### 物理代写|电动力学代写electromagnetism代考|ELEC2300

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• 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|Varieties of Stochastic Integral

This section provides a summary and overview of stochastic integrals. The notation for stochastic integrals in a Riemann setting is set out in section $8.2$ (pages $386-390$ ) of [MTRV]. The family of cells or intervals in a domain $\Omega$ is denoted by $\mathcal{I}(\Omega)$. So if $\Omega$ is, respectively, a real interval such as $] 0, t]$, a finitedimensional domain $\mathbf{R}^{M}$, or an infinite-dimensional domain $\mathbf{R}^{] 0, t]}$, then $\mathcal{I}(\Omega)$ consists of cells which are denoted, respectively, by
$$\text { 2, } I(M), \quad I[N],$$
where $M$ and $N$ are finite sets. For ease of reference, relevant content of section $8.2$ of [MTRV] is repeated here.

Suppose $\left.\mathbf{T}=] \tau^{\prime}, \tau\right]$ (closed on the right) and suppose $F_{X},=F_{X_{T}}$, is a distribution defined on $\mathcal{I}\left(\mathbf{R}^{\mathrm{T}}\right)$; so $X \simeq x\left[\mathbf{R}^{\mathrm{T}}, F_{X}\right]$ is a joint-basic observable.

Suppose $\left.\left.2,=i_{s},=\right] s, s^{\prime}\right] \in \mathcal{I}(\mathbf{T})$ and $f\left(x,\left{s, s^{\prime}\right}\right)=x\left(s^{\prime}\right)-x(s)=\mathbf{x}\left(\imath_{s}\right)$. We then have a contingent joint observable
$$f\left(X,\left{s, s^{\prime}\right}\right) \simeq f\left(x,\left{s, s^{\prime}\right}\right)\left[\mathbf{R}^{\mathbf{T}}, F_{X}\right], \quad \text { or } \quad \mathbf{X}\left(\imath_{s}\right) \simeq \mathbf{x}\left(\imath_{s}\right)\left[\mathbf{R}^{\mathbf{T}}, F_{X}\right] .$$
Suppose $g$ is a function of the elements $\mathbf{x}(\imath)$ for $\imath \in \mathcal{I}(\mathbf{T})$. For instance, $g(\mathbf{x}(\imath))$ could be the function
$$g(\mathbf{x}(2)),=g\left(\mathbf{x}\left(z_{s}\right)\right),=\mathbf{x}\left(z_{s}\right)^{2}=\left(x\left(s^{\prime}\right)-x(s)\right)^{2} .$$
The family of finite subsets of $\mathbf{T}$ is denoted by $\mathcal{N}(\mathrm{T})$. If $N=\left{t_{1}, t_{2}, \ldots, t_{n}\right} \in$ $\mathcal{N}(\mathrm{T})$ with $t_{0}=\tau^{\prime}$ and $t_{n}=\tau$, we $\operatorname{can}^{13}$ write $\left.\left.\imath_{j}=\right] t_{j-1}, t_{j}\right]$. Thus the cells $\left{\imath_{j}\right}$ form a partition of the domain $\mathbf{T}$. For simplicity, let the symbol $N$ denote:

• partition points $\left{t_{j}\right}$, or
• partition $\left{z_{j}\right}$, or
• division $\left.\left.\left{(\bar{s},] t_{j-1}, t_{j}\right]\right)\right$,$} , with associated points (or tag points) \bar{s}=t_{j-1}$ or $\bar{s}=t_{j} \cdot$

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

In Chapter 1 the classical or standard concept of stochastic integral, including Itô’s integral, is outlined. The mathematical need or motive for some concept of stochastic integration has been illustrated in preceding chapters by means of various examples. It is illustrated in particular by the manifestation, in the form of a stochastic integral, of the value at any time $t$ of a shareholding (or portfolio) of a quantity $g(s)$ of shares whose value at time $s(0 \leq s \leq t)$ is $x(s)$ :
$$\int_{0}^{t} g(s) d x(s),$$
where $g(s)$ is a deterministic or random function of time $s(0 \leq s \leq t)$ and $x(s)$ $(0<s \leq t)$ is a sample path of a process $X=\left(X_{s}\right)_{0<s \leq t^{*}}$

The latter expression $\mathcal{S}{\mathrm{T}}^{g}(X)$ corresponds to the classical $\int{0}^{t} g(s) d X(s)$ (which is the Itô integral if $\left(X_{s}\right)$ is a Brownian motion). The other three expressions are innovations. They are introduced in MTRV, which includes discussion of $\mathbf{s}{\mathrm{T}}^{g}(X)$ and $\mathbf{S}{\mathrm{T}}^{g}(X)$, along with a brief outline of the first one, $\mathcal{R}_{\mathrm{T}}^{g}(X, N)$.

In mathematical discussion of integration, including stochastic integration, it is customary to use a notation with three components
${$ integral symbol $}{$ point integrand $}{$ differential $}$ or $\left(\int\right)(f(y))(d y)$.
In the -complete integration of [MTRV], this is expressed as $\int f(y) k(I)$, where $k(I)=|I|$ corresponds to the traditional differential symbol $d y$.

Traditionally, a stochastic integral may take the form $\int f(X) d X$ where $X$ is a stochastic process. But [MTRV] breaks with this notation. Instead of $\int$ we have symbols s, $\mathbf{S}$, and $\mathcal{S}$; each used in particular contexts (see section 8.2, [MTRV] pages $386-390)$. Integrand elements such as $f(X) d X$ are denoted by some expression $g$ which is attached to the relevant integration symbol as a superscript, giving
$$\mathbf{s}^{g}, \quad \mathbf{S}^{g}, \quad \mathcal{S}^{g} .$$
[MTRV] also introduces another such functional, $\mathcal{R}^{g}$, which is a Riemann sum rather than an integral, and which is now to be written $f^{g}$.

As well as integration, these procedures have a functional aspect, in the sense that the final result depends on the choice of sample path $x_{T}$. The innovations in notation are intended to emphasize the functional rather than the integration aspect. So with the integrand $g$ safely relegated to superscript position, the functional dependence on $x_{\mathbf{T}}$ is denoted by
$$\mathbf{s}^{g}\left(x_{\mathbf{T}}\right), \quad \mathbf{S}^{g}\left(x_{\mathrm{T}}\right), \quad \mathcal{S}^{g}\left(x_{\mathrm{T}}\right)$$
and, whenever needed for clarity, the domain $\mathbf{T}$ is placed as subscript,
$$\mathbf{s}{\mathbf{T}}^{g}\left(x{\mathbf{T}}\right), \quad \mathbf{S}{\mathbf{T}}^{g}\left(x{\mathrm{T}}\right), \quad \mathcal{S}{\mathbf{T}}^{g}\left(x{\mathbf{T}}\right),$$
just as it is in $\int_{T}$. The same general idea is in the stochastic sum notation
$$\mathcal{R}{\mathbf{T}}^{g}\left(x{\mathbf{T}}\right), \quad \mathcal{f}{\mathbf{T}}^{g}\left(x{\mathbf{T}}\right)$$
These two symbols are equivalent, but the latter symbol is given precedence because of the suggestion it contains of “sum replacing integral”.

The aim of this chapter is to amplify and extend the ideas behind $\mathcal{R}{T}^{g}(X)-$ or $\mathcal{R}{\mathrm{T}}^{g}(X, \mathcal{N})$, or $\xi_{\mathbf{T}}^{g}\left(X_{\mathbf{T}}, \mathcal{N}\right)$-as a simpler and more comprehensive way of dealing with stochastic integration; so that the single formulation $f_{\mathbf{T}}^{g}\left(X_{\mathbf{T}}, \mathcal{N}\right)$ replaces each of $\mathbf{s}{\mathbf{T}}^{g}(X), \mathbf{S}{\mathbf{T}}^{g}(X), \mathcal{S}{\mathbf{T}}^{g}(X)$, and $\int{\mathbf{T}} f(X) d X$.

## 物理代写|电动力学代写electromagnetism代考|Review of Random Variability

To set the scene for this, here is an overview of the -complete approach to random variability, with emphasis on those aspects which reinforce and validate the replacement of stochastic integrals by stochastic sums.

Random variability is associated with observation or measurement of some quantity whose precise value is not known, but for which estimated values $x$ can be given. Suppose that, even though the precise or true value is not known, there is some method of assessing the accuracy of estimated values $x$. Suppose, in fact, that the degree or level of accuracy of the estimate $x$ can itself be estimated by means of a distribution function $F$ or $F_{X}$ defined on intervals $I$ of possible values of $x$ in domain $\Omega$ (called the sample space). Then the term observable is applied to the notion of measurement or estimate $X$, with possible values $x$ in sample space $\Omega$, equipped with accuracy function (or likelihood distribution function) $F_{X}$.
$$X \simeq x\left[\Omega, F_{X}\right] .$$
The measured value (or occurrence, or datum) $x$ can be a number (usually real), so $\Omega=\mathbf{R}$. Or $x$ can consist of jointly measured values $x=\left(x_{s}\right)$ where $s \in \mathbf{T}$; so if $\mathbf{T}$ is a finite set of cardinality $n$, then $\Omega=\mathbf{R}^{n}$ and $x=\left(x_{1}, \ldots, x_{n}\right)$. In that case, $X \simeq x\left[\mathbf{R}^{n}, F_{X}\right]$ is a joint-basic observable, with distribution function $F_{X}$ defined on cells
$$I=I_{1} \times \cdots \times I_{\mathrm{n}} \subset \mathbf{R}^{n}, \quad x_{j} \in I_{j}, \quad j=1, \ldots n .$$
If the measurement is a real value $f(x)$ formed by means of a deterministic function $f$ of the basic observable $x$, then $f(X)$ is a contingent observable
$$f(X) \simeq f(x)\left[\Omega, F_{X}\right] .$$
An event is a set of occurrences. An observable $f(X)$ is a random variable (or is an $F_{X}$-random variable) if its expected value $\mathrm{E}[f(X)]$ exists:
$$\mathrm{E}[f(X)]=\int_{\Omega} f(x) F_{X}(I) .$$

## 物理代写|电动力学代写electromagnetism代考|Varieties of Stochastic Integral

2, 我(米),我[ñ],

f\left(X,\left{s, s^{\prime}\right}\right) \simeq f\left(x,\left{s, s^{\prime}\right}\right)\left [\mathbf{R}^{\mathbf{T}}, F_{X}\right], \quad \text { 或 } \quad \mathbf{X}\left(\imath_{s}\right) \simeq \mathbf{x}\left(\imath_{s}\right)\left[\mathbf{R}^{\mathbf{T}}, F_{X}\right] 。f\left(X,\left{s, s^{\prime}\right}\right) \simeq f\left(x,\left{s, s^{\prime}\right}\right)\left [\mathbf{R}^{\mathbf{T}}, F_{X}\right], \quad \text { 或 } \quad \mathbf{X}\left(\imath_{s}\right) \simeq \mathbf{x}\left(\imath_{s}\right)\left[\mathbf{R}^{\mathbf{T}}, F_{X}\right] 。

G(X(2)),=G(X(和s)),=X(和s)2=(X(s′)−X(s))2.

• 分割点\左{t_{j}\右}\左{t_{j}\右}， 或者
• 分割\左{z_{j}\右}\左{z_{j}\右}， 或者
• 分配\left.\left.\left{(\bar{s},] t_{j-1}, t_{j}\right]\right)\right$,$} ，带有关联点（或标记点）\酒吧{s}=t_{j-1}\left.\left.\left{(\bar{s},] t_{j-1}, t_{j}\right]\right)\right$,$} ，带有关联点（或标记点）\酒吧{s}=t_{j-1}或者s¯=吨j⋅

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

∫0吨G(s)dX(s),

$一世n吨和Gr一个ls是米b○l$$p○一世n吨一世n吨和Gr一个nd$$d一世FF和r和n吨一世一个l$或者(∫)(F(是))(d是).

sG,小号G,小号G.
[MTRV] 还引入了另一个这样的功能，RG，这是黎曼和而不是积分，现在要写成FG.

sG(X吨),小号G(X吨),小号G(X吨)

s吨G(X吨),小号吨G(X吨),小号吨G(X吨),

R吨G(X吨),F吨G(X吨)

X≃X[Ω,FX].

F(X)≃F(X)[Ω,FX].

## 有限元方法代写

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## MATLAB代写

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