assignmentutor-lab™ 为您的留学生涯保驾护航 在代写随机微积分Stochastic calculus方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机微积分Stochastic calculus代写方面经验极为丰富，各种随机微积分Stochastic calculus相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
assignmentutor™您的专属作业导师

## 数学代写|随机微积分代写Stochastic calculus代考|Itô’s coupling procedure

Let $L$ be given by (1.2.5), and assume that $a-\sigma \sigma^{\top}$ for some $\sigma: \mathbb{R}^N \longrightarrow$ $\operatorname{Hom}\left(\mathbb{R}^M ; \mathbb{R}^N\right)$

We are now ready to describe Itô’s procedure, which is the pathspace implementation of the Euler approximation scheme used in $\S 1.2 .1$, and the first step is to construct the random variables used in his coupling procedure. Given $\mathbf{x} \in \mathbb{R}^N, n \geq 0$, and $w \in \mathbb{W}\left(\mathbb{R}^M\right)$, define $X_n(0, \mathbf{x})(w)=\mathbf{x}$ and \begin{aligned} X_n(t, \mathbf{x})(w)=X_n &\left(m 2^{-n}, \mathbf{x}\right)(w) \ +& \sigma\left(X_n\left(m 2^{-n}, \mathbf{x}\right)(w)\right)\left(w(t)-w\left(m 2^{-n}\right)\right) \ &+b\left(X_n\left(m 2^{-n}, \mathbf{x}\right)(w)\right)\left(t-m 2^{-n}\right) \end{aligned}
for $m \geq 0$ and $t \in I_{m, n}=\left[m 2^{-n},(m+1) 2^{-n}\right]$. Clearly $X_n(t, \mathbf{x})$ is $W_{\lfloor t\rfloor_n}{ }^{-}$ measurable. Hence, since $w(t)-w\left(\lfloor t\rfloor_n\right)$ is independent of $W_{\lfloor t\rfloor_n}$,
$$\begin{gathered} \mathcal{W}\left(X_n(t) \in \Gamma \mid W_{\lfloor t\rfloor_n}\right)=\bar{Q}\left(t-\lfloor t\rfloor_n, X_n\left(\lfloor t\rfloor_n\right), \Gamma\right) \ \text { where } Q(\tau, \mathbf{y})=\gamma_{\mathbf{y}+\tau b(\mathbf{y}), \tau a(\mathbf{y})} \end{gathered}$$
and so, using induction on $m \geq 0$, one can check that the distribution of $X_n(t, \mathbf{x})$ under $\mathcal{W}$ is the measure $\mu_{t, n}$ in (1.2.12) with $\nu=\delta_{\mathbf{x}}$. Now assume that $\sigma$ and $b$ are uniformly Lipschitz continuous, and set
\begin{aligned} |\sigma|_{\mathrm{u} L i p} &=\sup {\substack{\mathbf{y}, \mathbf{y}^{\prime} \in \mathbb{R}^N \ \mathbf{y}^{\prime} \neq \mathbf{y}}} \frac{\left|\sigma\left(\mathbf{y}^{\prime}\right)-\sigma(\mathbf{y})\right|{\mathrm{H} . \mathrm{S} .}}{\left|\mathbf{y}^{\prime}-\mathbf{y}\right|} \ |b|_{\mathrm{Lip}}=\sup _{\substack{\mathbf{y}, \mathbf{y}^{\prime} \in \mathbb{R}^N \ \mathbf{y}^{\prime} \neq \mathbf{y}}} \frac{\left|b\left(\mathbf{y}^{\prime}\right)-b(\mathbf{y})\right|}{\left|\mathbf{y}^{\prime}-\mathbf{y}\right|} \end{aligned}

## 数学代写|随机微积分代写Stochastic calculus代考|The Markov property

For each $\mathbf{x} \in \mathbb{R}^N$, let $\mathbb{P}{\mathbf{x}} \in \mathbf{M}_1\left(\mathcal{P}\left(\mathbb{R}^N\right)\right)$ be the distribution of $X(\cdot, \mathbf{x})$ under $\mathcal{W}$ Lemma 2.2.1. For each $\Phi \in C{\mathrm{b}}\left(\mathcal{P}\left(\mathbb{R}^N\right) ; \mathbb{R}\right)$,
$$\mathbb{E}^{\mathcal{W}}\left[\Phi\left(X_n(\cdot, \mathbf{x})\right)\right] \longrightarrow \mathbb{E}^{\Gamma \mathbf{x}}[\Phi]$$
uniformly for $\mathbf{x}$ in compact subsets. In particular, $\mathbf{x} \rightsquigarrow \mathbb{E}^{\mathbb{P}_{\mathbf{x}}}[\Phi]$ is continuous. Further, if $P_n(t, \mathbf{x})$ is the distribution of $X_n(t, \mathbf{x})$ and $P(t, \mathbf{x})$ is the distribution of $X(t, \mathbf{x})$ under $\mathcal{W}$, then $P_n \longrightarrow P$ in $C\left([0, \infty) \times \mathbb{R}^N ; \mathbf{M}_1\left(\mathbb{R}^N\right)\right)$.

Proof. Suppose that $\left{\mathbf{x}n: n \geq 0\right} \subseteq \mathbb{R}$ tends to $\mathbf{x}$, and let $\mathbb{P}_n$ be the distribution of $X_n\left(\cdot, \mathbf{x}_n\right)$ under $\mathcal{W}$. We need to show that $\mathbb{E}^{\mathbb{P}_n}[\Phi] \longrightarrow \mathbb{E}^{\mathbb{P}{\times}}[\Phi]$ for $\Phi \in C_{\mathrm{b}}\left(\mathcal{P}\left(\mathbb{R}^N\right) ; \mathbb{R}\right)$, and, by Theorem 9.1.5 in [20], it suffices to do so when $\Phi$ is uniformly continuous with respect to the metric $\rho$ in (2.1.3). To this end, note that, by (2.2.3),
$$\lim {n \rightarrow \infty} \sup {|\mathbf{y}| \leq R} \mathcal{W}\left(\rho\left(X_n(\cdot, \mathbf{y}), X(\cdot, \mathbf{y})\right) \geq \delta\right)=0$$
for all $\delta>0$. Hence, if $\Psi$ is uniformly continuous,

$\leq \sup {n \geq 1|\mathbf{y}| \leq R} \sup \mathbb{E}^{\mathcal{W}}\left[\left|\Phi\left(X_n(\cdot, \mathbf{y})\right)-\Phi(X(\cdot, \mathbf{y}))\right|, \rho\left(X_n(\cdot, \mathbf{y}), X(\cdot, \mathbf{y})\right) \leq \delta\right]$, for all $\delta>0$. Since the right hand side tends to 0 as $\delta \searrow 0$, this proves that $\mathbb{E}^{\mathbb{P}_n}[\Phi] \longrightarrow \mathbb{E}^{\mathbb{P}{\times}}[\Phi]$ for all $\Phi \in C_{\mathrm{b}}\left(\mathcal{P}\left(\mathbb{R}^N\right) ; \mathbb{R}\right)$. In addition, because, for each $n \geq 0, \mathbf{x} \rightsquigarrow \mathbb{E}^{\mathcal{W}}\left[\Phi\left(X_n(\cdot, \mathbf{x})\right)\right]$ is continuous, we have also shown that $\mathrm{x} \rightsquigarrow \mathbb{E}^{\mathbb{P}_x}[\Phi]$ is continuous.

Turning to the final assertion, apply the preceding to see that, for each $t \geq 0$ and $\varphi \in C_{\mathrm{b}}\left(\mathbb{R}^N ; \mathbb{R}\right),\left\langle\varphi, P_n(t, \mathbf{x})\right\rangle \longrightarrow\langle\varphi, P(t, \mathbf{x})\rangle$ uniformly for $\mathbf{x}$ in compact subsets. Further, using (2.2.2), one sees that, for each $T>0$ and $R>0$, there exists a $C(T, R)<\infty$ such that
$$\sup {n \geq 0} \sup {|\mathbf{x}| \leq R} \mathbb{E}^{\mathcal{W}}\left[\left|X_n(t, \mathbf{x})-X_n(s, \mathbf{x})\right|^2\right] \leq C(T, R)(t-s) \quad \text { for } 0 \leq s<t \leq T \text {. }$$

# 随机微积分代考

## 数学代写|随机微积分代写Stochastic calculus代考|Itô’s coupling procedure

$$X_n(t, \mathbf{x})(w)=X_n\left(m 2^{-n}, \mathbf{x}\right)(w)+\quad \sigma\left(X_n\left(m 2^{-n}, \mathbf{x}\right)(w)\right)\left(w(t)-w\left(m 2^{-n}\right)\right)+b\left(X_n\left(m 2^{-n}, \mathbf{x}\right)(w)\right)\left(t-m 2^{-n}\right)$$

$$|\sigma|{\mathrm{u} L i p}=\sup \mathbf{y}, \mathbf{y}^{\prime} \in \mathbb{R}^N \mathbf{y}^{\prime} \neq \mathbf{y} \frac{\left|\sigma\left(\mathbf{y}^{\prime}\right)-\sigma(\mathbf{y})\right| \text { H. S. }}{\left|\mathbf{y}^{\prime}-\mathbf{y}\right|}|b|{\text {Lip }}=\sup _{\mathbf{y}, \mathbf{y}^{\prime} \in \mathbb{R}^N \mathbf{y}^{\prime} \neq \mathbf{y}} \frac{\left|b\left(\mathbf{y}^{\prime}\right)-b(\mathbf{y})\right|}{\left|\mathbf{y}^{\prime}-\mathbf{y}\right|}$$

## 数学代写|随机微积分代写Stochastic calculus代考|The Markov property

$$\lim n \rightarrow \infty \sup |\mathbf{y}| \leq R \mathcal{W}\left(\rho\left(X_n(\cdot, \mathbf{y}), X(\cdot, \mathbf{y})\right) \geq \delta\right)=0$$

$\leq \sup n \geq 1|\mathbf{y}| \leq R \sup {\mathbb{E}^{\mathcal{W}}}\left[\left|\Phi\left(X_n(\cdot, \mathbf{y})\right)-\Phi(X(\cdot, \mathbf{y}))\right|, \rho\left(X_n(\cdot, \mathbf{y}), X(\cdot, \mathbf{y})\right) \leq \delta\right]$ ，对所有人 $\delta>0$. 由于右手边趋于 0 为 $\delta \searrow 0$, 这证明䄳 ${ }^{\mathrm{P} n}[\Phi] \longrightarrow \mathbb{E}^{\mathbb{P} \times}[\Phi]$ 转向最后一个断言，应用前面的来看看，对于每个 $t \geq 0$ 和 $\varphi \in C{\mathrm{b}}\left(\mathbb{R}^N ; \mathbb{R}\right),\left\langle\varphi, P_n(t, \mathbf{x})\right\rangle \longrightarrow\langle\varphi, P(t, \mathbf{x})\rangle$ 均匀地为 $\mathbf{x}$ 在紧失子集中。此外，使用 $(2.2 .2)$ ，可以看 到，对于每个 $T>0$ 和 $R>0$ ，存在一个 $C(T, R)<\infty$ 这样
$$\sup n \geq 0 \sup |\mathbf{x}| \leq R \mathbb{E}^{\mathcal{W}}\left[\left|X_n(t, \mathbf{x})-X_n(s, \mathbf{x})\right|^2\right] \leq C(T, R)(t-s) \quad \text { for } 0 \leq s<t \leq T$$

## 有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。