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## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|MARKOV PROCESSES

Markov processes are stochastic processes whose futures are conditionally independent of their pasts given their present values. More formally, a stochastic process $\left{X_t, t \in \mathscr{T}\right}$, with $\mathscr{T} \subseteq \mathbb{R}$, is called a Markov process if, for every $s>0$ and $t$,
$$\left(X_{t+s} \mid X_u, u \leqslant t\right) \quad \sim\left(X_{t+s} \mid X_t\right) .$$
In other words, the conditional distribution of the future variable $X_{t+s}$, given the entire past of the process $\left{X_u, u \leqslant t\right}$, is the same as the conditional distribution of $X_{t+s}$ given only the present $X_t$. That is, in order to predict future states, we only need to know the present one. Property (1.30) is called the Markov property.
Depending on the index set $\mathscr{T}$ and state space $\mathscr{E}$ (the set of all values the $\left{X_t\right}$ can take), Markov processes come in many different forms. A Markov process with a discrete index set is called a Markov chain. A Markov process with a discrete state space and a continuous index set (such as $\mathbb{R}$ or $\mathbb{R}_{+}$) is called a Markov jump process.

## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|Markov Chains

Consider a Markov chain $X=\left{X_t, t \in \mathbb{N}\right}$ with a discrete (i.e., countable) state space $\mathscr{E}$. In this case the Markov property (1.30) is
$$\mathbb{P}\left(X_{t+1}=x_{t+1} \mid X_0=x_0, \ldots, X_t=x_t\right)=\mathbb{P}\left(X_{t+1}=x_{t+1} \mid X_t=x_t\right)$$
for all $x_0, \ldots, x_{t+1}, \in \mathscr{E}$ and $t \in \mathbb{N}$. We restrict ourselves to Markov chains for which the conditional probabilities
$$\mathbb{P}\left(X_{t+1}=j \mid X_t=i\right), \quad i, j \in \mathscr{E}$$
are independent of the time $t$. Such chains are called time-homogeneous. The probabilities in (1.32) are called the (one-step) transition probabilities of $X$. The distribution of $X_0$ is called the initial distribution of the Markov chain. The one-step transition probabilities and the initial distribution completely specify the distribution of $X$. Namely, we have by the product rule (1.4) and the Markov property $(1.30)$
\begin{aligned} &\mathbb{P}\left(X_0=x_0, \ldots, X_t=x_t\right) \ &\quad=\mathbb{P}\left(X_0=x_0\right) \mathbb{P}\left(X_1=x_1 \mid X_0=x_0\right) \cdots \mathbb{P}\left(X_t=x_t \mid X_0=x_0, \ldots X_{t-1}=x_{t-1}\right) \ &\quad=\mathbb{P}\left(X_0=x_0\right) \mathbb{P}\left(X_1=x_1 \mid X_0=x_0\right) \cdots \mathbb{P}\left(X_t=x_t \mid X_{t-1}=x_{t-1}\right) \end{aligned}

## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|MARKOV PROCESSES

〈left 缺少或无法识别的分隔符 ， 和 $\mathscr{T} \subseteq \mathbb{R}$, 被称为马尔可夫过程，如果，对于每个 $s>0$ 和 $t$ ，
$$\left(X_{t+s} \mid X_u, u \leqslant t\right) \quad \sim\left(X_{t+s} \mid X_t\right) .$$

，与条件分布相同
$X_{t+s}$ 只给现在 $X_t$. 也就是说，为了预则末来的状态，我们只需要知道现在的状态。性质 (1.30) 称为马尔可夫性质。

## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|Markov Chains

$$\mathbb{P}\left(X_{t+1}=x_{t+1} \mid X_0=x_0, \ldots, X_t=x_t\right)=\mathbb{P}\left(X_{t+1}=x_{t+1} \mid X_t=x_t\right)$$

$$\mathbb{P}\left(X_{t+1}=j \mid X_t=i\right), \quad i, j \in \mathscr{E}$$

$$\mathbb{P}\left(X_0=x_0, \ldots, X_t=x_t\right) \quad=\mathbb{P}\left(X_0=x_0\right) \mathbb{P}\left(X_1=x_1 \mid X_0=x_0\right) \cdots \mathbb{P}\left(X_t=x_t \mid X_0=x_0, \ldots X_{t-1}=x_{t-1}\right) \quad=\mathbb{P}\left(X_0=x_0\right) \mathbb{P}\left(X_1=x_1 \mid X_0=\right.$$

## 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。

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## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|Linear Transformations

Let $\mathbf{x}=\left(x_1, \ldots, x_n\right)^{\top}$ be a column vector in $\mathbb{R}^n$ and $A$ an $m \times n$ matrix. The mapping $\mathbf{x} \mapsto \mathbf{z}$, with $\mathbf{z}=A \mathbf{x}$, is called a linear transformation. Now consider a random vector $\mathbf{X}=\left(X_1, \ldots, X_n\right)^{\top}$, and let
$$\mathbf{Z}=A \mathbf{X} \text {. }$$
Then $\mathbf{Z}$ is a random vector in $\mathbb{R}^m$. In principle, if we know the joint distribution of $\mathbf{X}$, then we can derive the joint distribution of $\mathbf{Z}$. Let us first see how the expectation vector and covariance matrix are transformed.

Theorem 1.8.1 If $\mathbf{X}$ has an expectation vector $\boldsymbol{\mu}{\mathbf{X}}$ and covariance matrix $\Sigma{\mathbf{X}}$, then the expectation vector and covariance matrix of $\mathbf{Z}=A \mathbf{X}$ are given by
$$\mu_{\mathbf{Z}}=A \mu_{\mathbf{X}}$$
and
$$\Sigma_{\mathbf{Z}}=A \Sigma_{\mathbf{X}} A^{\top} .$$
Proof: We have $\boldsymbol{\mu}{\mathbf{Z}}=\mathbb{E}[\mathbf{Z}]=\mathbb{E}[A \mathbf{X}]=A \mathbb{E}[\mathbf{X}]=A \boldsymbol{\mu}{\mathbf{X}}$ and
\begin{aligned} \Sigma_{\mathbf{Z}} &=\mathbb{E}\left[\left(\mathbf{Z}-\boldsymbol{\mu}{\mathbf{Z}}\right)\left(\mathbf{Z}-\boldsymbol{\mu}{\mathbf{Z}}\right)^{\top}\right]=\mathbb{E}\left[A\left(\mathbf{X}-\boldsymbol{\mu}{\mathbf{X}}\right)\left(A\left(\mathbf{X}-\boldsymbol{\mu}{\mathbf{X}}\right)\right)^{\top}\right] \ &=A \mathbb{E}\left[\left(\mathbf{X}-\boldsymbol{\mu}{\mathbf{X}}\right)\left(\mathbf{X}-\boldsymbol{\mu}{\mathbf{X}}\right)^{\top}\right] A^{\top} \ &=A \Sigma_{\mathbf{X}} A^{\top} \end{aligned}

## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|General Transformations

We can apply reasoning similar to that above to deal with general transformations $\mathbf{x} \mapsto \boldsymbol{g}(\mathbf{x})$, written out as
$$\left(\begin{array}{c} x_1 \ x_2 \ \vdots \ x_n \end{array}\right) \mapsto\left(\begin{array}{c} g_1(\mathbf{x}) \ g_2(\mathbf{x}) \ \vdots \ g_n(\mathbf{x}) \end{array}\right) .$$
For a fixed $\mathbf{x}$, let $\mathbf{z}=\boldsymbol{g}(\mathbf{x})$. Suppose that $\boldsymbol{g}$ is invertible; hence $\mathbf{x}=\boldsymbol{g}^{-1}$ (z). Any infinitesimal $n$-dimensional rectangle at $\mathbf{x}$ with volume $V$ is transformed into an $n$-dimensional parallelepiped at $\mathbf{z}$ with volume $V\left|J_{\mathbf{x}}(\boldsymbol{g})\right|$, where $J_{\mathbf{x}}(\boldsymbol{g})$ is the matrix of Jacobi at $\mathbf{x}$ of the transformation $\boldsymbol{g}$, that is,
$$J_{\mathbf{x}}(\boldsymbol{g})=\left(\begin{array}{ccc} \frac{\partial g_1}{\partial x_1} & \cdots & \frac{\partial g_1}{\partial x_n} \ \vdots & \cdots & \vdots \ \frac{\partial g_n}{\partial x_1} & \cdots & \frac{\partial g_n}{\partial x_n} \end{array}\right) .$$
Now consider a random column vector $\mathbf{Z}=\boldsymbol{g}(\mathbf{X})$. Let $C$ be a small cube around $\mathbf{z}$ with volume $h^n$. Let $D$ be the image of $C$ under $\boldsymbol{g}^{-1}$. Then, as in the linear case,
$$\mathbb{P}(\mathbf{Z} \in C) \approx h^n f_{\mathbf{Z}}(\mathbf{z}) \approx h^n\left|J_{\mathbf{z}}\left(\boldsymbol{g}^{-1}\right)\right| f_{\mathbf{X}}(\mathbf{x}) .$$
Hence we have the transformation rule
$$f_{\mathbf{Z}}(\mathbf{z})=f_{\mathbf{X}}\left(\boldsymbol{g}^{-1}(\mathbf{z})\right)\left|J_{\mathbf{z}}\left(\boldsymbol{g}^{-1}\right)\right|, \quad \mathbf{z} \in \mathbb{R}^n .$$
(Note: $\left.\left|J_{\mathbf{z}}\left(\boldsymbol{g}^{-1}\right)\right|=1 /\left|J_{\mathbf{x}}(\boldsymbol{g})\right| \cdot\right)$

## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|Linear Transformations

$$\mathbf{Z}=A \mathbf{X} .$$

$$\mu_{\mathbf{Z}}=A \mu_{\mathbf{X}}$$

$$\Sigma_{\mathbf{Z}}=A \Sigma_{\mathbf{X}} A^{\top} .$$

$$\Sigma_{\mathbf{Z}}=\mathbb{E}\left[(\mathbf{Z}-\boldsymbol{\mu} \mathbf{Z})(\mathbf{Z}-\boldsymbol{\mu} \mathbf{Z})^{\top}\right]=\mathbb{E}\left[A(\mathbf{X}-\boldsymbol{\mu} \mathbf{X})(A(\mathbf{X}-\boldsymbol{\mu} \mathbf{X}))^{\top}\right]=A \mathbb{E}\left[(\mathbf{X}-\boldsymbol{\mu} \mathbf{X})(\mathbf{X}-\boldsymbol{\mu} \mathbf{X})^{\top}\right] A^{\top}=A \Sigma_{\mathbf{X}} A^{\top}$$

## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|General Transformations

$$\left(x_1 x_2 \vdots x_n\right) \mapsto\left(g_1(\mathbf{x}) g_2(\mathbf{x}) \vdots g_n(\mathbf{x})\right)$$

$$J_{\mathbf{x}}(\boldsymbol{g})=\left(\begin{array}{lllllll} \frac{\partial g_1}{\partial x_1} & \cdots & \frac{\partial g_1}{\partial x_n} & \ldots & \vdots \frac{\partial g_n}{\partial x_1} & \cdots & \frac{\partial g_n}{\partial x_n} \end{array}\right) .$$

$$\mathbb{P}(\mathbf{Z} \in C) \approx h^n f \mathbf{Z}(\mathbf{z}) \approx h^n\left|J_{\mathbf{z}}\left(\boldsymbol{g}^{-1}\right)\right| f_{\mathbf{X}}(\mathbf{x}) .$$

$$f_{\mathbf{Z}}(\mathbf{z})=f_{\mathbf{X}}\left(\boldsymbol{g}^{-1}(\mathbf{z})\right)\left|J_{\mathbf{z}}\left(\boldsymbol{g}^{-1}\right)\right|, \quad \mathbf{z} \in \mathbb{R}^n .$$
(笔记: $\left.\left|J_{\mathrm{z}}\left(\boldsymbol{g}^{-1}\right)\right|=1 /\left|J_{\mathbf{x}}(\boldsymbol{g})\right| \cdot\right)$

## 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:Monte Carlo Method, 数学代写, 蒙特卡罗模拟

## avatest™帮您通过考试

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## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|RANDOM EXPERIMENTS

The basic notion in probability theory is that of a random experiment: an experiment whose outcome cannot be determined in advance. The most fundamental example is the experiment where a fair coin is tossed a number of times. For simplicity suppose that the coin is tossed three times. The sample space, denoted $\Omega$, is the set of all possible outcomes of the experiment. In this case $\Omega$ has eight possible outcomes:
$$\Omega={H H H, H H T, H T H, H T T, T H H, T H T, T T H, T T T},$$
where, for example, HTH means that the first toss is heads, the second tails, and the third heads.

Subsets of the sample space are called events. For example, the event $A$ that the third toss is heads is
$$A={H H H, H T H, T H H, T T H} .$$
We say that event A occurs if the outcome of the experiment is one of the elements in $A$. Since events are sets, we can apply the usual set operations to them. For example, the event $A \cup B$, called the union of $A$ and $B$, is the event that $A$ or $B$ or both occur, and the event $A \cap B$, called the intersection of $A$ and $B$, is the event that $A$ and $B$ both occur. Similar notation holds for unions and intersections of more than two events. The event $A^c$, called the complement of $A$, is the event that $A$ does not occur. Two events $A$ and $B$ that have no outcomes in common, that is, their intersection is empty, are called disjoint events. The main step is to specify the probability of each event.

## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|CONDITIONAL PROBABILITY AND INDEPENDENCE

How do probabilities change when we know that some event $B \subset \Omega$ has occurred? Given that the outcome lies in $B$, the event $A$ will occur if and only if $A \cap B$ occurs, and the relative chance of $A$ occurring is therefore $\mathbb{P}(A \cap B) / \mathbb{P}(B)$. This leads to the definition of the conditional probability of $A$ given $B$ :
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} .$$
For example, suppose that we toss a fair coin three times. Let $B$ be the event that the total number of heads is two. The conditional probability of the event $A$ that the first toss is heads, given that $B$ occurs, is $(2 / 8) /(3 / 8)=2 / 3$.

Rewriting (1.3) and interchanging the role of $A$ and $B$ gives the relation $\mathbb{P}(A \cap$ $B)=\mathbb{P}(A) \mathbb{P}(B \mid A)$. This can be generalized easily to the product rule of probability, which states that for any sequence of events $A_1, A_2, \ldots, A_n$,
$$\mathbb{P}\left(A_1 \cdots A_n\right)=\mathbb{P}\left(A_1\right) \mathbb{P}\left(A_2 \mid A_1\right) \mathbb{P}\left(A_3 \mid A_1 A_2\right) \cdots \mathbb{P}\left(A_n \mid A_1 \cdots A_{n-1}\right),$$
using the abbreviation $A_1 A_2 \cdots A_k \equiv A_1 \cap A_2 \cap \cdots \cap A_k$.
Suppose that $B_1, B_2, \ldots, B_n$ is a partition of $\Omega$. That is, $B_1, B_2, \ldots, B_n$ are disjoint and their union is $\Omega$. Then, by the sum rule, $\mathbb{P}(A)=\sum_{i=1}^n \mathbb{P}\left(A \cap B_i\right)$ and hence, by the definition of conditional probability, we have the law of total probability:
$$\mathbb{P}(A)=\sum_{i=1}^n \mathbb{P}\left(A \mid B_i\right) \mathbb{P}\left(B_i\right)$$

## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|RANDOM EXPERIMENTS

$\Omega=H H H, H H T, H T H, H T T, T H H, T H T, T T H, T T T$

$$A=H H H, H T H, T H H, T T H .$$

## 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|CONDITIONAL PROBABILITY AND INDEPENDENCE

$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} .$$

$$\mathbb{P}\left(A_1 \cdots A_n\right)=\mathbb{P}\left(A_1\right) \mathbb{P}\left(A_2 \mid A_1\right) \mathbb{P}\left(A_3 \mid A_1 A_2\right) \cdots \mathbb{P}\left(A_n \mid A_1 \cdots A_{n-1}\right),$$

$$\mathbb{P}(A)=\sum_{i=1}^n \mathbb{P}\left(A \mid B_i\right) \mathbb{P}\left(B_i\right)$$

## 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。