### 统计代写|随机过程代写stochastic process代考|Multiplicative Additive Functionals, Excessive Functions

statistics-lab™ 为您的留学生涯保驾护航 在代写随机过程stochastic process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过程stochastic process代写方面经验极为丰富，各种代写随机过程stochastic process相关的作业也就用不着说。

## 统计代写|随机过程代写stochastic process代考|Multiplicative Additive Functionals, Excessive Functions

Definition and basic properties of additive and multiplicative functionals. Let $\left{\mathscr{F}, \mathcal{N}, \mathrm{P}_x\right}$ be a homogeneous Markov process in a certain phase space ${\mathscr{X}, \mathfrak{B}}$. We shall consider special families of $\mathscr{N}_t$-measurable numerical variables. Since they are $\mathscr{N}_t$-measurable the shift-operator $\theta_h$ is applicable to them.

A family of variables $\alpha_t$ defined for $t \geqslant 0$ on $\mathscr{F}$ is called a homogeneous multiplicative functional if it satisfies the conditions:
M1) $\alpha_t$ is measurable with respect to $\mathscr{N}t$ for all $t \geqslant 0$, M2) for all $t \geqslant 0$ and $h>0$ relation $\alpha{t+h}=\alpha_h \theta_h \alpha_t$ is satisfied for all $x \in \mathscr{X}$ with probability $P_x=1$.

A family of variables $\varphi_t$ defined for $t \geqslant 0$ on $\mathscr{F}$ is called a homogeneous additive functional provided:
A 1) for all $t \geqslant 0$ the variable $\varphi_t$ is measurable with respect to $\mathcal{N}t$. A 2) for all $t \geqslant 0$ and $h>0$ relation $\varphi{t+h}=\varphi_h+\theta_h \varphi_t$ is satisfied for all $x \in \mathscr{X}$ with probability $P_x=1$.
We present examples of multiplicative functionals.
I. Let $\mathscr{X}$ be a topological space and let the sample function of the process $x(t)$ be continuous from the right, and let $g(x)$ be a bounded continuous function on $\mathscr{x}$. Then for all $t \geqslant 0$ the quantity
$$\int_0^t g\left(x_s\right) d s$$

is defined and is $\mathscr{N}t$-measurable. Set $$\alpha_t=\exp \left{\int_0^t g\left(x_s\right) d s\right}$$ Since and $$\theta_h \alpha_t=\exp \left{\int_0^t \theta_h g\left(x_s\right) d s\right}=\exp \left{\int_h^{t+h} g\left(x_s\right) d s\right}$$ $$\alpha_h \theta_h \alpha_t=\exp \left{\int_0^h g\left(x_s\right) d s\right} \exp \left{\int_h^{t+h} g\left(x_s\right) d s\right}=\alpha{t+h},$$
it follows that $\alpha_t$ is a multiplicative functional.

## 统计代写|随机过程代写stochastic process代考|Random time substitution

(This quantity is well defined if $\varphi(+\infty)>t$.) Thus we have a family of random processes: a random process $x\left(\tau_t\right)$ on the probability space $\left{\mathscr{F}, \mathcal{N}, \mathrm{P}_x\right}$ corresponds to each $x$. It turns out that this family is a Markov family of random processes. We construct a homogeneous Markov process with marginal distributions of the following structure: given that the initial value of the process is $x$, the marginal distributions of the process coincide with the marginal distributions of the process $x\left(\tau_t\right)$ on the probability space $\left{\mathscr{F}, \mathcal{N}, \mathrm{P}_x\right}$.

We introduce the set of functions $\mathscr{F}: x(t)=x(\tau)$ on $[0, \tilde{\zeta})$ where the variables $\tau_t$ are defined by relation (25) for all $x(\cdot) \in \mathscr{F}$ and $t \in[0, \tilde{\zeta}), \tilde{\zeta}=\varphi(+\infty)$. Denote by $\tilde{\mathcal{N}}$ the $\sigma$-algebra generated by the variables $\tilde{x}(t)$, by $\tilde{N}t$ the $\sigma$-algebra generated by $\tilde{x}(s)$ for $s \leqslant t$. Finally denote by $\tilde{\mathrm{P}}_x$ the measure on $\mathscr{N}$ defined by relation $$\tilde{\mathrm{P}}_x(C)=\mathrm{P}_x\left{x\left(\tau_0\right) \in C\right}$$ for any cylinder $C$ in $\mathscr{N}$; since $x\left(\tau_t\right)$ is an $\mathscr{N}$-measurable variable it follows that the set $\left{x(\cdot): x\left(\tau_0\right) \in C\right}$ is $\mathcal{N}$-measurable for any cylinder $C$. To verify that $\left{\tilde{\mathscr{F}}, \tilde{\mathcal{N}}, \tilde{\mathrm{P}}_x\right}$ is a Markov process it is sufficient to show that $$\tilde{\mathrm{P}}_x\left{\tilde{x}(t+h) \in B \mid \tilde{\mathscr{N}}_t\right}=\tilde{\mathrm{P}}{\tilde{x}(t)}{\tilde{x}(h) \in B}$$
with probability $\mathrm{P}x=1$ for any $x \in \mathscr{X}, B \in \mathcal{B}, t>0$ and $h>0$. Relation (26) is equivalent to the following: $$\mathrm{P}_x\left{x\left(\tau{t+h}\right) \in B \mid \mathcal{N}{\tau_t}\right}=\mathrm{P}{x\left(\tau_t\right)}\left{x\left(\tau_h\right) \in B\right}$$
with probability $P_x=1$. It follows from the strong Markov property of the process $\left{\mathscr{F}, \mathcal{N}, P_x\right}$ that $(27)$ is satisfied provided
$$\theta_{\tau_t}\left[x\left(\tau_h\right)\right]=x\left(\tau_{h+t}\right)$$
with probability $P_x=1$.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Multiplicative Additive Functionals, Excessive Functions

1)对于所有$t \geqslant 0$，变量$\varphi_t$相对于$\mathcal{N}t$是可测量的。A 2)对于所有$t \geqslant 0$和$h>0$关系$\varphi{t+h}=\varphi_h+\theta_h \varphi_t$，对于所有$x \in \mathscr{X}$都以$P_x=1$的概率满足。

1、设$\mathscr{X}$为拓扑空间，设过程$x(t)$的样本函数从右连续，设$g(x)$为$\mathscr{x}$上的有界连续函数。然后对于所有$t \geqslant 0$数量
$$\int_0^t g\left(x_s\right) d s$$

## 统计代写|随机过程代写stochastic process代考|Random time substitution

(这个数量定义得很好，如果$\varphi(+\infty)>t$。)这样我们就有了一组随机过程:概率空间$\left{\mathscr{F}, \mathcal{N}, \mathrm{P}_x\right}$上的一个随机过程$x\left(\tau_t\right)$对应于每个$x$。这个族是随机过程的马尔可夫族。我们构造了一个齐次马尔可夫过程，其边缘分布具有如下结构:给定过程的初值为$x$，过程的边缘分布与过程$x\left(\tau_t\right)$在概率空间$\left{\mathscr{F}, \mathcal{N}, \mathrm{P}_x\right}$上的边缘分布重合。

## 有限元方法代写

tatistics-lab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。