### 物理代写|电动力学代写electromagnetism代考|PHYSICS2534

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
• (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|Review of Brownian Probability

About half of [MTRV] is taken up with $(5.8),(5.12)$, and related expressions. When motivational and explanatory material is included, these expressions, along with their properties and implications, constitute almost all of the present

book whenever the Feynman quantum mechanical expression (8.29) of Section $8.6$ below is included.
[MTRV] uses the symbol $\mathcal{G}$ for geometric Brownian distribution function (5.12). while the symbol $G_{c}(I[N])$ is used to denote the joint distribution function for $c$-Brownian motion. So $G_{-\frac{1}{2}}$ gives standard or classical Brownian motion (5.8); while $G_{\frac{1}{2}}$ is used in [MTRV] for the Feynman theory of (8.29).
But in this book the symbol $G$ is used without subscript, allowing the context (Brownian motion or quantum mechanics) to show which meaning is intended.
To sum up, the theory of $(5.8),(5.12)$, and $(8.29)$ is covered in chapters 6,7 , and 8 of [MTRV]. It is not proposed to rehearse this theory here; but, instead, to highlight some aspects of it which are particularly relevant to the topics of this book.

In [MTRV], Brownian motion, geometric Brownian motion, and Feynman path integration (for single particle mechanical phenomena) are united in a single theory based on a version of Fresnel’s integral using a parameter $c=a+\iota b$, where $\iota=\sqrt{-1}, a \leq 0$, and $c \neq 0$ (so $a$ and $b$ are not both zero.)

The case $c=-\frac{1}{2}$ (real, negative) leads to Brownian and geometric Brownian motion. The case $c=\frac{\sqrt{-1}}{2}$ (pure imaginary) gives Feynman path integrals. The designation $c$-Brownian motion is intended to cover all cases, including those which are “intermediate” between real negative and pure imaginary.

The Fresnel evaluation is in theorem 133 (pages 261-262 of [MTRV]). For $c<0$ (real, negative) lemmas 12 and 13 (page 262) show that finite compositions (or addition) of normal distributions are normal, so that these distributions are additive in some sense. And provided the real part of $c$ is non-positive (with $c \neq 0$ ), lemmas 12 and 13 are valid for complex-valued $c$.

These results are crucial in going from finite compositions of distributions to infinite compositions, giving a theory of infinitely many (and “infinitely divisible” ${ }^{n}$ or continuum) of normal distributions (or c-normal distributions), leading to the theory of $c$-Brownian motion.

Joint probability distributions are defined on domains $\mathbf{R}^{\mathbf{T}}$ where $\mathbf{T}$ is typically a real interval open on the left and closed on the right, such as
$$] 0, t], \quad] 0, \tau], \quad] \tau^{\prime}, \tau\right] \text {. }$$
A joint distribution such as $(5.8),(5.12)$, or $(8.29)$, is constructed for samples
$$N=\left{t_{1}, t_{2}, \ldots, t_{n-1}, t_{n}\right} \subset \mathbf{T}$$
with $t_{0}$ taken to be the left hand boundary of interval $\mathbf{T}$, and $t_{n}$ the right hand boundary point.

## 物理代写|电动力学代写electromagnetism代考|Brownian Stochastic Integration

A stochastic integral with respect to Brownian processes can have forms
$$\int_{0}^{t} g(s) d X(s), \quad \int_{0}^{t} f(X(s)) d X(s), \quad \int_{0}^{t} Z(s) d X(s) .$$
Each random variable $X(s)(0<s \leq t)$ is normally distributed, so individually they are not too difficult.

But $\int_{0}^{t}$ involves a continuum of such normal distributions. Section $4.4$ mentions step functions, cylinder functions, and sampling functions as a progression of stages in dealing with this problem. This section applies a cylinder function approach to Brownian stochastic integrals. The idea is to replace the continuum ] $0, t]$ by discrete times $0=\tau_{0}<\tau_{1}<\cdots<\tau_{n}=t$. (In Part II, R. Feynman’s path integrals of cylinder functions use a countable infinity of discrete times $\tau_{j}$-)
For $0<t \leq \tau$ let $\mathbf{T}$ denote $] 0, t]$, closed at boundary $t$; and let $T$ denote $] 0, t[$, open at boundary $t$. Suppose an asset price process $X_{T}$ can be represented as
$$X_{\mathrm{T}} \simeq x_{\mathrm{T}}\left[\mathbf{R}^{\mathbf{T}}, G\right]$$

where $x(0)=0$ for all sample paths, and $G$ is the joint probability distribution function $G(I[N])$ of $(5.8)$ for standard Brownian motion. Assume the asset is some portfolio which (unlike shares) can take unbounded positive and negative values. In other words the “asset” (or portfolio) can also be a liability.

A distinction can be made between the value of the portfolio at any time $t$, and the earnings of the portfolio at time $t$.

The latter is intended to denote the stake of the investor or holder of the portfolio, taking account of the initial expenditure (denoted below by $\beta$ ) paid out by the investor in order to acquire possession of the portfolio to begin with.
The former represents a third party view of the portfolio, disregarding any cost of acquisition. If $w(t)$ is the value of the portfolio then earnings equal $w(t)-\beta$ for all $t$; where $\beta$ denotes the upfront cost to the investor of acquiring the portfolio at time $t=0$.

The value of the portfolio at any time $s$ depends on the size $\nu(s)$ of (or number of units of the assets/liabilities in) the portfolio. Then the value of the portfolio at time $s$ is $\nu(s) x(s)$. For the purpose of investigating stochastic integrals, the number $\nu(s)$ can have various interpretations, such as
$$g(s), \quad Z(s), \quad f(X(s))$$
where, for $0 \leq s \leq t, g(s)$ is a deterministic 4 function, $(Z(s))$ is a random process independent of the Brownian motion $(X(s))$, and $(f(X(s)))$ is a process which depends on $(X(s))$.

## 物理代写|电动力学代写electromagnetism代考|Some Features of Brownian Motion

Example 19 above suggests there is a need to consider some extreme behaviour of Brownian paths.

Mathematical Brownian motion is very “bad”. A stereotypical pictorial representation of a sample element of Brownian motion is a “jagged-path” graph consisting of straight line segments adjoining each other consecutively with sharp corners at the points where each one adjoins the next one.

Mathematically, however, a typical sample path is nowhere differentiable. This is much “worse” than the jagged-path graphical representation. Except for their end points, line segments are smooth, or differentiable. So the class of all such jagged paths are a $G$-null subset of the sample space $\Omega=\mathbf{R}^{\mathbf{T}}$.

The reason for this “badness” is that, typically, the increments or transitions $x\left(s^{\prime}\right)-x(s)$ vary as the square root of the time increment $s^{\prime}-s$. Calculating a derivative for $x(s)$ at $s$ involves
$$\frac{x\left(s^{\prime}\right)-x(s)}{s^{\prime}-s}=\frac{1}{\sqrt{s^{\prime}-s}}\left(\frac{x\left(s^{\prime}\right)-x(s)}{\sqrt{s^{\prime}-s}}\right)$$
which diverges as $s^{\prime} \rightarrow s$ for “typical” $x$ of Brownian motion, since the final factor remains finite for such $x$.

From a different perspective, mathematical Brownian motion is very “good”. This is because, typically ${ }^{10}$, its sample paths are uniformly continuous. The reason for this “goodness” is that, typically, the increments or transitions $x\left(s^{\prime}\right)$ $x(s)$ vary as the square root of the time increment $s^{\prime}-s$. So if $s^{\prime} \rightarrow s$ then $\sqrt{s^{\prime}-s} \rightarrow 0$ and hence $x\left(s^{\prime}\right) \rightarrow x(s) .$

These issues are discussed in detail in chapter 6 of [MTRV], and in many other presentations of the subject

Brownian motion includes sample paths which resemble the Dirichlet function of Example 13, and it includes straight lines, and it includes everything in between these two extremes.

In this book stochastic integrals have been presented as some kind of Stieltjes integral, involving integration of one point function $h_{1}(s)$ with respect to a different point function $h_{2}(s)$. (In Section $5.4$ the integrator function $h_{2}\left(s^{\prime}\right)$ $h_{2}(s)$ was supposed to be a Brownian sample path increment $x\left(s^{\prime}\right)-x(s)$; while the integrand function $h_{1}(s)$ was generally designated as $g(s)$.)

A basic Riemann-Stieltjes integral has Riemann sum approximations of the form
$$\sum h_{1}\left(s^{\prime \prime}\right)\left(h_{2}\left(s^{\prime}\right)-h_{2}(s)\right)$$
where $s^{\prime \prime}$ satisfies $s \leq s^{\prime \prime} \leq s^{\prime}$; so $s^{\prime \prime}$ could be taken to be $s$ for every term of the Riemann sum. This fits in with the usual form of the stochastic integral (notably in finance) where $s^{\prime \prime}$ is taken to be the initial $s$ of the time increment $\left[s, s^{\prime}[\right.$.

## 物理代写|电动力学代写electromagnetism代考|Review of Brownian Probability

[MTRV] 使用符号G对于几何布朗分布函数（5.12）。而符号GC(我[ñ])用于表示联合分布函数C-布朗运动。所以G−12给出标准或经典布朗运动 (5.8)；尽管G12在 [MTRV] 中用于 (8.29) 的费曼理论。

] 0, t], \quad] 0, \tau], \quad] \tau^{\prime}, \tau\right] \text {. }] 0, t], \quad] 0, \tau], \quad] \tau^{\prime}, \tau\right] \text {. }

N=\left{t_{1}, t_{2}, \ldots, t_{n-1}, t_{n}\right} \subset \mathbf{T}N=\left{t_{1}, t_{2}, \ldots, t_{n-1}, t_{n}\right} \subset \mathbf{T}

## 物理代写|电动力学代写electromagnetism代考|Brownian Stochastic Integration

∫0吨G(s)dX(s),∫0吨F(X(s))dX(s),∫0吨从(s)dX(s).

X吨≃X吨[R吨,G]

G(s),从(s),F(X(s))

## 物理代写|电动力学代写electromagnetism代考|Some Features of Brownian Motion

X(s′)−X(s)s′−s=1s′−s(X(s′)−X(s)s′−s)

[MTRV] 的第 6 章以及该主题的许多其他介绍中详细讨论了这些问题

∑H1(s′′)(H2(s′)−H2(s))

## 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。