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# 数学代写|随机过程代写Stochastic Porcess代考|Basic Definitions

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## 数学代写|随机过程代写Stochastic Porcess代考|Basic Definitions

In the previous section a general definition of a Markov process was given and the basic properties of these processes were studied. The present chapter is devoted to the most important class of Markov processes-the homogeneous (or more precisely homogeneous in time) Markov processes. Roughly speaking a Markov process is homogeneous if its transition probabilities $P(t, x, s, B)$ depends only on the difference $s-t$. However, in the modern theory of Markov processes a more restrictive definition is given. This definition involves certain restrictions on the set of sample functions of the processes also. We shall, however, utilize an even more restrictive definition which is convenient for solving basic problems of the theory and which is at the same time a natural definition for the construction of a process with a given transition probability. The basic difference between the definition presented below and the general definition is that here we consider Markov processes with respect to a current of $\sigma$-algebras generated by the values of the sample functions of the process and not with respect to an arbitrary current of $\sigma$-algebras.
First we shall present a definition for the case of a non-cut-off Markov process.
A homogeneous Markov process is defined if the following objects are given:
A certain measurable space ${\mathscr{X}, \mathfrak{B}}$ which is called the phase space of the process is defined.
Next is defined a space of trajectories (sample functions) $\mathscr{F}$-a certain set of functions $x(t)$ defined on $[0, \infty)$ and taking on values in $\mathscr{X}-$ (note that $\mathscr{F}$ does not necessarily coincide with the set of all functions defined on $[0, \infty)$ with values in $\mathscr{X})$; it is assumed that the set $\mathscr{F}$ satisfies the condition: for all $x(\cdot) \in \mathscr{F}$ and $h>0$ the function $\theta_h x(\cdot)$ defined by the relation $\theta_h x(t)=x(t+h)$ also belongs to $\mathscr{F}$.
Operators $\theta_h$ defined in this manner are called shift operators. Clearly these operators form a semigroup: $\theta_{h+s}=\theta_h \theta_s$.

## 数学代写|随机过程代写Stochastic Porcess代考|The Resolvent and the Generating Operator of a Weakly Measurable Markov Process

The transition probability $P(t, x, B)$ of a Markov process in the phase space ${\mathscr{X}, \mathfrak{B}}$ is called measurable if for each $B \in \mathfrak{B}$, this probability is a measurable function of the pair $(t, x)$ on the product of measurable spaces $\left{\mathscr{R}{+}, \mathfrak{A}{+}\right} \times{\mathscr{X}, \mathfrak{B}}$, where $\mathscr{R}{+}=[0, \infty)$ and $\mathfrak{U}{+}$is the $\sigma$-algebra of Borel sets in $\mathscr{R}_{+}$.

A homogeneous Markov process is called weakly measurable if its transition probability is measurable. In this section only weakly measurable processes are considered.

Let $\mathbf{T}t$ be the semi-group of operators associated with a weakly measurable process. Define the resolvent of the process $\mathbf{R}\lambda$ as the family of operator on $B$ defined for all complex $\lambda$ such that $\operatorname{Re} \lambda>0$ by formula
$$\mathbf{R}_\lambda f(x)=\int_0^{\infty} e^{-\lambda t} \int f(y) P(t, x, d y) d t=\int_0^{\infty} e^{-\lambda t} \mathbf{T}_t f(x) d t$$

It is easy to verify that $\mathbf{R}\lambda f(x)$ is a complex-valued measurable function. If $\mathrm{B}^$ denotes the space of complex valued $\mathfrak{B}$-measurable bounded functions $g(x)$ defined on $\mathscr{X}$ with the norm $|g|=\sup _x|g(x)|$ then $\mathbf{R}\lambda$ becomes a linear operator acting from $B$ into $B^$.
We also define the resolvent kernel by
$$R_\lambda(x, B)=\int_0^{\infty} e^{-\lambda t} P(t, x, B) d t$$
The function $R_\lambda(x, B)$, for each $x$ and $\lambda(\operatorname{Re} \lambda>0)$ is a complex-valued countably additive set function on $\mathfrak{B}$. Therefore, the integral
$$\int f(y) R_\lambda(x, d y)$$
is defined for all $f \in \mathrm{B}$. This integral is a linear bounded operator acting from $B$ into $B^*$. Since this operator coincides with $\mathbf{R}\lambda$ on the indicators of the sets in $\mathfrak{B}$, we deduce that $$\mathbf{R}\lambda f(x)=\int f(y) R_\lambda(x, d y) .$$

# 随机过程代写

## 数学代写|随机过程代写Stochastic Porcess代考|The Resolvent and the Generating Operator of a Weakly Measurable Markov Process

$$\mathbf{R}_\lambda f(x)=\int_0^{\infty} e^{-\lambda t} \int f(y) P(t, x, d y) d t=\int_0^{\infty} e^{-\lambda t} \mathbf{T}_t f(x) d t$$

$$\int f(y) R_\lambda(x, d y)$$

## 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。