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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


从一些比较麻烦的问题开始,示例路径(X(s))价格过程X吨在示例 15 和 16 中包括极端振荡反映狄利克雷函数的路径d(s)例 13 的。

  • 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。

另一方面,无限维域上的积分有时会简化为更熟悉的有限维积分。这种现象的某些方面可以概括如下。认为吨是一个无限的标签集,例如 ]0,吨], 并假设

  • 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.

什么时候吨是无限的(也就是说,当吨具有无限基数)细胞我是圆柱形的,如符号所示我=我[ñ]对于有限子集N=\left{t_{1}, \ldots, t_{n}\right}N=\left{t_{1}, \ldots, t_{n}\right}的 T. 因此,有限笛卡尔积的某些方面


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



然而,它们更简单、更容易表述的场景在数学方面不一定是最易于管理的——因为例如示例 13 中所示的狄利克雷振荡。此外,前面的示例虽然可能有助于为主题提供一种感觉, 不是在实践中重要的过程。


在实际定义布朗运动之前,下面的示例 17 是它的一个版本,它演示了如何避免示例 12、15、16 的狄利克雷型振荡。它旨在成为将这些示例与本章将要讨论的标准布朗运动连接起来的桥梁。

前面的例子是位于形式的布朗运动域R吨. 但这有点人为。实际上,它们的随机变异性仅扩展到[−1,1], 不R=]−∞,∞[. 为了强调这一点,下面的例子使用了域Ω=]−1,1[吨, 不是R吨. 此外,[MTRV] 和 [website] 的符号和论点更加突出。
示例 17 与吨=]0,τ]假设一个资产价格过程X吨表示为

X吨≃X吨[Ω,FX吨] 在哪里 Ω=(]−1,1[)吨

因此,对于0<s≤τ, 资产价格X(s)可以取一个之间的值−1和+1,

[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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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





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