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# 数学代写|蒙特卡罗模拟代考Monte Carlo Method代考|NE591 MARKOV PROCESSES

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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。