### 物理代写|电动力学代写electromagnetism代考|PHYS2016

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## 物理代写|电动力学代写electromagnetism代考|Introduction to ∫T(dXs)2 = t

The basis of traditional Itô calculus is the isometry property $\int_{\mathrm{T}}\left(d X_{s}\right)^{2}=t$. For this to be valid for Brownian motion $X=X_{\mathbf{T}}=\left(X_{s}: 0<s \leq t\right)$, and if an appropriate meaning or interpretation can be given to the “integral” expression of the isometry property, then the statement $\int_{T}\left(d x_{s}\right)^{2}=t$ must in some sense be valid for “typical” Brownian sample paths $x=x_{\mathbf{T}}=\left(x_{s}\right)$.

Traditional Itô calculus provides such an interpretation. The following discussion aims to provide a sense of what is involved.

In Example 24, every sample path $(x(s))$ satisfies $\int_{\mathrm{T}} d x_{s}=x_{t}$ provided the Stieltjes-complete definition of $\int_{T}$ is used. Examples in section $8.4$ of [MTRV] (pages 398-399) show that this approach does not work for $\int_{\mathbf{T}}\left(d x_{s}\right)^{2}$.

If $\int_{T}\left(d x_{s}\right)^{2}$ has some meaning as an integral then it is not unreasonable to seek to approximate it by means of some kind of Riemann sum expression of the form
(N) $\sum\left(x_{s^{\prime}}-x_{s}\right)^{2},=\sum_{j=1}^{n}\left(x_{j}-x_{j-1}\right)^{2}$, where $N=\left{s_{1}, \ldots, s_{n-1}, s_{n}\right}$
is a partition of $\mathbf{T}=] 0, t]$ with $s=s_{j-1}<s_{j}=s^{\prime}, j=1, \ldots, n ; 0=s_{0}, s_{n}=t$.
Such “typical” sample paths $\left(x_{s}\right)$ have unbounded variation (so the Lebesgue-style $\int_{0}^{t}\left|d x_{s}\right|$ typically diverges to $\left.+\infty\right)$. But ” $d x_{s}^{2}$ ” is typically less than $”\left|d x_{s}\right|$ “, so an aggregation of the form $\int_{0}^{t} d x_{s}^{2}$ may turn out to have a finite value. The expression $\sum_{j=1}^{n}\left(x_{j}-x_{j-1}\right)^{2}$ is a sample occurrence of a stochastic sum $f_{\mathrm{T}}^{g_{2}}\left(X_{\mathrm{T}}, \mathcal{N}\right)$ where the summand $g_{2}$ is
$$\left.\left.\left.\left.g_{2}\left(z_{s}\right)=\left(x\left(s^{\prime}\right)-x(s)\right)^{2} \text { for } \imath_{s}=\right] s, s^{\prime}\right], \quad x_{j}=\right] s_{j-1}, s_{j}\right], \quad x_{s_{j}}=x\left(s_{j}\right)=x_{j} .$$
For all partitions $N$ we have
$$X_{t}=\sum_{j=1}^{n}\left(X_{j}-X_{j-1}\right)=f_{\mathbf{T}}^{g_{1}}\left(X_{\mathbf{T}}, \mathcal{N}\right)$$
as in Example 24, and
$$\mathrm{E}\left[X_{t}\right]=\mathrm{E}\left[f_{\mathbf{T}}^{g_{1}}\left(X_{\mathbf{T}}, \mathcal{N}\right)\right]=\mathrm{E}\left[\sum_{j=1}^{n}\left(X_{j}-X_{j-1}\right)\right]=\sum_{j=1}^{n} \mathrm{E}\left[\left(X_{j}-X_{j-1}\right)\right]=0$$ with $\mathrm{E}\left[\left(X_{j}-X_{j-1}\right)\right]=0$ for each $j$ since the increments of standard Brownian motion have mean 0. Again, according to the theory of Brownian motion the increments $X_{j}-X_{k}$ are independent for all choices of $j, k$, including $k=0$ and $j=n$, with variance $t_{j}-t_{k}$ in each case. Recall that, for any random variable $Y$, the variance $\operatorname{Var}[Y]$ is $\mathrm{E}\left[(Y-\mathrm{E}[Y])^{2}\right]$.

## 物理代写|电动力学代写electromagnetism代考|Isometry Preliminaries

Some properties of finite-dimensional Gaussian integrals can be used to establish a version of the isometry property of Brownian motion.

P1 Assume $c<0$. Consider the one-dimensional integral $h(I)=\int_{u}^{v} y^{2} e^{c y^{2}} d y$ with $I$ a cell such as ] $u, v]$. In [MTRV] (page 263) integration by parts is applied, giving
\begin{aligned} \int_{u}^{v} y^{2} e^{c y^{2}} d y &=\frac{1}{2 c} \int_{u}^{v} y\left(e^{c y^{2}} 2 c y\right) d y \ &=\frac{1}{2 c}\left[y e^{c y^{2}}-\int e^{c y^{2}} d y\right]{u}^{v} \ &=\frac{1}{2 c}\left(v e^{c v^{2}}-u e^{c u^{2}}-\int{u}^{v} e^{c y^{2}} d y\right) \end{aligned}

P2 Suppose $x \in \mathbf{R}, f(x)=x^{2}$, and, for cells $\left.\left.I=\right] u, v\right]$ in $\mathbf{R}$, the cell function $g(I)$ is $\int_{I} e^{c y^{2}} d y$. For associated $(x, I)$ in $\mathbf{R}$, consider integrand $f(x) g(I)$ in domain $\mathbf{R}$. It is easy to show that $f(x) g(I)$ is variationally equivalent ${ }^{5}$ to $h_{0}(I)=\int_{u}^{v} y^{2} e^{c y^{2}} d y$. Since the latter is an additive cell function, it is the indefinite integral ${ }^{6}$ of the integrand $f(x) g(I)$; and, by the preceding calculation, the indefinite integral of $f(x) g(I)$ can be expressed as the additive cell function
$$h_{0}(I)=\frac{1}{2 c}\left(v e^{c v^{2}}-u e^{c u^{2}}-\int_{u}^{v} e^{c y^{2}} d y\right)$$
The purpose of presenting the indefinite integral of integrand $x^{2} \int_{I} e^{c y^{2}} d y$ in the form $(6.8)$ is to establish the isometry property of Brownian motion.
P3 Next, consider finite-dimensional domain $\mathbf{R}^{m}$ with points and cells
$$x=\left(x_{1}, \ldots, x_{m-1}, x_{m}\right), \quad I=I_{1} \times \cdots \times I_{m-1} \times I_{m},$$
respectively. Let $h_{1}(I)=$
$$=\int_{I}\left(y_{1}^{2}+\cdots+y_{m-1}^{2}+y_{m}^{2}\right) e^{\left(c_{1} y_{1}^{2}+\cdots+c_{m-1} y_{m-1}^{2}+c_{m} y_{m}^{2}\right)} d y_{1} \ldots d y_{m-1} d y_{m}$$
if the integral exists. Assume $c_{j}<0$ for $j=1 \ldots, m$. Regarding existence, for any $k(1 \leq k \leq m)$, with $\left.\left.I_{k}=\right] u_{k}, v_{k}\right],(6.8)$ implies
\begin{aligned} & \int_{I}\left(y_{k}^{2} e^{\sum_{j=1}^{m} c_{j} y_{j}^{2}}\right) d y_{1} \ldots d y_{m}=\ =& \int_{I_{k}} y_{k}^{2} e^{c_{k} y_{k}^{2}} d y_{k} \prod\left(\int_{I_{j}} e^{c_{j} y_{j}^{2}} d y_{j}: j=1,2, \ldots, m, j \neq k\right) \ =& \frac{1}{2 c_{k}}\left(v_{k} e^{c_{k} v_{k}^{2}}-u_{k} e^{c_{k} u_{k}^{2}}-\int_{I_{k}} e^{c_{k} y^{2}} d y_{k}\right) \prod_{j \neq k}\left(\int_{I_{j}} e^{c_{j} y_{j}^{2}} d y_{j}\right) \end{aligned}
so the first integral $\int_{I} \cdots$ exists. Thus $h_{1}(I)=$
$$=\sum_{k=1}^{m}\left(\frac{1}{2 c_{k}}\left(v_{k} e^{c_{k} v_{k}^{2}}-u_{k} e^{c_{k} u_{k}^{2}}-\int_{I_{k}} e^{c_{k} y_{k}^{2}} d y_{k}\right) \prod_{j \neq k}\left(\int_{I_{j}} e^{c_{j} y_{j}^{2}} d y_{j}\right)\right)$$
and $h_{1}(I)$ is finitely additive on disjoint cells $I$. This ensures that $h_{1}(I)$ is integrable on $\mathbf{R}^{m}$. Now define $h(I):=$
$$:=h_{1}(I) \prod_{j=1}^{m}\left(\frac{\pi}{-c_{j}}\right)^{-\frac{1}{2}}=\int_{I_{1} \times \cdots \times I_{m}}\left(\sum_{j=1}^{m} y_{j}^{2} e^{\sum_{j=1}^{m} c_{j} y_{j}^{2}}\right) \frac{d y_{1}}{\sqrt{\frac{\pi}{-c_{1}}}} \cdots \frac{d y_{m}}{\sqrt{\frac{\pi}{-c_{m}}}}$$

(where $\int_{\mathbf{R}} e^{c_{j} y_{j}^{2}} \frac{d y_{j}}{\sqrt{\frac{\pi}{-c_{j}}}}=1$ for each $j$ by theorem 133, [MTRV] page 261). Note that each of $v_{k} e^{c_{k} v_{k}^{2}}$ and $u_{k} e^{c_{k} u_{k}^{2}}$ tends to zero as $\left|v_{k}\right|,\left|u_{k}\right|$ tend to infinity. Therefore, using the -complete integral construction on $\mathbf{R}$ ([MTRV] pages 69-78, corresponding to improper Riemann integration),
$$\int_{\mathbf{R}^{m}} h(I)=h\left(\mathbf{R}^{m}\right)=\sum_{k=1}^{m} \frac{-1}{2 c_{k}} .$$

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

The second integrand/summand in the lists $(5.31)$ and $(5.32)$ is the function $g_{2}$. By adding more detail to Section $6.5$ the formulation $\int_{0}^{t} d X_{s}^{2}=t$ can now be brought into a framework of stochastic sums.

In $(6.7)$ the partition points $\tau_{j}$ of $M$ are taken to be fixed times for the purpose of calculating the expected value $\mathrm{E}\left[\sum_{j=1}^{m}\left(X_{j}-X_{j-1}\right)^{2}\right]$ in a finitedimensional sample space $\mathbf{R}^{M}$, with $M=\left{\tau_{1}, \ldots, \tau_{m-1}, \tau_{m}\right}$. In contrast, Sections $6.3$ and $6.4$ have provided expressions such as $\sum_{j=1}^{n}\left(X_{j}-X_{j-1}\right)^{2}$ in equation (6.7) with an enhanced meaning as a new kind of observable or random

variable,
$$f_{\mathbf{T}}^{g_{2}}\left(X_{\mathbf{T}}, \mathcal{N}\right)=\sum_{j=1}^{n}\left(X_{j}-X_{j-1}\right)^{2} .$$
Here, $f_{\mathbf{T}}^{g_{2}}\left(X_{\mathbf{T}}, \mathcal{N}\right)$ is an observable in sample space $\mathbf{R}^{\mathbf{T}}$ with distribution function $G(I[N])$ for times $N \subset \mathbf{T}$ :
$$\mathscr{f}{\mathbf{T}}^{g{2}}\left(X_{\mathbf{T}}, \mathcal{N}\right) \simeq \mathscr{f}{\mathbf{T}}^{g{2}}\left(x_{\mathbf{T}}, N\right)\left[\mathbf{R}^{\mathrm{T}}, G\right]$$
in which $N$ is variable, so sample values $f_{\mathbf{T}}^{g_{2}}\left(x_{\mathbf{T}}, N\right)$ are constructed from samples of times $s_{j} \in N \subset \mathbf{T}$, with corresponding sample values $x_{j},=x\left(s_{j}\right)$ of the random variables $X_{j},=X\left(s_{j}\right)$, of the process $X_{\mathbf{T}}$.

If observable $f_{\mathbf{T}}^{g_{2}^{2}}\left(X_{\mathbf{T}}, \mathcal{N}\right)$ has expected value it is a random variable. And, in that case, it is an absolute random variable (and therefore measurable) since its sample values are non-negative. (See theorems 76 and 250 , [MTRV] pages 193 and 494.) Example 25 below confirms these properties, with
$$\mathrm{E}\left[f_{\mathbf{T}}^{g_{2}}\left(X_{\mathbf{T}}, \mathcal{N}\right)\right]=\int_{\mathbf{R}^{\mathbf{T}}}\left(f_{\mathbf{T}}^{g_{2}}\left(x_{\mathrm{T}}, N\right)\right) G(I[N])=t$$
Thus
$$f_{\mathbf{T}}^{g_{2}}\left(X_{\mathbf{T}}, \mathcal{N}\right), \quad=\sum_{s_{j} \in N}\left(X\left(s_{j}\right)-X\left(s_{j-1}\right)\right)^{2} \text { with variable } N \in \mathcal{N},$$
is the meaning we ascribe to $\int_{\mathrm{T}} d X_{s}^{2}$, validating the latter as a random variable contingent on the Brownian process $X_{\mathbf{T}}$, so
$$\mathrm{E}\left[\int_{\mathbf{T}} d X_{s}^{2}\right]=\int_{\mathbf{R}^{\mathbf{T}}}\left(\int_{\mathrm{T}} d x_{s}^{2}\right) G(I[N])=t .$$
In this way, Example 25 supports the traditional Itô calculus interpretation of ” $\int_{\mathrm{T}} d X_{s}^{2} “$ as a weak integral which converges “in the mean” to value $t$.

## 物理代写|电动力学代写electromagnetism代考|Introduction to ∫T(dXs)2 = t

(N)∑(Xs′−Xs)2,=∑j=1n(Xj−Xj−1)2， 在哪里N=\left{s_{1}, \ldots, s_{n-1}, s_{n}\right}N=\left{s_{1}, \ldots, s_{n-1}, s_{n}\right}

G2(和s)=(X(s′)−X(s))2 为了 我s=]s,s′],Xj=]sj−1,sj],Xsj=X(sj)=Xj.

X吨=∑j=1n(Xj−Xj−1)=F吨G1(X吨,ñ)

## 物理代写|电动力学代写electromagnetism代考|Isometry Preliminaries

P1 假设C<0. 考虑一维积分H(我)=∫在在是2和C是2d是和我像 ] 这样的单元格在,在]. 在 [MTRV]（第 263 页）中，应用了按部分集成，给出

∫在在是2和C是2d是=12C∫在在是(和C是22C是)d是 =12C[是和C是2−∫和C是2d是]在在 =12C(在和C在2−在和C在2−∫在在和C是2d是)

P2 假设X∈R,F(X)=X2, 并且, 对于细胞我=]在,在]在R, 细胞函数G(我)是∫我和C是2d是. 对于关联(X,我)在R, 考虑被积函数F(X)G(我)在域中R. 很容易证明F(X)G(我)是变分等价的5至H0(我)=∫在在是2和C是2d是. 由于后者是一个加法单元函数，它是不定积分6被积函数的F(X)G(我); 并且，通过前面的计算，不定积分F(X)G(我)可以表示为加性细胞函数

H0(我)=12C(在和C在2−在和C在2−∫在在和C是2d是)

P3 接下来，考虑有限维域R米带有点和单元格

X=(X1,…,X米−1,X米),我=我1×⋯×我米−1×我米,

=∫我(是12+⋯+是米−12+是米2)和(C1是12+⋯+C米−1是米−12+C米是米2)d是1…d是米−1d是米

∫我(是ķ2和∑j=1米Cj是j2)d是1…d是米= =∫我ķ是ķ2和Cķ是ķ2d是ķ∏(∫我j和Cj是j2d是j:j=1,2,…,米,j≠ķ) =12Cķ(在ķ和Cķ在ķ2−在ķ和Cķ在ķ2−∫我ķ和Cķ是2d是ķ)∏j≠ķ(∫我j和Cj是j2d是j)

=∑ķ=1米(12Cķ(在ķ和Cķ在ķ2−在ķ和Cķ在ķ2−∫我ķ和Cķ是ķ2d是ķ)∏j≠ķ(∫我j和Cj是j2d是j))

:=H1(我)∏j=1米(圆周率−Cj)−12=∫我1×⋯×我米(∑j=1米是j2和∑j=1米Cj是j2)d是1圆周率−C1⋯d是米圆周率−C米

（在哪里∫R和Cj是j2d是j圆周率−Cj=1对于每个j由定理 133，[MTRV] 第 261 页）。请注意，每个在ķ和Cķ在ķ2和在ķ和Cķ在ķ2趋于零|在ķ|,|在ķ|趋于无穷大。因此，使用 -complete 积分构造R（[MTRV] 第 69-78 页，对应于不正确的黎曼积分），

∫R米H(我)=H(R米)=∑ķ=1米−12Cķ.

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

F吨G2(X吨,ñ)=∑j=1n(Xj−Xj−1)2.

F吨G2(X吨,ñ)≃F吨G2(X吨,ñ)[R吨,G]

F吨G2(X吨,ñ),=∑sj∈ñ(X(sj)−X(sj−1))2 有变量 ñ∈ñ,

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