物理代写|电动力学代写electromagnetism代考|PHYC20011

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• 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 Integrability Issues

In the course of the preceding discussion, while some challenging features were encountered, there were occasions when integrability was fairly easily established.

Starting with some of the more troublesome issues, the sample paths $(x(s))$ of the price process $X_{\mathbf{T}}$ in Examples 15 and 16 include paths whose extreme oscillation mirrors that of the Dirichlet function $d(s)$ of Example 13 .
Here are some further issues:

• Suppose a domain $\Omega$ can be partitioned into sub-domains $\Omega_{j}$, and suppose an integrand $g$ is a step function taking constant values $\kappa_{j}$ in domain $\Omega_{j}$ for each $j$. Then, even if $\kappa_{j}$ is integrable on $\Omega_{j}$ for each $j$, it is not necessarily the case that $f$ is integrable on $\Omega$.
• If a sequence of such step functions converges pointwise to a function $f$ it is not necessarily the case that $f$ is integrable.
• Dirichlet-type oscillation can occur in the sample functions $x_{\mathrm{T}}$ of a stochastic process $X_{T}$. This phenomenon presents integrability problems.

On the other hand, integrals on infinite-dimensional domains sometimes reduce to more familiar finite-dimensional integrals. Some aspects of this phenomenon can be summarized as follows. Suppose $\mathbf{T}$ is an infinite labelling set such as ] $0, t]$, and suppose

• $x_{\mathrm{T}} \in \mathbf{R}^{\mathbf{T}}$
• $f\left(x_{\mathbf{T}}\right)$ is an integrand in $\mathbf{R}^{\mathbf{T}}$
• $F(I)$ is an integrator function defined on the cells $I$ of $\mathbf{R}^{\mathrm{T}}$.
The integral on $\mathbf{R}^{\mathbf{T}}$ of $f\left(x_{\mathbf{T}}\right)$ with respect to $F(I)$ (if it exists) is $\int_{\mathbf{R}^{\mathbf{T}}} f\left(x_{\mathbf{T}}\right) F(I)$, which for present purposes can be denoted as $\int_{\mathbf{R}^{\infty}} f(x) d F$.

When $\mathbf{T}$ is infinite (that is, when $\mathbf{T}$ has infinite cardinality) the cells $I$ are cylindrical, as indicated in the notation $I=I[N]$ for finite subsets $N=\left{t_{1}, \ldots, t_{n}\right}$ of T. Accordingly, some aspects of the finite Cartesian product
$$\mathbf{R}^{n},=\mathbf{R} \times \cdots \times \mathbf{R}=\mathbf{R}{t{1}} \times \cdots \times \mathbf{R}{t{n}}$$
already make an appearance in the integration.

物理代写|电动力学代写electromagnetism代考|Introduction to Brownian Motion

Section $5.6$ below provides a summary of various different kinds of stochastic integral, whose intuitive meaning can be obtained from the elementary examples and illustrations in preceding chapters, and which are presented here in terms of the theory provided in [MTRV].

The stochastic processes of the previous sections are somewhat artificial and selective. They were chosen because they are fairly easily intelligible and relatively straightforward.

Nevertheless, their simpler and more easily formulated scenarios are not necessarily the most manageable in mathematical terms -because, for instance, of Dirichlet oscillation as illustrated in Example 13. Also, the preceding examples, though they may help to provide a feel for the subject, are not the kind of processes which are important in practice.

One of the most important stochastic processes is Brownian motion. It is not so easy to formulate; it reflects some of the complexity of real random phenomena. Nonetheless it is relatively amenable to some well-established mathematical techniques.

Before actually defining Brownian motion, Example 17 below is a version of it which demonstrates how the Dirichlet-type oscillation of Examples 12,15 , 16 may be evaded. It is intended to be a bridge joining those examples to the standard Brownian motion to be discussed in this chapter.

The preceding examples are located-like Brownian motion-in domains of the form $\mathbf{R}^{T}$. But this was somewhat artificial. In reality their random variability extended only to $[-1,1]$, not to $\mathbf{R}=]-\infty, \infty[$. To emphasize this point, the following example uses domain $\Omega=]-1,1\left[{ }^{T}\right.$, not $\mathbf{R}^{T}$. Also, the notation and arguments of [MTRV] and [website] are given more prominence.
Example 17 With $\mathbf{T}=] 0, \tau]$ suppose an asset price process $X_{\mathbf{T}}$ is represented as
$$X_{\mathbf{T}} \simeq x_{\mathbf{T}}\left[\Omega, F_{X_{\mathrm{T}}}\right] \text { where } \Omega=(]-1,1[)^{\mathbf{T}}$$

Thus, for $0<s \leq \tau$, the asset price $x(s)$ can take a value between $-1$ and $+1$,
$$-1<x(s)<1, \quad 0<s \leq \tau .$$
Suppose the probability distribution function $F_{X_{T}}$ satisfies the following conditions:
[S1] For $0<s \leq \tau, F_{X_{n}}(]-1,1[)=1$
[S2] For any $s(0<s \leq \tau), \mathrm{E}\left[X_{s}\right],=\int_{-1}^{1} x_{s} F_{X_{n}}\left(I_{s}\right),=\mu$, a constant for all $s \in \mathbf{T}$. (For instance, the distribution functions $F_{X_{*}}$ can be the same for all s.)

物理代写|电动力学代写electromagnetism代考|Brownian Motion Preliminaries

Chapter 7 of [MTRV] contains a mathematical account of Brownian motion as a random variation phenomenon, from the -complete standpoint rather than the classical Itô/Kolmogorov/Lebesgue standpoint. Without repeating all the technicalities, some aspects can be reviewed here with the stochastic integral issue in view.

Small but visible particles suspended in some medium such as gas or water are seen to undergo rapid, irregular motion. Successive impacts on such a particle by invisible molecular-scale particles of the medium produce successive spatial transitions of the visible particle. Under molecular particle impact, the visible particle follows a straight line trajectory or transition until the next molecular impact produces a new trajectory or transition. The successive transitions are small, but whenever observable by sight they are seen to follow a zig-zag course made up of continuous straight line segments, or polygonal-type paths through space.

• The length of any one transition does not depend on the length of the immediately preceding transition or, indeed, on any of the preceding transitions.
• The lengths of individual line segments or transitions are mostly small, but longer segments or transitions occur less frequently.
• It is observed that that the square of net distance traversed by a visible particle from some initial starting point is, on average, proportional to the time elapsed.

Comparable behaviour was observed in the changes or movements of share prices in stock markets over any given time period:

• Price changes, like Brownian particle transitions, are uncertain or unpredictable.
• Over any given time period the range or spread of possible price change tends on average to correlate with the time elapsed.
• There tend to be many small price changes, with larger price changes being rarer.

Some of the examples and illustrations in the preceding sections show that, for a system involving only a finite number of transitions, or even a countable number of discrete transitions (i.e. discrete times), a mathematical representation is not too difficult to find.

物理代写|电动力学代写electromagnetism代考|Review of Integrability Issues

• 假设一个域Ω可以划分为子域Ωj, 并假设一个被积函数G是一个取常数值的阶跃函数ķj在域中Ωj对于每个j. 那么，即使ķj可积在Ωj对于每个j, 不一定是这样F可积在Ω.
• 如果一系列这样的阶跃函数逐点收敛到一个函数F不一定是这样F是可积的。
• 样本函数中可能出现狄利克雷型振荡X吨一个随机过程X吨. 这种现象存在可积性问题。

• X吨∈R吨
• F(X吨)是一个被积函数R吨
• F(我)是在单元格上定义的积分函数我的R吨.
积分在R吨的F(X吨)关于F(我)（如果存在）是∫R吨F(X吨)F(我), 对于目前的目的可以表示为∫R∞F(X)dF.

Rn,=R×⋯×R=R吨1×⋯×R吨n

物理代写|电动力学代写electromagnetism代考|Introduction to Brownian Motion

X吨≃X吨[Ω,FX吨] 在哪里 Ω=(]−1,1[)吨

−1<X(s)<1,0<s≤τ.

[S1] 对于0<s≤τ,FXn(]−1,1[)=1
[S2] 对于任何s(0<s≤τ),和[Xs],=∫−11XsFXn(我s),=μ, 一个常数s∈吨. （例如，分布函数FX∗所有 s 都可以相同。）

物理代写|电动力学代写electromagnetism代考|Brownian Motion Preliminaries

[MTRV] 的第 7 章包含对作为随机变化现象的布朗运动的数学说明，从完全的观点而不是经典的 Itô/Kolmogorov/Lebesgue 观点。在不重复所有技术细节的情况下，可以在这里回顾随机积分问题的某些方面。

• 任何一个过渡的长度不依赖于前一个过渡的长度，或者实际上，不依赖于任何前面的过渡。
• 单个线段或过渡的长度大多很小，但较长的线段或过渡出现的频率较低。
• 可以观察到，可见粒子从某个初始起点经过的净距离的平方平均与经过的时间成正比。

• 价格变化，如布朗粒子跃迁，是不确定或不可预测的。
• 在任何给定的时间段内，可能的价格变化的范围或分布平均倾向于与经过的时间相关。
• 往往会有许多小的价格变化，较大的价格变化很少见。

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