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# 数学代写|随机过程Stochastic Porcesses代考|STAT251b Diffusion Processes

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## 数学代写|随机过程Stochastic Porcesses代考|The Wiener process

We already mentioned the Wiener process twice in Chapter 2. In this section, we will first present a classic way of obtaining this process from a random walk. Then we will give its main properties.

Consider the discrete-time Markov chain $\left{X_n, n=0,1, \ldots\right}$ whose state space is the set of all integers $\mathbf{Z}:={0, \pm 1, \pm 2, \ldots}$ and whose one-step transition probabilities are given by
$$p_{i, i+1}=p_{i, i-1}=1 / 2 \quad \text { for all } i \in \mathbf{Z}$$
This Markov chain is a symmetric random walk (see p. 48). A possible interpretation of this process is the following: suppose that a particle moves randomly among all the integers. At each time unit, for example, each minute, a fair coin is tossed. If “tails” (respectively, “heads”) appears, then the particle moves one integer (that is, one unit of distance) to the right (resp., left).

To obtain the stochastic process called the Brownian motion, we accelerate the random walk. The displacements are made every $\delta$ unit of time, and the distance traveled by the particle is equal to $\epsilon$ unit of distance to the left or to the right, where, by convention, $\delta>0$ and $\epsilon>0$ are real numbers that can be chosen as small as we want. As the Wiener process is a continuous-time and continuous-state process, we will take the limit as $\delta$ and $\epsilon$ decrease to 0 , so that the particle will move continuously, but will travel an infinitesimal distance on each displacement. However, as will be seen subsequently, we cannot allow the constants $\delta$ and $\epsilon$ to decrease to 0 independently from each other; otherwise, the variance of the limiting process is equal either to zero or to infinity, so that this limiting process would be devoid of interest.

## 数学代写|随机过程Stochastic Porcesses代考|Diffusion processes

Continuous-time and continuous-state Markovian processes are, under certain conditions, diffusion processes. The Wiener process is the archetype of this type of process. One way, which can be made even more rigorous, of defining a diffusion process is as follows.

Definition 4.2.1. The continuous-time and continuous-state Markovian stochastic process ${X(t), t \geq 0}$, whose state space is an interval $(a, b)$, is a diffusion process if
$$\left.\lim {\epsilon \downarrow 0} \frac{1}{\epsilon} P \llbracket X(t+\epsilon)-X(t)|>\delta| X(t)=x\right]=0$$ $\forall \delta>0$ and $\forall x \in(a, b)$, and if its infinitesimal parameters defined by (see p. 63) $$m(x ; t)=\lim {\epsilon \downarrow 0} \frac{1}{\epsilon} E[X(t+\epsilon)-X(t) \mid X(t)=x]$$
and
$$v(x ; t)=\lim _{\epsilon \downarrow 0} \frac{1}{\epsilon} E\left[(X(t+\epsilon)-X(t))^2 \mid X(t)=x\right]$$
are continuous functions of $x$ and of $t$.

## 数学代写|随机过程Stochastic processes代考|The Wiener process

. The Wiener process

$$p_{i, i+1}=p_{i, i-1}=1 / 2 \quad \text { for all } i \in \mathbf{Z}$$

## 数学代写|随机过程Stochastic processes代考|Diffusion processes

.

，则为扩散过程$$\left.\lim {\epsilon \downarrow 0} \frac{1}{\epsilon} P \llbracket X(t+\epsilon)-X(t)|>\delta| X(t)=x\right]=0$$ $\forall \delta>0$ 和 $\forall x \in(a, b)$，如果它的无穷小参数定义为(见第63页) $$m(x ; t)=\lim {\epsilon \downarrow 0} \frac{1}{\epsilon} E[X(t+\epsilon)-X(t) \mid X(t)=x]$$

$$v(x ; t)=\lim _{\epsilon \downarrow 0} \frac{1}{\epsilon} E\left[(X(t+\epsilon)-X(t))^2 \mid X(t)=x\right]$$

## 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。