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## 数学代写|矩阵方法代写Applied Matrix Theory代考|Continuous-Time Martingales

Consider a stochastic process with index set $[0, \infty)$ taking values in $\mathbb{R}$ (or $\mathbb{R}^d$ ), i.e., a collection of random variables (or vectors) $\left{X_t\right}_{t \geq 0}$. The process is said to be adapted to a $\sigma$-algebra $\left{\mathscr{F}t\right}{t \geq 0}$ if $X_t$ is $\mathscr{F}t$-measurable for all $t \geq 0$. Let $(\Omega, \mathscr{F}, \mathbb{P})$ denote a basic probability space on which the process $\left{X_t\right}{t \geq 0}$ is defined. Then $\left(\Omega, \mathscr{F},\left{\mathscr{F}t\right}{t \geq 0}, \mathbb{P}\right)$ is called a filtered probability space, where $\mathscr{F}t \subseteq \mathscr{F}$ for all $t \geq 0$. We think of $\mathscr{F}$ as the collection of all possible events, while $\mathscr{F}_t$ contains only all possible events up to time $t$. The filtration $\left{\mathscr{F}_t\right}{t \geq 0}$ is called complete if $\mathscr{F}_0$, and hence all the $\sigma$-algebras including $\mathscr{F}$, contains all the sets $A$ for which $\mathbb{P}(A)=0$. Completion of filtrations is often important in proving that certain properties are valid almost surely.

Definition 2.2.30. Let $\left{\mathscr{F}t\right}{t \geq 0}$ be a filtration. Let $\left{X_t\right}_{t \geq 0}$ be an adapted stochastic process with values in $\mathbb{R}$ such that $\mathbb{E}\left|X_t\right|<\infty$ for all $t \geq 0$. Then we say that $\left{X_t\right}_{t \geq 0}$ is an $\left{\mathscr{F}s\right}{s \geq 0}$
(a) submartingale if
$$s \leq t \Rightarrow X_s \leq \mathbb{E}\left(X_t \mid \mathscr{F}_s\right) \text { a.s. }$$
(b) supermartingale if
$$s \leq t \Rightarrow X_s \geq \mathbb{E}\left(X_t \mid \mathscr{F}_s\right) a . s .$$
(c) martingale if it is both a sub- and supermartingale, i.e., if
$$s \leq t \Rightarrow X_s=\mathbb{E}\left(X_t \mid \mathscr{F}_s\right) \text { a.s. }$$
The condition $\mathbb{E}\left|X_t\right|<\infty$ ensures the existence of the conditional expectations to be used in (b), and it is referred to as the integrability condition.

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Regularization of submartingales

Assume that the filtration $\left{\mathscr{F}t\right}_t \geq 0$ is right continuous and complete, i.e., contains all sets of measure zero. Then the submartingale $\left{X_t\right}{t \geq 0}$ has a càdlàg modification, i.e., with probability one the process is right continuous and has limits from the left.
Proof. From the proof of Theorem 2.2.31 we recall that the total number of upcrossings in $[0, t]$ for fixed $t>0$ satisfies $U<\infty$ a.s. Now let $\Omega_0={\omega \in \Omega \mid U(\omega)<\infty} \subseteq$ $\Omega$. Then both right and left limits exist on this subset. Since $\Omega_0^C$ is a set of measure zero, by completeness it is contained in all $\sigma$-algebras $\mathscr{F}t$ and so is $\Omega_0$. Now define $$\tilde{X}_t(\omega)=\left{\begin{array}{cc} X{t+}(\omega) & \omega \in \Omega_0, \ 0 & \omega \notin \Omega_0 \end{array}\right.$$
Then $\tilde{X}t$ is measurable, since $\Omega_0$ is contained in the $\sigma$-algebras. By definition, $X{t+}$ is right continuous, and since $\Omega_0$ has probability one, $\tilde{X}t$ is a right-continuous modification of $X ;\left{X{t+}\right}$ is an $\left{\mathscr{F}{t+}\right}$-submartingale, and by right continuity of the filtration, $\left{X{t+}\right}$ is hence also an $\left{\mathscr{F}_t\right}$-submartingale. By completeness of the filtration, then $\left{\tilde{X}_t\right}$ is an $\mathscr{F}_t$-submartingale. That the process also possesses left limits at all points is clear from the construction using the upcrossing inequality.

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Continuous-Time Martingales

\left 的分隔符汻失或无法识别 . 据兑该过程适用于 $\sigma$-代数lleft 的分隔符秝失或无法识别

\left 的分隔符缺失或无法识别 称为过澞概

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Regularization of submartingales

\left 的分隔符秝失或无法识别 有一个 càdlàg 修改，即概率为 1 的过程是右连续的并且从左开始有限制。 证明。从定理 2.2.31 的证明中，我们回杝起在 $[0, t]$ 对于固定 $t>0$ 满足 $U<\infty$ 现在让 $\Omega_0=\omega \in \Omega \mid U(\omega)<\infty \subseteq \Omega$. 那 $\angle$ 这 个子集上同时存在左右限制。自从 $\Omega_0^C$ 是一组零测度，通过完整生它包含在所有 $\sigma-$ 代数 $\mathscr{F} t$ 也是如此 $\Omega_0$. 现在定义 $\$ \$$\mid tilde {\mathrm{x}} t(\mid lomega )=\mid left {$$
$$【正确的。 \ \$$

\left 的分隔符缺失或无法识别 是一个〈left 的分隔符午失或无法识别

## 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:Applied Matrix Theory, 数学代写, 矩阵方法

## avatest™帮您通过考试

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## 数学代写|矩阵方法代写Applied Matrix Theory代考|Discrete-Time Martingales

Let $\left{X_n\right}_{n \in \mathbb{N}}$ be a (discrete-time) stochastic process, $\left{\mathscr{F}n\right}{n \in \mathbb{N}}$ a filtration such that $X_n$ is $\mathscr{F}n$-measurable. Then we say that $\left{X_n\right}{n \in \mathbb{N}}$ is adapted to the filtration $\left{\mathscr{F}n\right}{n \in \mathbb{N}}$. This would, for example, be the case if $\mathscr{F}_n=\sigma\left(X_0, X_1, \ldots, X_n\right)$, i.e., the smallest $\sigma$-algebra that makes $X_0, \ldots, X_n$ measurable.

Definition 2.2.1. Let $\left{X_n\right}_{n \in \mathbb{N}}$ be an $\left{\mathscr{F}n\right}$-adapted stochastic process. Assume that $\mathbb{E}\left|X_n\right|<\infty$ for all $n \in \mathbb{N}$. Then we say that $\left{X_n\right}$ is (a) a submartingale if $$X_n \leq \mathbb{E}\left(X{n+1} \mid \mathscr{F}n\right) \quad \text { a.s. },$$ (b) a supermartingale if $$X_n \geq \mathbb{E}\left(X{n+1} \mid \mathscr{F}n\right) \quad \text { a.s. },$$ (c) a martingale if it is both a super- and a submartingale, i.e., $$X_n=\mathbb{E}\left(X{n+1} \mid \mathscr{F}n\right) \text { a.s. }$$ In case of ambiguity, we may explicitly specify the filtration to be used, for example by calling $\left{X_n\right}{n \in \mathbb{N}}$ an $\left{\mathscr{F}n\right}{n \in \mathbb{N}}$-martingale.

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Uniformly Integrable Martingales

The optional stopping theorems can be significantly strengthened by assuming that the martingales involved are uniformly integrable.

Definition 2.2.22. We say that a sub- or supermartingale $\left{X_n\right}_{n \in \mathbb{N}}$ is bounded in $L^1$ if
$\sup n \mathbb{E}\left|X_n\right|<\infty .$ If furthermore, $$\sup _n \mathbb{E}\left(\left|X_n\right| 1\left{\left|X_n\right|>x\right}\right) \rightarrow 0 \text { as } x \rightarrow \infty \text {, }$$ then we say that $\left{X_n\right}$ is uniformly integrable. It is well known (Dunford-Pettis) that uniform integrability provides the strongest possible convergence criterion in the sense that if $X_n \rightarrow X{\infty}$ in probability, then
$X_n \rightarrow X_{\infty}$ in $L^1 \Longleftrightarrow\left{X_n\right}$ is uniformly integrable.
We recall that $X_n \rightarrow X_{\infty}$ in $L^p$ if $\mathbb{E}\left|X_n-X_{\infty}\right|^p \rightarrow 0$ for $n \rightarrow \infty$.

Since we now know that $L^1$ bounded submartingales or supermartingales converge a.s., it follows that uniformly integrable submartingales and supermartingales also converge in $L^1$. In fact, we may conclude that they converge in $L^1$ if and only if they are uniformly integrable, since both of these conditions imply the $L_1$ boundedness conditions of Theorem 2.2.20. We shall now identify uniformly integrable martingales.

## 数学代写矩阵方法代写Applied Matrix Theory代考|Discrete-Time Martingales

$$X_n \leq \mathbb{E}\left(X n+1 \mid \mathscr{F}n\right) \quad \text { a.s. },$$ (b) 上䜯, 如果 $$X_n \geq \mathbb{E}\left(X n+1 \mid \mathscr{F}_n\right) \quad \text { a.s. },$$ $$X_n=\mathbb{E}(X n+1 \mid \mathscr{F} n) \text { a.s. }$$ 如果有歧义，我们可以明确指定要使用的过滤，例如通过调用\left 的分隔符缺失或无法识别 \left 的分隔符缺失或无法识别

## 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:Applied Matrix Theory, 数学代写, 矩阵方法

## avatest™帮您通过考试

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## 数学代写|矩阵方法代写Applied Matrix Theory代考|Discrete Phase-Type Distributions

Let $\left{X_n\right}_{n \in \mathbb{N}}$ be a Markov chain with state space ${1,2, \ldots, p, p+1}$, where the states $1,2, \ldots, p$ are transient, and consequently, state $p+1$ is absorbing. Then $\left{X_n\right}_{n \in \mathbb{N}}$ has a transition matrix $\boldsymbol{P}$ of the form
$$\boldsymbol{P}=\left(\begin{array}{ll} \boldsymbol{T} & \boldsymbol{t} \ \mathbf{0} & 1 \end{array}\right)$$

where $T$ is a $p \times p$ subtransition matrix (i.e., a matrix of nonnegative numbers in which the rows sum to numbers less than or equal to one, written as $\boldsymbol{T e} \leq \boldsymbol{e}$ ), and $\boldsymbol{t}$ is a p-dimensional column vector. Since $t_i$ is the probability of jumping to an absorbing state directly from state $i$, we shall refer to these probabilities as exit probabilities. Since the rows sum to 1 , we must have that
$$\boldsymbol{t}=\boldsymbol{e}-\boldsymbol{T e}=(\boldsymbol{I}-\boldsymbol{T}) \boldsymbol{e},$$
where $\boldsymbol{e}^{\prime}=(1,1, \ldots, 1)$ is the column vector of ones. Thus $\boldsymbol{t}$ can be obtained from $\boldsymbol{T}$ and hence discarded when the necessary parameters are specified. Let $\pi_i=$ $\mathbb{P}\left(X_0=i\right), \boldsymbol{\pi}=\left(\pi_1, \ldots, \pi_p\right)$ and assume that $\boldsymbol{\pi} \boldsymbol{e}=\pi_1+\cdots+\pi_p=1$.

Definition 1.2.54. Let $\tau=\inf \left{n \geq 1 \mid X_n=p+1\right}$ be the time until absorption. Then we say that $\tau$ has a (discrete) phase-type distribution with initial distribution $\pi$ and subtransition matrix $\boldsymbol{T}$, and we write
$$\tau \sim \mathrm{DPH}_p(\boldsymbol{\pi}, \boldsymbol{T})$$

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Markov Jump Processes

In this section we consider Markov processes in continuous time that take values in a discrete (finite or at most countable) state space. By nature, such processes are piecewise constant, and transitions occur via jumps. They are often referred to as Markov jump processes or continuous-time Markov chains. Which value the process takes at the time of a jump can be assigned arbitrarily, however, we will always assume that the process takes the value of the state to which it jumps. This assumption makes Markov jump processes continuous from the right (and with limits from the left), i.e., they are so-called càdlàg processes.

Definition 1.3.1. A continuous-time stochastic process $\left{X_t\right}_t \geq 0$ taking values in a countable set $E$ is called a Markov jump process with state space $E$ if for all $t_n>$ $t_{n-1}>\cdots>t_1>0$ and $i_n, i_{n-1}, \ldots, i_0 \in E$, we have that
$$\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}, \ldots, X_{t_1}=i_1, X_0=i_0\right)=\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}\right) .$$
The process is called time-homogeneous if the transition probabilities $\mathbb{P}\left(X_{t+h}=\right.$ $\left.j \mid X_t=i\right)$ depend only on $h$, in which case it is denoted by $p_{i j}(h)$ and referred to as an $h$-step transition probability. Throughout, we assume that all Markov jump processes are time-homogeneous.
The transition probabilities are then arranged in transition matrices
$$\boldsymbol{P}(h)=\left{p_{i j}(h)\right}_{i, j \in E}, \quad h \geq 0 .$$

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Discrete Phase-Type Distributions

$$\boldsymbol{P}=\left(\begin{array}{lll} \boldsymbol{T} & \boldsymbol{t} 0 & 1 \end{array}\right)$$

$$t=e-T e=(I-T) e,$$
$\mathbb{P}\left(X_0=i\right), \pi=\left(\pi_1, \ldots, \pi_p\right)$ 并假设 $\pi e=\pi_1+\cdots+\pi_p=1$.

$$\tau \sim \operatorname{DPH}p(\pi, \boldsymbol{T})$$

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Markov Jump Processes

$$\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}, \ldots, X_{t_1}=i_1, X_0=i_0\right)=\mathbb{P}\left(X_{t_n}=i_n \mid X_{t_{n-1}}=i_{n-1}\right) .$$

〈left 的分隔符缺失或无法识别

## 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。