### 物理代写|电动力学代写electromagnetism代考|PHYS2213

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电动力学代写electromagnetism代考|A Basic Stochastic Integral

The following is similar to Example $2 .$
Example 5 Suppose $s=1,2,3, \ldots$ is time, measured in days. Suppose a share, or unit of stock, has value $x(s)$ on day s; suppose $z(s)$ is the number of shares held on day $s$; and suppose $c(s)$ is the change in the value of the shareholding on day $s$ as a result of the change in share value from the previous day so $c(s)=$ $z(s-1)(x(s)-x(s-1))$. Let $w(s)$ be the cumulative change in shareholding value at end of day $s$, so $w(s)=w(s-1)+c(s)$. If share value $x(s)$ and stockholding $z(s)$ are subject to random variability, how is the gain (or loss) from the stockholding to be estimated?

Take initial value (at time $s=0)$ of the share to be $x(0)$ (or $\left.x_{0}\right)$, take the initial shareholding or number of shares owned to be $z(0)$ (or $\left.z_{0}\right)$. Then, at end of day $1(s=1)$,
$$c(1)=z(0) \times(x(1)-x(0)), \quad w(1)=w(0)+c(1)=c(1)$$
At end of day $s$,
$$c(s)=z(s-1) \times(x(s)-x(s-1)), \quad w(s)=w(s-1)+c(s)$$
After $t$ days,
$$w(t)=\sum_{s=1}^{t} z(s-1)(x(s)-x(s-1)) .$$
If the time increments are reduced to arbitrarily small size (so $s$ represents number of “time ticks” -fractions of a second, say), with the meaning of the other variables adjusted accordingly, then
$$w(t)=\sum_{j=1}^{n} z\left(s_{j-1}\right)\left(x\left(s_{j}\right)-x\left(s_{j-1}\right)\right), \quad \text { or } \quad w(t)=\sum z(s) \Delta x(s)$$
The latter expressions are Riemann sum estimates of $\int_{0}^{t} z(s) d x(s)$ (a Stieltjestype integral) whenever the latter exists.
Each of the expressions in (2.4) is sample value of a random variable
$$W(t)=\sum_{j=1}^{n} Z\left(s_{j-1}\right)\left(X\left(s_{j}\right)-X\left(s_{j-1}\right)\right) \text { or } \int_{0}^{t} Z(s) d X(s)$$constructed from the random variables $X, Z$, and $W$. These notations symbolize in a “naive” or “realistic” way-the stochastic integral of the process $Z$ with respect to the process $X$. In chapter 8 of [MTRV], symbols s, or $\mathbf{S}$, or $\mathcal{S}$ are used (in place of the symbol $\int$ ) for various kinds of stochastic integral. In the context described here, S would be the appropriate notation. (See (5.28) below.)

## 物理代写|电动力学代写electromagnetism代考|Choosing a Sample Space

It was mentioned earlier that there are many alternative ways of producing a sample space $\Omega$ (along with the linked probability measure $P$ and family $\mathcal{A}$ of measurable subsets of $\Omega$ ). The set of numbers
$${-5,-4,-3,-2,-1,0,1,2,3,4,5,10}$$
was used as sample space for the random variability in the preceding example of stochastic integration. The measurable space $\mathcal{A}$ was the family of all subsets of $\Omega$, and the example was illustrated by means of two distinct probability measures $P$, one of which was based on Up and Down transitions being equally likely, where for the other measure an Up transition was twice as likely as a Down.
An alternative sample space for this example of random variability is
$$\Omega=\Omega \Omega_{1} \times \Omega \Omega_{2} \times \Omega \Omega_{3} \times \Omega \Omega_{4}$$
where $\Omega_{j}={U, D}$ for $j=1,2,3,4$; so the elements $\omega$ of $\Omega$ consist of sixteen 4-tuples of the form
$$\omega=(\cdot, \cdot \cdot \cdot), \quad \text { such as } \omega=(U, D, D, U) \text { for example. }$$
Let the measurable space $\mathcal{A}$ be the family of all subsets $A$ of $\Omega$; so $\mathcal{A}$ contains $2^{16}$ members, one of which (for example) is
$$A={(D, U, U, D),(U, D, D, U),(D, U, D, U),(U, U, U, U),(D, D, D, D)}$$

with $A$ consisting of five individual four-tuples. Assume that Up transitions and Down transitions are equally likely, and that they are independent events. Then, as before,
$$P({\omega})=\frac{1}{16}$$
for each $\omega \in \Omega$. For $A$ above, $P(A)=\frac{5}{16}$.
To relate this probability structure to the shareholding example, let $\mathbf{R}^{4}=$ $\mathbf{R} \times \mathbf{R} \times \mathbf{R} \times \mathbf{R}$, and let
$$f: \Omega \mapsto \mathbf{R}^{4}, \quad f(\omega)=((x(1), x(2), x(3), x(4)),$$
using Table 2.4; so, for instance,
$$f(\omega)=f((U, D, D, U))=(11,10,9,10)=(x(1), x(2), x(3), x(4)),$$
and so on. Next, let $\mathbf{S}$ denote the stochastic integrals of the preceding section, so for $x=(x(1), x(2), x(3), x(4)) \in \mathbf{R}^{4}$,
$$\mathbf{S}(x)=\int_{0}^{4} z(s) d x(s)=\sum_{s=1}^{4} z(s-1)(x(s)-x(s-1)),$$
so $\mathbf{S}(x)$ gives the values $w(4)$ of Table 2.4. As described in Section $2.3$, the rationale for deducing the probabilities of outcomes $\mathbf{S}(x)$, = $w(4)$, from the probabilities on $\Omega$ is the relationship
$$P(w(4))=P\left(f^{-1}\left(\mathbf{S}^{-1}(w(4))\right)\right) .$$

## 物理代写|电动力学代写electromagnetism代考|More on Basic Stochastic Integral

The constructions in Sections $2.3$ and $2.4$ purported to be about stochastic integration. While a case can be made that (2.6) and (2.7) are actually stochastic integrals, such simple examples are not really what the standard or classical theory of Chapter 1 is all about. The examples and illustrations in Sections $2.3$ and $2.4$ may not really be much help in coming to grips with the standard theory of stochastic integrals outlined in Chapter $1 .$

This is because Chapter 1, on the definition and meaning of classical stochastic integration, involves subtle passages to a limit, whereas (2.6) and (2.7) involve only finite sums and some elementary probability calculations.

From the latter point of view, introducing probability measure spaces and random-variables-as-measurable-functions seems to be an unnecessary complication. So, from such a straightforward starting point, why does the theory become so challenging and “messy”, as portrayed in Chapter $1 ?$

As in Example 2, the illustration in Section $2.3$ involves dividing up the time period (4 days) into 4 sections; leading to sample space $\Omega=\mathbf{R}^{4}$ in (2.15). Why not simply continue in this vein, and subdivide the time into 40 , or 400 , or 4 million steps instead of just 4 ; using sample spaces $\mathbf{R}^{40}$, or $\mathbf{R}^{400}$, or $\mathbf{R}^{4000000}$, respectively? The computations may become lengthier, but no new principle is involved; each of the variables changes in discrete steps at discrete points in time. ${ }^{5}$

Other simplifications can be similarly adopted. For instance, only two kinds of changes are contemplated in Section 2.3: increase (Up) or decrease (Down).

## 物理代写|电动力学代写electromagnetism代考|A Basic Stochastic Integral

C(1)=和(0)×(X(1)−X(0)),在(1)=在(0)+C(1)=C(1)

C(s)=和(s−1)×(X(s)−X(s−1)),在(s)=在(s−1)+C(s)

（2.4）中的每个表达式都是随机变量的样本值

## 物理代写|电动力学代写electromagnetism代考|Choosing a Sample Space

−5,−4,−3,−2,−1,0,1,2,3,4,5,10

Ω=ΩΩ1×ΩΩ2×ΩΩ3×ΩΩ4

ω=(⋅,⋅⋅⋅), 如 ω=(在,D,D,在) 例如。

F:Ω↦R4,F(ω)=((X(1),X(2),X(3),X(4)),

F(ω)=F((在,D,D,在))=(11,10,9,10)=(X(1),X(2),X(3),X(4)),

## 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。