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# 金融代写|随机微积分代写Stochastic Calculus代考|IMSE760 Independence

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## 金融代写|随机微积分代写Stochastic Calculus代考|Independence

Two events $A, B$ in a probability space $(\Omega, \mathcal{F}, \mathrm{P})$, i.e. $A, B \in \mathcal{F}$, are said to be independent if
$$\mathrm{P}(A \cap B)=\mathrm{P}(A) \mathrm{P}(B) .$$
For $j=1,2, \ldots m$, let $X_{j}$ be an $\mathbb{R}^{d}$-valued random variable on a probability space $(\Omega, \mathcal{F}, \mathrm{P})$. Then $X_{1}, X_{2}, \ldots X_{m}$ are said to be independent if for all $A_{j} \in \mathcal{B}\left(\mathbb{R}^{d}\right)$, $1 \leq j \leq m$
$$\mathrm{P}\left(X_{j} \in A_{j}, 1 \leq j \leq m\right)=\prod_{j=1}^{m} \mathrm{P}\left(X_{j} \in A_{j}\right)$$
A sequence $\left{X_{n}\right}$ of random variables is said to be a sequence of independent random variables if $X_{1}, X_{2}, \ldots, X_{m}$ are independent for every $m \geq 2$.

Let $\mathcal{G}$ be a sub- $\sigma$-field of $\mathcal{F}$. An $\mathbb{R}^{d}$-valued random variable $X$ is said to be independent of the $\sigma$-field $\mathcal{G}$ if for all $A \in \mathcal{B}\left(\mathbb{R}^{d}\right), D \in \mathcal{G}$,
$$\mathrm{P}({X \in A} \cap D)=\mathrm{P}({X \in A}) \mathrm{P}(D) .$$

## 金融代写|随机微积分代写Stochastic Calculus代考|Filtration

Suppose $X_{n}$ denotes a signal being transmitted at time $n$ over a noisy channel (such as voice over telephone lines), and let $N_{n}$ denote the noise at time $n$ and $Y_{n}$ denote the noise-corrupted signal that is observed. Under the assumption of additive noise, we get
$$Y_{n}=X_{n}+N_{n}, \quad n \geq 0 .$$
Now the interest typically is in estimating the signal $X_{n}$ at time $n$. Since the noise as well the true signal is not observed, we must require that the estimate $\widehat{X}{n}$ of the signal at time $n$ be a function of only observations up to time $n$, i.e. $\widehat{X}{n}$ must only be a function of $\left{Y_{k}: 0 \leq k \leq n\right}$, or $\widehat{X}{n}$ is measurable with respect to the $\sigma$-field $\mathcal{G}{n}=\sigma\left{Y_{k}: 0 \leq k \leq n\right}$. A sequence of random variables $X=\left{X_{n}\right}$ will also be referred to as a process. Usually, the index $n$ is interpreted as time. This is the framework for filtering theory. See Kallianpur [31] for more on filtering theory.
Consider a situation from finance. Let $S_{n}$ be the market price of shares of a company UVW at time $n$. Let $A_{n}$ denote the value of the assets of the company, $B_{n}$ denote the value of contracts that the company has bid and $C_{n}$ denote the value of contracts that the company is about to sign. The process $S$ is observed by the public, but the processes $A, B, C$ are not observed by the public at large. Hence, while making a decision on investing in shares of the company UVW, on the $n$th day, an investor can only use information $\left{S_{k}: 0 \leq k<n\right}$ (we assume that the investment decision is to be made before the price on $n$th day is revealed). Indeed, in trying to find an optimal investment policy $\pi=\left(\pi_{n}\right)$ (optimum under some criterion), the class of all investment strategies must be taken as all processes $\pi$ such that for each $n$, $\pi_{n}$ is a (measurable) function of $\left{S_{k}: 0 \leq k<n\right}$. In particular, the strategy cannot be a function of the unobserved processes $A, B, C$.

Let $\mathcal{G}{n}$ be the $\sigma$-field generated by all the random variables observable before time $n$, namely $S{0}, S_{1}, S_{2}, \ldots, S_{n-1}$. It is reasonable to require that any action to be taken at time $n$ (say investment decision) is measurable with respect to $\mathcal{G}_{n}$. These observations lead to the following definitions.

## 金融代写|随机微积分代写STOCHASTIC CALCULUS代 考|INDEPENDENCE

$$\mathrm{P}(A \cap B)=\mathrm{P}(A) \mathrm{P}(B) .$$

$$\mathrm{P}\left(X_{j} \in A_{j}, 1 \leq j \leq m\right)=\prod_{j=1}^{m} \mathrm{P}\left(X_{j} \in A_{j}\right)$$

$$\mathrm{P}(X \in A \cap D)=\mathrm{P}(X \in A) \mathrm{P}(D) .$$

## 金融代写|随机微积分代写STOCHASTIC CALCULUS代 考|FILTRATION

$$Y_{n}=X_{n}+N_{n}, \quad n \geq 0 .$$

31

weassumethattheinvestmentdecisionistobemadebeforethepriceon $\$ n \ thdayistevealed. 事实上，在试图找到一个最优的投资政策 $\pi=\left(\pi_{n}\right)$ 不能是末观尓到的过程的函数 $A, B, C$.

## 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。