{2^n}\int_t^T\bigg({\mathbb E}[|Y_{s^+}^n-Y_{s^+}|^2]\bigg)^{\frac{1}{2}}\bigg( {\mathbb E}[|\dot{B}_s^n|^2]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ & \leq& C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{eqnarray} $

$ G_2 $可写为

$ \begin{eqnarray} G_2&=&2\int_t^T(Y_{s^+}^n-Y_{s^+})[g'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))-g'(Y_{s^+}^n)]\times \bigg[\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg]{{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^T(Y_{s^+}^n-Y_{s^+})g'(Y_{s^+}^n)\bigg[\int_s^{s^+}(g(Y_u^n)-g(Y_{s^+}^n)){{\mathord{{{\rm{d}}}}}} B_u^n\bigg]{{\mathord{{{\rm{d}}}}}} B_s^n\\ &&+2\int_t^T(Y_{s^+}^n-Y_{s^+})g'(Y_{s^+}^n)g(Y_{s^+}^n)(B_s^n-B_{s^+}^n){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&G_{21}+G_{22}+G_{23}. \end{eqnarray} $

利用$ g' $的Lipschitz连续性, 得

$ \begin{eqnarray} G_{21}&\leq&C\int_t^T|Y_{s^+}^n-Y_{s^+}||Y_s^n-Y_{s^+}^n|\bigg|\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg| |\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\int_t^T|Y_{s^+}-Y_{s^+}|\bigg|\int_s^{s^+}f(v, Y_v^n, Z_v^n, U_v^n, Y_{v+\mu(v)}^n, Z_{v+\nu(v)}^n, U_{v+\delta(v)}^n){{\mathord{{{\rm{d}}}}}} v\bigg|\\ &&\times\bigg|\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg|\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg|^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|\bigg|\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg| |\dot{B}_s^n|{\rm d}s\\ &&+C\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg|\bigg|\int_s^{s^+}g(Y_u^n){{\mathord{{{\rm{d}}}}}} B_u^n\bigg| |\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ :&=&G_{21, 1}+G_{21, 2}+G_{21, 3}+G_{21, 4}. \end{eqnarray} $

由Hölder不等式和(2.88)式, (2.89)式可得

$ \begin{eqnarray} {\mathbb E}[G_{21, 1}]&\leq& C\frac{1}{2^n}{\mathbb E}\bigg[\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg|\int_s^{s^+}\bigg| \dot{B}_u^n|{{\mathord{{{\rm{d}}}}}} u||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C\frac{1}{2^n}(2^n)^2{\mathbb E}\bigg[\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg|\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u\bigg| \sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C{\mathbb E}\bigg[\int_t^T|Y_{s^+}^n-Y_{s^+}|\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C\int_t^T\bigg({\mathbb E}[|Y_{s^+}^n-Y_{s^+}|^2]\bigg)^{\frac{1}{2}}\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}} |B_r-B_v|^4\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^{1-\delta}. \end{eqnarray} $

类似地

$ \begin{eqnarray} {\mathbb E}[G_{21, 2}]&\leq& C{\mathbb E}\bigg[\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg|\int_s^{s^+}{{\mathord{{{\rm{d}}}}}} u\bigg|^2 \sup\limits_s|\dot{B}_s^n|^3{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^2(2^n)^3{\mathbb E}\bigg[\int_t^T|Y_{s^+}^n-Y_{s^+}|\sup\limits_{|r-v|\leq \frac{1}{2^n}}|B_r-B_v|^3{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C2^n\int_t^T\bigg({\mathbb E}[|Y_{s^+}^n-Y_{s^+}|^2]\bigg)^{\frac{1}{2}}\bigg( {\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^6\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}, \end{eqnarray} $

$ \begin{eqnarray} {\mathbb E}[G_{21, 3}]&\leq& C{\mathbb E}\bigg[\int_t^T|Y_{s^+}-Y_{s^+}|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|\bigg(\int_s^{s^+} |\dot{B}_u^n|{{\mathord{{{\rm{d}}}}}} u\bigg)|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C\frac{1}{2^n}(2^n)^2{\mathbb E}\bigg[\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg| \sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C2^n\int_t^T\bigg({\mathbb E}[|Y_{s^+}^n-Y_{s^+}|^4]\bigg)^{\frac{1}{4}}\bigg({\mathbb E}\bigg[\bigg| \int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|^2\bigg]\bigg)^{\frac{1}{2}}\\ &&\times\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}} |B_r-B_v|^8\bigg]\bigg)^{\frac{1}{4}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^{1-\delta}2^n\int_t^T\bigg({\mathbb E}[|Y_{s^+}^n-Y_{s^+}|^4]\bigg)^{\frac{1}{4}} \bigg({\mathbb E}\bigg[\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^{1-\delta}2^n\bigg(\int_t^T\bigg({\mathbb E}[|Y_{s^+}^n-Y_{s^+}|^4]\bigg)^{\frac{1}{2}} {{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\times\bigg(\int_t^T{\mathbb E}\bigg[\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^{1-\delta}2^n\bigg({\mathbb E}\bigg[\int_t^T(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\int_{v^-}^v{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg)^{\frac{1}{2}}\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

类似可得

$ \begin{eqnarray} {\mathbb E}[G_{21, 4}] &\leq& C{\mathbb E}\bigg[\int_t^T|Y_{s^+}-Y_{s^+}|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg|\bigg(\int_s^{s^+} |\dot{B}_u^n|{{\mathord{{{\rm{d}}}}}} u\bigg)|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C\frac{1}{2^n}(2^n)^2{\mathbb E}\bigg[\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg| \sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C2^n\int_t^T\bigg({\mathbb E}[|Y_{s^+}^n-Y_{s^+}|^4]\bigg)^{\frac{1}{4}}\bigg({\mathbb E}\bigg[\bigg| \int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg|^2\bigg]\bigg)^{\frac{1}{2}}\\ &&\times\bigg({\mathbb E}\bigg[ \sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^8\bigg]\bigg)^{\frac{1}{4}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\lambda\bigg(\frac{1}{2^n}\bigg)^{1-\delta}2^n\int_t^T\bigg({\mathbb E}[|Y_{s^+}^n-Y_{s^+}|^4]\bigg)^{\frac{1}{4}} \bigg({\mathbb E}\bigg[\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\lambda\bigg(\frac{1}{2^n}\bigg)^{1-\delta}2^n\bigg(\int_t^T\bigg({\mathbb E}[|Y_{s^+}^n-Y_{s^+}|^4]\bigg)^ {\frac{1}{2}} {{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &&\times\bigg(\int_t^T{\mathbb E}\bigg[\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq& C\lambda\bigg(\frac{1}{2^n}\bigg)^{1-\delta}2^n\bigg({\mathbb E}\bigg[\int_t^T(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\int_{v^-}^v{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg) ^{\frac{1}{2}} \leq C\lambda\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

由(2.100)–(2.103)式可得

$ \begin{equation} {\mathbb E}[G_{21}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

$ G_{22} $, 有

$ \begin{eqnarray} G_{22}&\leq& C\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg(\int_s^{s^+}|Y_u^n-Y_{s^+}^n||\dot{B}_u^n|{{\mathord{{{\rm{d}}}}}} u\bigg) |\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg(\int_s^{s^+}\bigg|\int_u^{s^+}f(v, Y_v^n, Z_v^n, U_v^n, Y_{v+\mu(v)}^n, Z_{v+\nu(v)}^n, U_{v+\delta(v)}^n){{\mathord{{{\rm{d}}}}}} v\bigg| \\ && \times|\dot{B}_u^n|{{\mathord{{{\rm{d}}}}}} u\bigg)|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg(\int_s^{s^+}\bigg|\int_u^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg| |\dot{B}_u^n|{{\mathord{{{\rm{d}}}}}} u\bigg)|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg(\int_s^{s^+}\bigg|\int_u^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg| |\dot{B}_u^n|{{\mathord{{{\rm{d}}}}}} u\bigg)|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_{s^+}^n-Y_{s^+}|\bigg(\int_s^{s^+}\bigg|\int_u^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg| |\dot{B}_u^n|{{\mathord{{{\rm{d}}}}}} u\bigg)|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ :&=&G_{22, 1}+G_{22, 2}+G_{22, 3}+G_{22, 4}. \end{eqnarray} $

类似地, 可得

$ \begin{equation} {\mathbb E}[G_{22, j}]\leq C\bigg(\frac{1}{2^n}\bigg)^{1-\delta}, \quad j=1, 2, 3, 4. \end{equation} $

于是

$ \begin{equation} {\mathbb E}[G_{22}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

$ G_{23} $, 我们将区间$ [t, T] $分割为一些小区间$ [\frac{k}{2^n}, \frac{k+1}{2^n}] $的和, 有

$ \begin{eqnarray} G_{23}&=&2\sum\limits_k\int_{\frac{k}{2^n}}^{\frac{k+1}{2^n}}\bigg(Y_{\frac{k+2}{2^n}}^n-Y_{\frac{k+2}{2^n}}\bigg) gg'(Y_{\frac{k+2}{2^n}}^n)2^n\bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+3}{2^n}}\bigg)\bigg(B_{\frac{k+2}{2^n}} -B_{\frac{k+1}{2^n}}\bigg)\overrightarrow{{{\mathord{{{\rm{d}}}}}} s}\\ &&+2\sum\limits_k\int_{\frac{k}{2^n}}^{\frac{k+1}{2^n}}\bigg(Y_{\frac{k+2}{2^n}}^n-Y_{\frac{k+2}{2^n}}\bigg) gg'(Y_{\frac{k+2}{2^n}}^n)\times(2^n)^2\bigg(s-\frac{k+1}{2^n}\bigg)\bigg(B_{\frac{k+2}{2^n}}- B_{\frac{k+1}{2^n}}\bigg)^2\overrightarrow{{{\mathord{{{\rm{d}}}}}} s}\\ &=&-2\sum\limits_k\bigg(Y_{\frac{k+2}{2^n}}^n-Y_{\frac{k+2}{2^n}}\bigg)gg'\bigg(Y_{\frac{k+2}{2^n}}^n\bigg) \bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+3}{2^n}}\bigg)\bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg)\\ &&+\sum\limits_k\bigg(Y_{\frac{k+2}{2^n}}^n-Y_{\frac{k+2}{2^n}}\bigg)gg'\bigg(Y_{\frac{k+2}{2^n}}^n\bigg)\bigg\{ \bigg(B_{\frac{k+2}{2^n}}-B_{\frac{k+1}{2^n}}\bigg)^2-\frac{1}{2^n}\bigg\}\\ &&+\int_t^T(Y_{s^+}^n-Y_{s^+})gg'(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s. \end{eqnarray} $

由全期望公式和鞅的性质可得

$ \begin{eqnarray} {\mathbb E}[G_{23}]={\mathbb E}\bigg[\int_t^T(Y_{s^+}^n-Y_{s^+})gg'(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg]. \end{eqnarray} $

由(2.104), (2.107)和(2.109)式可得

$ \begin{eqnarray} {\mathbb E}[G_2]&\leq& C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}+{\mathbb E}\bigg[\int_t^T(Y_{s^+}^n-Y_{s^+}) gg'(Y_{s^+}^n){{\mathord{{{\rm{d}}}}}} s\bigg], \end{eqnarray} $

$ \begin{eqnarray} G_3&=&-2\bigg\{\int_t^T[(Y_{s^+}^n-Y_{s^+})g'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))-(Y_s^n-Y_s)g'(Y_s^n)] \\ && \times\bigg(\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg){{\mathord{{{\rm{d}}}}}} B_s^n\bigg\} -2\int_t^T(Y_s^n-Y_s)g'(Y_s^n)\bigg(\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&G_{31}+G_{32}. \end{eqnarray} $

首先

$ \begin{eqnarray} {\mathbb E}[G_{32}]=-2\int_t^T{\mathbb E}\bigg[(Y_s^n-Y_s)g'(Y_s^n)\dot{B}_s^n{\mathbb E}\bigg[\bigg (\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg)\Big|{\cal F}_s\bigg]\bigg]{{\mathord{{{\rm{d}}}}}} s =0. \end{eqnarray} $

$ G_{31} $, 由$ g' $的Lipschitz连续性, 得

$ \begin{eqnarray} G_{31}&\leq& C\int_t^T|Y_{s^+}^n-Y_s^n|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s+C\int_t^T|Y_{s^+}-Y_s| \bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_s^n-Y_s||Y_{s^+}^n-Y_s^n|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ :&=&G_{31, 1}+G_{31, 2}+G_{31, 3}. \end{eqnarray} $

$ \begin{eqnarray} G_{31, 1}&\leq& C\int_t^T\bigg|\int_s^{s^+}f(v, Y_v^n, Z_v^n, U_v^n, Y_{v+\mu(v)}^n, Z_{v+\nu(v)}^n, U_{v+\delta(v)}^n){{\mathord{{{\rm{d}}}}}} v\bigg| \bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s +C\int_t^T\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ :&=&G_{31, 11}+G_{31, 12}+G_{31, 13}+G_{31, 14}. \end{eqnarray} $

$ G_{31, 11} $, 由$ f $的有界性和Itô等距可得

$ \begin{eqnarray} {\mathbb E}[G_{31, 11}]\leq C\frac{1}{2^n}\int_t^T\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg|\bigg]\bigg)^{\frac{1}{2}} ({\mathbb E}[|\dot{B}_s^n|^2])^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s \leq C\frac{1}{2^n}. \end{eqnarray} $

由(2.88)式, (2.89)式, Itô等距和$ |\dot{B}_s^n| $关于$ {\cal F}_s $可测得, 可得

$ \begin{eqnarray} {\mathbb E}[G_{31, 12}] {} &\leq& C\frac{1}{2^n}\int_t^T{\mathbb E}\bigg[\sup\limits_s|\dot{B}_s^n|^2\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\frac{1}{2^n}(2^n)^2\int_t^T\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^4\bigg]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\frac{1}{2^n}(2^n)^2\bigg(\frac{1}{2^n}\bigg)^{1-\delta}\bigg(\int_t^T{\mathbb E}\bigg[ \int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq& C\frac{1}{2^n}(2^n)^2\bigg(\frac{1}{2^n}\bigg)^{1-\delta}\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}} \bigg(\int_t^T{\mathbb E}[(Z_v^n)^2]{{\mathord{{{\rm{d}}}}}} v\bigg)^{\frac{1}{2}} \leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

$ \begin{eqnarray} {\mathbb E}[G_{31, 13}]&\leq& C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg)^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg] {}\\ &=&C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}(|\dot{B}_s^n|)^{\frac{1}{2}}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &=&C{\mathbb E}\bigg[\int_t^T\int_s^{s^+}(|\dot{B}_s^n|)(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq& C(2^n){\mathbb E}\bigg[\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|\bigg)\int_t^T\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq &C(2^n)\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2\bigg]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg( \int_t^T\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

$ G_{31, 14} $, 由(2.88)式, (2.89)式, Fubini定理和Itô等距可得

$ \begin{eqnarray} {\mathbb E}[G_{31, 14}] {}&\leq& C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}|\dot{B}_s^n|^{\frac{1}{2}}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg) \bigg(\int_s^{s^+}|\dot{B}_s^n|^{\frac{1}{2}}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg){{\mathord{{{\rm{d}}}}}} s\bigg]\nonumber\\ &\leq &C\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}|\dot{B}_s^n|^{\frac{1}{2}}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg]^2\bigg)^{1/2} \bigg({\mathbb E}\bigg[\int_s^{s^+}|\dot{B}_s^n|^{\frac{1}{2}}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg]^2\bigg)^{1/2}{{\mathord{{{\rm{d}}}}}} s\\ &=&C\lambda\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}(|\dot{B}_s^n|)(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{1/2} \bigg({\mathbb E}\bigg[\int_s^{s^+}|\dot{B}_s^n|(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{1/2}{{\mathord{{{\rm{d}}}}}} s\\ &\leq &C\lambda\int_t^T\bigg({\mathbb E}\bigg[(2^n)\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|\bigg)\int_s^{s^+} (U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{1/2}\\ &&\times\bigg({\mathbb E}\bigg[(2^n)\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|\bigg)\int_s^{s^+} (Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{1/2}{{\mathord{{{\rm{d}}}}}} s\\ &\leq &C\lambda\int_t^T\bigg({\mathbb E}\bigg[(2^n)\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2\bigg]\bigg)^{\frac{1}{4}} \bigg({\mathbb E}\bigg(\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg)^2\bigg)^{\frac{1}{4}}\\ &&\times\bigg({\mathbb E}\bigg[(2^n)\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2\bigg]\bigg)^{\frac{1}{4}} \bigg({\mathbb E}\bigg(\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg)^2\bigg)^{\frac{1}{4}}{{\mathord{{{\rm{d}}}}}} s\\ {}&\leq &C\lambda(2^n)(\frac{1}{2^n})^{\frac{1}{2}-\delta}\int_t^T\bigg({\mathbb E}\bigg(\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg)^2 \bigg)^{\frac{1}{4}}\bigg({\mathbb E}\bigg(\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg)^2\bigg)^{\frac{1}{4}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C(\frac{1}{2^n})^{\frac{1}{2}-\delta}. \end{eqnarray} $

由(2.115)–(2.118)式可推出

$ \begin{equation} {\mathbb E}[G_{31, 1}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

$ G_{31, 2} $, 有

$ \begin{eqnarray} G_{31, 2}&\leq& C\int_t^T\bigg|\int_s^{s^+}f(v, Y_v, Z_v, U_v, Y_{v+\mu(v)}, Z_{v+\nu(v)}, U_{v+\delta(v)}){{\mathord{{{\rm{d}}}}}} v\bigg|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}g(Y_v){{\mathord{{{\rm{d}}}}}} B_v\bigg|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}gg'(Y_v){{\mathord{{{\rm{d}}}}}} v\bigg|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}Z_v{{\mathord{{{\rm{d}}}}}} W_v\bigg|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}U_v{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ :&=&G_{31, 21}+G_{31, 22}+G_{31, 23}+G_{31, 24}+G_{31, 25}. \end{eqnarray} $

通过(2.115)式的处理方法得

$ \begin{equation} {\mathbb E}[G_{31, 21}]\leq\frac{C}{2^n}. \end{equation} $

$ G_{31, 22} $, 由Fubini定理, $ g $的有界性和Itô等距可得

$ \begin{eqnarray} {\mathbb E}[G_{31, 22}]&\leq& C\int_t^T\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}Z_u^n(|\dot{B}_s^n|){{\mathord{{{\rm{d}}}}}} W_u\bigg|^2\bigg]\bigg) ^{\frac{1}{2}}\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}g(Y_u){{\mathord{{{\rm{d}}}}}} B_u\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq &C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}(Z_u^n)^2|\dot{B}_s^n|^2 {{\mathord{{{\rm{d}}}}}} u\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s. \end{eqnarray} $

与(2.118)式类似, 可得

$ \begin{equation} {\mathbb E}[G_{31, 22}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

类似地

$ \begin{equation} {\mathbb E}[G_{31, 23}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

$ G_{31, 24} $, 有

$ \begin{equation} G_{31, 24}\leq C\int_t^T\bigg(\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg)^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s+C\int_t^T\bigg(\int_s^{s^+} Z_v{{\mathord{{{\rm{d}}}}}} W_v\bigg)^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s. \end{equation} $

与(2.118)式方法类似可得

$ \begin{equation} {\mathbb E}[G_{31, 24}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}, \end{equation} $

$ \begin{eqnarray} {\mathbb E}[G_{31, 25}]&\leq &C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}U_v{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg)^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg] +C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg)^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &=&C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}U_v|\dot{B}_s^n|^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg)^2{{\mathord{{{\rm{d}}}}}} s\bigg] +C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}Z_v^n|\dot{B}_s^n|^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} W_v\bigg)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &=&\lambda C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}(U_v)^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} v\bigg){{\mathord{{{\rm{d}}}}}} s\bigg] +C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}(Z_v^n)^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} v\bigg){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\lambda(2^n){\mathbb E}\bigg[\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|\bigg)\int_t^T\int_s^{s^+}(U_v)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &&+C(2^n){\mathbb E}\bigg[\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|\bigg)\int_t^T\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\lambda(2^n)\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2\bigg]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg(\int_t^T\int_s^{s^+}(U_v)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg)^{\frac{1}{2}}\\ &&+C(2^n)\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2\bigg]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg(\int_t^T\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq&C\lambda\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}+C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

由(2.121)–(2.127)式可推出

$ \begin{equation} {\mathbb E}[G_{31, 2}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

最后估计$ {\mathbb E}[G_{31, 3}] $, 注意到

$ \begin{eqnarray} G_{31, 3}&\leq&C\int_t^T|Y_s^n-Y_s|\bigg|\int_s^{s^+}f(v, Y_v^n, Z_v^n, U_v^n, Y_{v+\mu(v)}^n, Z_{v+\nu(v)}^n, U_{v+\delta(v)}^n){{\mathord{{{\rm{d}}}}}} v\bigg|\\ &&\times\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_s^n-Y_s|\bigg|\int_s^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg| |\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_s^n-Y_s|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_s^n-Y_s|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg| |\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ :&=&G_{31, 31}+G_{31, 32}+G_{31, 33}+G_{31, 34}. \end{eqnarray} $

$ G_{31, 31} $, 由(2.88)式, (2.89)式, Fubini定理和Itô等距可得

$ \begin{eqnarray} {\mathbb E}[G_{31, 31}]&\leq &C\frac{1}{2^n}\int_t^T({\mathbb E}[|Y_s^n-Y_s|^2])^{\frac{1}{2}}\bigg({\mathbb E}\bigg[\bigg| \int_s^{s^+}(|\dot{B}_s^n|)Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq& C\frac{1}{2^n}\bigg(\int_t^T{\mathbb E}[|Y_s^n-Y_s|^2]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\bigg(\int_t^T{\mathbb E}\bigg[ \bigg|\int_s^{s^+}|\dot{B}_s^n|^2(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg|\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq &C\frac{1}{2^n}(2^n)\bigg({\mathbb E}\bigg[\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2\bigg)\int_t^T\int_s^{s^+} (Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^4\bigg]\bigg) ^{\frac{1}{4}}\bigg({\mathbb E}\bigg[\bigg(\int_t^T\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg)^{\frac{1}{4}}\\ &\leq& C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}\bigg({\mathbb E}\bigg[\bigg(\int_t^T(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg)^2 \bigg]\bigg)^{\frac{1}{4}}\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

由Itô等距和$ |Y_s^n-Y_s| $, $ |\dot{B}_s^n| $都是关于$ {\cal F}_s $可测的, 得

$ \begin{eqnarray} {\mathbb E}[G_{31, 33}]&=&C\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|^2|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ & =&C\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg(\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg)|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C2^n{\mathbb E}\bigg[\bigg(\sup\limits_{0\leq s\leq T}|Y_s^n-Y_s|\bigg)\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|\bigg) \int_t^T\bigg(\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C2^n\bigg({\mathbb E}\bigg[\sup\limits_{0\leq s\leq T}|Y_s^n-Y_s|^4\bigg]\bigg)^{\frac{1}{4}}\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}} |B_r-B_v|^4\bigg]\bigg)^{\frac{1}{4}}\\ &&\times\bigg({\mathbb E}\bigg[\bigg(\int_t^T\bigg(\int_s^{s^+}(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg){{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg) ^{\frac{1}{2}}\\ &\leq&C2^n\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}\bigg({\mathbb E}\bigg[\bigg(\int_t^T(Z_v^n)^2\bigg( \int_{v^-}^v{{\mathord{{{\rm{d}}}}}} s\bigg){{\mathord{{{\rm{d}}}}}} v\bigg)^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq&C2^n\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}\frac{1}{2^n}\bigg({\mathbb E}\bigg[\bigg( \int_t^T(Z_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg)^2\bigg]\bigg)^{\frac{1}{2}} \leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

$ G_{31, 34} $, 有

$ \begin{eqnarray} {\mathbb E}[G_{31, 34}]&=&C\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg| \bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg||\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\int_t^T{\mathbb E}\bigg[|Y_s^ n-Y_s|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg|^2|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ & &+C\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|^2|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s, \end{eqnarray} $

其中

$ \begin{eqnarray} &&C\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg|^2|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ & =&C\lambda\int_t^T\bigg[{\mathbb E}|Y_s^n-Y_s|\bigg|\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg||\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\lambda2^n{\mathbb E}\bigg[\bigg(\sup\limits_{0\leq s\leq T}|Y_s^n-Y_s|\bigg)\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|\bigg)\int_t^T \bigg(\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg){{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\lambda2^n\bigg({\mathbb E}\bigg[\sup\limits_{0\leq s\leq T}|Y_s^n-Y_s|^4\bigg]\bigg)^{\frac{1}{4}}\bigg({\mathbb E}\sup\limits_{|r-v|\leq\frac{1}{2^n}} |B_r-B_v|^4\bigg)^{\frac{1}{4}}\\ & &\times\bigg({\mathbb E}\bigg[\bigg(\int_t^T\bigg(\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg){{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg) ^{\frac{1}{2}}\\ &\leq&C\lambda2^n\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}\bigg({\mathbb E}\bigg[\bigg(\int_t^T(U_v^n)^2\bigg( \int_{v^-}^v{{\mathord{{{\rm{d}}}}}} s\bigg){{\mathord{{{\rm{d}}}}}} v\bigg)^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq&C\lambda2^n\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}\frac{1}{2^n}\bigg({\mathbb E}\bigg[\bigg( \int_t^T(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg)^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq& C\lambda\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

因此

$ \begin{equation} {\mathbb E}[G_{31, 34}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

$ G_{31, 32} $, 有

$ \begin{eqnarray} {\mathbb E}[G_{31, 32}]&\leq&C\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|^2|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_s^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg|\int_s^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg|^2|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s. \end{eqnarray} $

对上式中第二项, 有

$ \begin{eqnarray} & &\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg|\int_s^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg|^2|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C(2^n)^3{\mathbb E}\bigg[\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^3\bigg)\int_t^T|Y_s^n-Y_s|\bigg(\int_s^{s^+} |g(Y_v^n)|{{\mathord{{{\rm{d}}}}}} v\bigg)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C(2^n)\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^6\bigg]\bigg)^{\frac{1}{2}}\bigg( {\mathbb E}\bigg[\bigg(\int_t^T|Y_s^n-Y_s|{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

从(2.131)–(2.135)式可得

$ \begin{eqnarray} {\mathbb E}[G_{31, 32}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

由(2.130)–(2.136)式得

$ \begin{eqnarray} {\mathbb E}[G_{31, 3}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

于是

$ \begin{equation} {\mathbb E}[G_3]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

$ G_4 $, 有

$ \begin{eqnarray} G_4&=&-2\bigg\{\int_t^T\bigg[(Y_{s^+}^n-Y_{s^+})g'(Y_{s^+}^n+\eta(Y_s^n-Y_{s^+}^n))- (Y_s^n-Y_s)g'(Y_s^n)\bigg] \\ && \times\bigg(\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg){{\mathord{{{\rm{d}}}}}} B_s^n\bigg\} -2\int_t^T(Y_s^n-Y_s)g'(Y_s^n)\bigg(\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg){{\mathord{{{\rm{d}}}}}} B_s^n\\ :&=&G_{41}+G_{42}, \end{eqnarray} $

其中

$ \begin{eqnarray} {\mathbb E}[G_{42}]=-2\int_t^T{\mathbb E}\bigg[(Y_s^n-Y_s)g'(Y_s^n){\mathbb E}\bigg[\bigg(\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg) \Big|{\cal F}_s\bigg]\bigg]{{\mathord{{{\rm{d}}}}}} s=0. \end{eqnarray} $

$ G_{41} $, 有

$ \begin{eqnarray} G_{41}&\leq&C\int_t^T|Y_{s^+}^n-Y_s^n|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s +C\int_t^T|Y_{s^+}-Y_s|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_s^n-Y_s||Y_{s^+}^n-Y_s^n|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ :&=&G_{41, 1}+G_{41, 2}+G_{41, 3}. \end{eqnarray} $

$ G_{41, 1} $, 有

$ \begin{eqnarray} G_{41, 1}&\leq&C\int_t^T\bigg|\int_s^{s^+}f(v, Y_v^n, Z_v^n, U_v^n, Y_{v+\mu(v)}^n, Z_{v+\nu(v)}^n, U_{v+\delta(v)}^n){{\mathord{{{\rm{d}}}}}} v\bigg| \bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg| |\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg|^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ :&=&G_{41, 11}+G_{41, 12}+G_{41, 13}+G_{41, 14}. \end{eqnarray} $

$ G_{41, 11} $, 有

$ \begin{eqnarray} {\mathbb E}[G_{41, 11}]&\leq&C\lambda\frac{1}{2^n}\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[|\dot{B}_s^n|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\leq C\frac{1}{2^n}. \end{eqnarray} $

$ G_{41, 12} $, 有

$ \begin{eqnarray} {\mathbb E}[G_{41, 12}]&\leq&C\frac{1}{2^n}\int_t^T{\mathbb E}\bigg[\sup\limits_s|\dot{B}_s^n|^2\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\frac{1}{2^n}(2^n)^2\int_t^T\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^4\bigg] \bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\lambda\frac{1}{2^n}(2^n)^2\bigg(\frac{1}{2^n}\bigg)^{1-\delta}\bigg(\int_t^T{\mathbb E}\bigg[ \int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq&C\lambda\frac{1}{2^n}(2^n)^2\bigg(\frac{1}{2^n}\bigg)^{1-\delta}\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}} \bigg(\int_t^T{\mathbb E}[(U_v^n)^2]{{\mathord{{{\rm{d}}}}}} v\bigg)^{\frac{1}{2}} \leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

$ G_{41, 14} $, 有

$ \begin{eqnarray} {\mathbb E}[G_{41, 14}]&=&C{\mathbb E}\bigg[\int_t^T\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg|^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ & =&C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}|\dot{B}_s^n|^{\frac{1}{2}}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &=&C\lambda {\mathbb E}\bigg[\int_t^T\int_s^{s^+}|\dot{B}_s^n|(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\lambda(2^n){\mathbb E}\bigg[\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|\bigg)\int_t^T\int_s^{s^+} (U_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\lambda(2^n)\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2\bigg]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg(\int_t^T\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

$ G_{41, 13} $, 有

$ \begin{equation} {\mathbb E}[G_{41, 13}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

于是

$ \begin{equation} {\mathbb E}[G_{41, 1}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

$ G_{41, 2} $, 有

$ \begin{eqnarray} G_{41, 2}&\leq&C\int_t^T\bigg|\int_s^{s^+}f(v, Y_v, Z_v, U_v, Y_{v+\mu(v)}, Z_{v+\nu(v)}, U_{v+\delta(v)}){{\mathord{{{\rm{d}}}}}} v\bigg| \bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ & &+C\int_t^T\bigg|\int_s^{s^+}g(Y_v){{\mathord{{{\rm{d}}}}}} B_v\bigg|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}gg'(Y_v){{\mathord{{{\rm{d}}}}}} v\bigg|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T\bigg|\int_s^{s^+}Z_v{{\mathord{{{\rm{d}}}}}} W_v\bigg|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s +C\int_t^T\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg|^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ :&=&G_{41, 21}+G_{41, 22}+G_{41, 23}+G_{41, 24}+G_{41, 25}. \end{eqnarray} $

与(2.142)式处理方法相似, 可得

$ \begin{equation} {\mathbb E}[G_{41, 21}]\leq C\frac{1}{2^n}, \end{equation} $

$ \begin{equation} {\mathbb E}[G_{41, 23}]\leq C\frac{1}{2^n}. \end{equation} $

$ G_{41, 22} $, 有

$ \begin{eqnarray} {\mathbb E}[G_{41, 22}]&\leq&C\int_t^T\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}U_v^n|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg|^2\bigg]\bigg) ^{\frac{1}{2}}\bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}g(Y_v){{\mathord{{{\rm{d}}}}}} B_v\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\lambda\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}\int_t^T\bigg({\mathbb E}\bigg[\int_s^{s^+}(U_v^n)^2 |\dot{B}_s^n|^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\lambda\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}(2^n)\int_t^T\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}} |B_r-B_v|^2\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\lambda\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}(2^n)\int_t^T\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}} |B_r-B_v|^4\bigg]\bigg)^{\frac{1}{4}}\bigg({\mathbb E}\bigg[\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg]^2\bigg)^{\frac{1}{4}}{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

$ G_{41, 25} $, 有

$ \begin{equation} {\mathbb E}[G_{41, 25}]\leq C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg)^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg] +C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg)^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg], \end{equation} $

其中

$ \begin{eqnarray} &&C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg)^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\bigg]\\ & =&C{\mathbb E}\bigg[\int_t^T\bigg(\int_s^{s^+}|\dot{B}_s^n|^{\frac{1}{2}}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg)^2{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &=&C\lambda {\mathbb E}\bigg[\int_t^T\int_s^{s^+}|\dot{B}_s^n|(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg]\\ & \leq&C\lambda(2^n){\mathbb E}\bigg[\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|\bigg)\int_t^T\int_s^{s^+} (U_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg]\\ &\leq&C\lambda(2^n)\bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^2\bigg]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg(\int_t^T\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

$ \begin{equation} {\mathbb E}[G_{41, 25}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

类似地

$ \begin{equation} {\mathbb E}[G_{41, 24}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

因此

$ \begin{equation} {\mathbb E}[G_{41, 2}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

$ G_{41, 3} $, 有

$ \begin{eqnarray} G_{41, 3}&\leq&C\int_t^T|Y_s^n-Y_s|\bigg|\int_s^{s^+}f(v, Y_v^n, Z_v^n, U_v^n, Y_{v+\mu(v)}^n, Z_{v+\nu(v)}^n, U_{v+\delta(v)}^n){{\mathord{{{\rm{d}}}}}} v\bigg|\\ &&\times\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ & &+C\int_t^T|Y_s^n-Y_s|\bigg|\int_s^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg| \bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_s^n-Y_s|\bigg|\int_s^{s^+}Z_v^n{{\mathord{{{\rm{d}}}}}} W_v\bigg| \bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg||\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T|Y_s^n-Y_s|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg|^2|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}} s\\ & :=&G_{41, 31}+G_{41, 32}+G_{41, 33}+G_{41, 34}. \end{eqnarray} $

$ G_{41, 31} $, 可得

$ \begin{eqnarray} {\mathbb E}[G_{41, 31}]&\leq&C\frac{1}{2^n}\int_t^T\bigg({\mathbb E}\bigg[|Y_s^n-Y_s|^2\bigg]\bigg)^{\frac{1}{2}} \bigg({\mathbb E}\bigg[\bigg|\int_s^{s^+}U_v^n|\dot{B}_s^n|{{\mathord{{{\rm{d}}}}}}\hat{N}_v\bigg|^2\bigg]\bigg)^{\frac{1}{2}}{{\mathord{{{\rm{d}}}}}} s\\ & \leq&C\lambda\frac{1}{2^n}\bigg(\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|^2\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}} \bigg(\int_t^T{\mathbb E}\bigg[\bigg|\int_s^{s^+}|\dot{B}_s^n|^2(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg|\bigg]{{\mathord{{{\rm{d}}}}}} s\bigg)^{\frac{1}{2}}\\ &\leq&C\lambda\frac{1}{2^n}(2^n)\bigg({\mathbb E}\bigg[\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|\bigg) \int_t^T\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v{{\mathord{{{\rm{d}}}}}} s\bigg]\bigg)^{\frac{1}{2}}\\ &\leq&C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}}. \end{eqnarray} $

$ G_{41, 34} $, 有

$ \begin{eqnarray} {\mathbb E}[G_{41, 34}]&=&C\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg|^2|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &=&C\lambda\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg(\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg)|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &\leq&C\lambda(2^n){\mathbb E}\bigg[\bigg(\sup\limits_{0\leq s\leq T}|Y_s^n-Y_s|\bigg)\bigg(\sup\limits_{|r-v|\leq\frac{1}{2^n}} |B_r-B_v|\bigg)\int_t^T\bigg(\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg){{\mathord{{{\rm{d}}}}}} s\bigg]\\ & \leq&C\lambda(2^n)\bigg({\mathbb E}\bigg[\sup\limits_{0\leq s\leq T}|Y_s^n-Y_s|^4\bigg]\bigg)^{\frac{1}{4}} \bigg({\mathbb E}\bigg[\sup\limits_{|r-v|\leq\frac{1}{2^n}}|B_r-B_v|^4\bigg]\bigg)^{\frac{1}{4}}\\ &&\times\bigg({\mathbb E}\bigg[\bigg(\int_t^T\bigg(\int_s^{s^+}(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg){{\mathord{{{\rm{d}}}}}} s\bigg)^2\bigg]\bigg)^ {\frac{1}{2}}\\ &\leq&C\lambda(2^n)\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}\bigg({\mathbb E}\bigg[\bigg(\int_t^T (U_v^n)^2\bigg(\int_{v^-}^v{{\mathord{{{\rm{d}}}}}} s\bigg){{\mathord{{{\rm{d}}}}}} v\bigg)^2\bigg]\bigg)^{\frac{1}{2}}\\ &\leq&C\lambda(2^n)\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}\frac{1}{2^n} \bigg({\mathbb E}\bigg[\bigg(\int_t^T(U_v^n)^2{{\mathord{{{\rm{d}}}}}} v\bigg)^2\bigg]\bigg)^{\frac{1}{2}} \leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

$ G_{41, 33} $, 有

$ \begin{equation} {\mathbb E}[G_{41, 33}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

$ G_{41, 32} $, 有

$ \begin{eqnarray} {\mathbb E}[G_{41, 32}]&\leq&C\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg|\int_s^{s^+}U_v^n{{\mathord{{{\rm{d}}}}}} \hat{N}_v\bigg|^2|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s\\ &&+C\int_t^T{\mathbb E}\bigg[|Y_s^n-Y_s|\bigg|\int_s^{s^+}g(Y_v^n){{\mathord{{{\rm{d}}}}}} B_v^n\bigg|^2|\dot{B}_s^n|\bigg]{{\mathord{{{\rm{d}}}}}} s{}\\ & \leq & C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{eqnarray} $

于是

$ \begin{equation} {\mathbb E}[G_{41, 3}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}, \end{equation} $

$ \begin{equation} {\mathbb E}[G_{41}]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}, \end{equation} $

$ \begin{equation} {\mathbb E}[G_4]\leq C\bigg(\frac{1}{2^n}\bigg)^{\frac{1}{2}-\delta}. \end{equation} $

利用(2.97), (2.110), (2.139)和(2.165)式, 可得引理2.5成立.

参考文献

Hao T, Li J. Mean-field SDEs with jumps and nonlocal integral-PDEs. Nonlinear Differ Equ Appl, 2016, 23, Article number: 17

Hu F , Chen Z .

$L^p$ solutions of anticipated backward stochastic differential equations under monotonicity and general increasing conditions

Stochastics : An International Journal of Probability and Stochastic Processes, 2016, 88 (2): 267- 284

DOI:10.1080/17442508.2015.1052810      [本文引用: 1]

Hu Y , Matoussi A , Zhang T .

Wong-Zakai approximations of backward doubly stochastic differential equations

Stochastic Process Appl, 2015, 125: 4375- 4404

DOI:10.1016/j.spa.2015.07.003      [本文引用: 5]

Li S, Xu X. Anticipated backward stochastic differential equations and their applications to zero-sum stochastic differential games. Communications in Statistics-Simulation and Computation, 2019. DOI:10.1080/03610918.2019.1694950

[本文引用: 1]

Ma J , Zhang J .

Representation theorems for backward stochastic differential equations

Ann Probab, 2002, 12 (4): 1390- 1418

[本文引用: 1]

Pardoux E , Peng S .

Adapted solution of a backward stochastic differential equation

Systems Control Lett, 1990, 14: 55- 61

DOI:10.1016/0167-6911(90)90082-6      [本文引用: 1]

Pardoux E , Peng S .

Backward doubly SDEs and systems of quasilinear SPDEs

Probab Theory Related Fields, 1994, 98: 209- 227

DOI:10.1007/BF01192514      [本文引用: 1]

Peng S , Yang Z .

Anticipated backward stochastic differential equations

Ann Probab, 2009, 37 (3): 877- 902

Shen X , Jiang L , Tian D .

$L^p$ solutions of anticipated BSDEs with weak monotonicity and general growth generators

Communications in Statistics-Simulation and Computation, 2019, 48 (1): 73- 90

DOI:10.1080/03610918.2017.1373812      [本文引用: 1]

Wen J , Shi Y .

Mean-field anticipated BSDEs driven by fractional Brownian motion and related stochastic control problem

Applied Mathematics and Computation, 2019, 35: 282- 298

Wen J , Shi Y .

Solvability of anticipated backward stochastic Volterra integral equations

Stat Probabil Lett, 2020, 156: 108599

DOI:10.1016/j.spl.2019.108599      [本文引用: 1]

Wong E , Zakai M .

On the relation between ordinary and stochastic differential equations

Internat J Engrg Sci, 1965, 3: 213- 229

DOI:10.1016/0020-7225(65)90045-5     

Wu H , Wang W , Ren J .

Anticipated backward stochastic equations with non-Lipschitz coefficients

Stat Probabil Lett, 2012, 82: 672- 682

DOI:10.1016/j.spl.2011.12.008     

Xu W .

Backward doubly stochastic equations with jumps and comparision theorems

J Math Anal Appl, 2016, 443 (1): 596- 624

DOI:10.1016/j.jmaa.2016.05.050      [本文引用: 1]

Xu X .

Anticipated backward doubly stochastic differential equations

Appl Math Comput, 2013, 220: 53- 62

[本文引用: 3]

Yang Z , Elliott R .

Some properties of generalized anticipated backward stochastic differential equations

Electron Commun Probab, 2013, 18 (63): 1- 10

[本文引用: 1]

Zhang T .

Wong-Zakai approximations to SDEs with reflection

Potential Anal, 2014, 41 (3): 783- 815

DOI:10.1007/s11118-014-9394-9     

Zhu Q , Shi Y .

Backward doubly stochastic differential equations with jumps and stochastic partial differential-integral equations

Chin Ann Math Ser B, 2012, 33: 127- 142

DOI:10.1007/s11401-011-0686-8     

/