Posted on Categories:Stochastic Porcesses, 数学代写, 随机过程

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

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

Let us consider a continuous time and continuous state space stochastic process ${X(t), t \in T}$ with $T=[0, \infty), S=(-\infty,+\infty)$.

More specifically let ${X(t), t \in T}$ be a Markov process on a probability space $(\Omega, \mathscr{D}, P)$ assuming values in a set $S$. Denote by $\mathscr{L}$ a $\sigma$-algebra of subsets of $S$. The measure space $(S, \mathscr{L})$ will be called the “state space” of the stochastic process $X(t)$. The process $X(t)$ is said to be a homogeneous Markov process if its transition probability has the property $P(X(t+s) \in B \mid X(s)=x)=P(x, t, B)$, $B \in \mathscr{L}$. One may imagine a continuous parameter Markov process ${X(t), t \in T}$ that is not a process with independent increments. Suppose that given $X(s)=x$, for small times $t$, the displacement $X(s+t)-X(s)=X(s+t)-x$ has mean and variance approximately $t \mu(x)$ and $t \sigma^2(x)$, respectively. Here $\mu(x)$ and $\sigma^2(x)$ are functions of the state of $x$, and not constants as in the case of Brownian motion $W(t)$. The distinction between ${W(t)}$ and ${X(t)}$ is analogous to that between simple random walk and a birth-death chain. More precisely, suppose
$$\left.\begin{array}{l} E(X(s+t)-X(s) \mid X(s)=x)=t \mu(x)+o(t) \ E\left(\left(X(s+t)-X(s)^2 \mid X(s)=x\right)=t \sigma^2(x)+o(t)\right. \ E\left(|X(s+t)-X(s)|^3 \mid X(s)=x\right)=o(t) \end{array}\right}$$
hold, as $t \downarrow 0$ for every $x \in \mathscr{L}$.
Note that (8.1) holds for Brownian motion.

## 数学代写|随机过程Stochastic Porcesses代考|Kolmogorov Backward and Forward Diffusion Equations

There are various methods for determining transition probability function or transition distribution of a Markov process, ranging from purely analytical to purely probabilistic. The method presented here was developed by Kolmogorov in 1931.

Theorem 8.1 Let ${X(t), t \in T}$ be a diffusion process. Assuming that $\frac{\partial}{\partial y} F(y, s ; x, t)$ and $\frac{\partial^2}{\partial y^2} F(y, s ; x, t)$ exist and are jointly continuous in all the variables, we get
$$\frac{\partial}{\partial s} F(y, s ; x, t)=-a(y, s) \frac{\partial}{\partial y} F(y, s ; x, t)-\frac{1}{2} b(y, s) \frac{\partial^2}{\partial y^2} F(y, s ; x, t),$$
the Generalized Heat equation or the Backward Kolomogorov’s Differential equation.

Assume that $f(y, s ; x, t)=\frac{\partial}{\partial x} F(y, s ; x, t)$ exists and that $f(x, t)=f(y, s ; x, t)$. Also assume that $\frac{\partial f}{\partial t}, \frac{\partial}{\partial x}[a(x, t) f(x, t)]$ and $\frac{\partial^2}{\partial x^2}[b(x, t) f(x, t)]$ exist and are continuous in all the variables. Then we get
$$\frac{\partial f(x, t)}{\partial t}=-\frac{\partial}{\partial x}[a(x, t) f(x, t)]+\frac{1}{2} \frac{\partial^2}{\partial x^2}[b(x, t) f(x, t)],$$
the Kolmogorov’s forward differential equation or the Fokker-Planck equation. Proof of Backward Equation
\begin{aligned} F(y, s-\Delta s ; x, t)-F(y, s ; x, t)= & \int_{-\infty}^{\infty} F(z, s ; x, t) d F_z(y, s-\Delta s ; z, s) \ & -F(y, s ; x, t) \int_{-\infty}^{\infty} d F_z(y, s-\Delta s ; z, s) \end{aligned}
Let $\quad I=\int_{-\infty}^{\infty}[F(z, s ; x, t)-F(y, s ; x, t)] d F_z(y, s-\Delta s ; z, s)$
By assumptions for $|z-y| \leq \delta$ and for some $\delta>0$,
$$\begin{gathered} F(z, s ; x, t)=F(y, s ; x, t)+(z-y) \frac{\partial F(y, s ; x, t)}{\partial y} \ +\frac{1}{2}(z-y)^2 \frac{\partial^2 F(y, s ; x, t)}{\partial y^2}+o\left((z-y)^2\right) \end{gathered}$$

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

}right 缺少或无法识别的分隔符

## 数学代写|随机过程Stochastic Porcesses代考|Kolmogorov Backward and Forward Diffusion Equations

$$\frac{\partial}{\partial s} F(y, s ; x, t)=-a(y, s) \frac{\partial}{\partial y} F(y, s ; x, t)-\frac{1}{2} b(y, s) \frac{\partial^2}{\partial y^2} F(y, s ; x, t)$$

$$\frac{\partial f(x, t)}{\partial t}=-\frac{\partial}{\partial x}[a(x, t) f(x, t)]+\frac{1}{2} \frac{\partial^2}{\partial x^2}[b(x, t) f(x, t)]$$
Kolmogorov 的正微分方程或 Fokker-Planck 方程。倒向方程的证明
$$F(y, s-\Delta s ; x, t)-F(y, s ; x, t)=\int_{-\infty}^{\infty} F(z, s ; x, t) d F_z(y, s-\Delta s ; z, s) \quad-F(y, s ; x, t) \int_{-\infty}^{\infty} d F_z(y, s-\Delta s ; z, s)$$

$$F(z, s ; x, t)=F(y, s ; x, t)+(z-y) \frac{\partial F(y, s ; x, t)}{\partial y}+\frac{1}{2}(z-y)^2 \frac{\partial^2 F(y, s ; x, t)}{\partial y^2}+o\left((z-y)^2\right)$$

## 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。

Posted on Categories:Stochastic Porcesses, 数学代写, 随机过程

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|随机过程Stochastic Porcesses代考|Birth and Death Processes

Definition 1 A birth and death process is a conservative M.P. with infinitesimal generator
$$A=\left(\begin{array}{cccccc} -\lambda_0 & \lambda_0 & 0 & 0 & 0 & \cdots \ \mu_1 & -\left(\lambda_1+\mu_1\right) & \lambda_1 & 0 & 0 & \cdots \ 0 & \mu_2 & -\left(\mu_2+\lambda_2\right) & \lambda_2 & 0 & \cdots \ \vdots & \vdots & \vdots & \vdots & & \end{array}\right)$$
where $\lambda_i$ and $\mu_j, i=0,1,2, \ldots, j=1,2,3 \ldots$ are the birth and death rates, respectively.

Definition 2 (Karlin) A birth and death process is a continuous time Markov process on the state space $S={0,1,2, \ldots}$ with stationary transition functions $\left.p_{i j}(t)=P(X(t+s))=j \mid X(s)=i\right)$ and such that
(i) $p_{i j}(0)=\delta_{i j}$
(ii) $P_{i i+1}(h)=\lambda_r h+o(h)$ for $i \geq 0$ and $h \downarrow 0$
(iii) $p_{i i-1}(h)=\mu_i h+o(h)$ for $i \geq 0$ and $h \downarrow 0$
(iv) $p_{i i}(h)=1-\left(\lambda_i+\mu_i\right) h+o(h)$ for $i \geq 0$ and $h \downarrow 0$ and $p_{i j}(h)=o(h)$ for $j \neq i+1, i-1, i$ as $h \downarrow 0$ and all $i \geq 1$.
(v) $\lambda_i>0$ for $i \geq 0$ and $\mu_i>0$ for $i \geq 1$ and $\mu_0=0$ are the birth and death rates, respectively. $\lambda_i$ and $\mu_i$ are also called parameters of birth and death processes.
Note These two definitions can be proved to be identical. The backward Kolmogorov equation will be given by
$$p_{i j}^{\prime}(t)=\mu_i p_{i-1, j}(t)-\left(\lambda_j+\mu_i\right) p_{i j}(t)+\lambda_i p_{i+1, j}(t), i \geq 1$$
and $p_{o j}^{\prime}(t)=-\lambda_0 p_{o j}(t)+\lambda_0 p_{1 j}(t)$ with initial conditions $p_{i j}(0)=\delta_{i j}$. The forward Kolmogorov equation for birth and death processes are
$$p_{i j}^{\prime}(t)=\lambda_{j-1} p_{i, j-1}(t)-\left(\lambda_j+\mu_j\right) p_{i j}(t)+\mu_{j+1} p_{i, j+1}(t), j \geq 1$$
and $p_{i o}^{\prime}(t)=-\lambda_0 p_{i \omega}(t)+\mu_1 p_{i 1}(t)$ with initial conditions $p_{i j}(0)=\delta_{i j}$.

## 数学代写|随机过程Stochastic Porcesses代考|The Yule Process

This a pure birth process (that arises in physics and biology) with parameters $\lambda_i=\beta(N+i)$ and $\mu_i=0, \beta>0$. Kolmogorov’s backward equation becomes (putting $i=n$ )
$$p_{N+n}^{\prime}(t)=-\beta(N+n) p_{N+n}(t)+\beta(N+n-1) p_{N+n-1}(t), n \geq 1$$
and
$$p_N^{\prime}(t)=-\beta N p_N(t)$$
where $p_{N+n}(t)=p_{N, N+n}(t)=P(X(t)=N+n / x(0)=N)$.
If $N=1,(6.22)$ becomes
$$p_{n+1}(t)-\beta(n+1) p_{n+1}(t)+\beta n p_n(t)$$
with initial conditions $p_1(0)=1, p_n(0)=0, n=2,3, \ldots$.
By induction, $p_1^{\prime}(t)=-\beta p_1(t)$ implies $p_1(t)=c e^{-\beta t}$ and initial conditions give the solution of (2) as
$$p_n(t)=e^{-\beta t}\left(1-e^{-\beta t}\right)^{n-1}, n \geq 0 .$$
To get a general solution of (6.22) (i.e. when $N>1$ ) we shall first express the Kolmogorov’s differential equation for the Yule process as a partial differential equation in terms of the G.F. of the process.

## 数学代写|随机过程Stochastic Porcesses代考|Birth and Death Processes

(i) $p_{i j}(0)=\delta_{i j}$
(二) $P_{i i+1}(h)=\lambda_r h+o(h)$ 为了 $i \geq 0$ 和 $h \downarrow 0$
(三) $p_{i i-1}(h)=\mu_i h+o(h)$ 为了 $i \geq 0$ 和 $h \downarrow 0$
(四) $p_{i i}(h)=1-\left(\lambda_i+\mu_i\right) h+o(h)$ 为了 $i \geq 0$ 和 $h \downarrow 0$ 和 $p_{i j}(h)=o(h)$ 为了 $j \neq i+1, i-1, i$ 作为 $h \downarrow 0$ 和所有 $i \geq 1$.
(在) $\lambda_i>0$ 为了 $i \geq 0$ 和 $\mu_i>0$ 为了 $i \geq 1$ 和 $\mu_0=0$ 分别是出生率和死亡率。 $\lambda_i$ 和 $\mu_i$ 也称为出生和死亡过程的参数。

$$p_{i j}^{\prime}(t)=\mu_i p_{i-1, j}(t)-\left(\lambda_j+\mu_i\right) p_{i j}(t)+\lambda_i p_{i+1, j}(t), i \geq 1$$

$$p_{i j}^{\prime}(t)=\lambda_{j-1} p_{i, j-1}(t)-\left(\lambda_j+\mu_j\right) p_{i j}(t)+\mu_{j+1} p_{i, j+1}(t), j \geq 1$$

## 数学代写|随机过程Stochastic Porcesses代考|The Yule Process

Kolmogorov 的逆向方程变为 (将 $i=n$ )
$$p_{N+n}^{\prime}(t)=-\beta(N+n) p_{N+n}(t)+\beta(N+n-1) p_{N+n-1}(t), n \geq 1$$

$$p_N^{\prime}(t)=-\beta N p_N(t)$$

$$p_{n+1}(t)-\beta(n+1) p_{n+1}(t)+\beta n p_n(t)$$

$$p_n(t)=e^{-\beta t}\left(1-e^{-\beta t}\right)^{n-1}, n \geq 0 .$$

## 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。