### 物理代写|电动力学代写electromagnetism代考|PHYS2001

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

This section seeks to extend the Riemann sum stratagem described above in order to simplify and unify various conceptions of strong and weak stochastic integration; and to replace stochastic integrals by stochastic sums.

A stochastic process is a family of random variables $X=X_{\mathbf{T}}=\left(X_{s}\right), s \in \mathbf{T}$, where $\mathbf{T}$ is an infinite set such as $] 0, t]$. Stochastic integration is a device which constructs a random variable $Z$ from a process $X_{\mathrm{T}}$; such as
$$Z=\int_{0}^{t} X_{s} d X_{s} .$$
Example 23 Constructions of this kind have been given a variety of interpretations and meanings in chapter 8 (pages 383-446) of [MTRV], such as strong and weak stochastic integrals:
$$\mathbf{S}{T}^{g}\left(X{\mathbf{T}}\right), \quad \mathcal{S}{\mathbf{T}}^{g}\left(X{\mathrm{T}}\right)$$
where (in this case) the integrand $g$ is $X_{s}\left(X_{s^{\prime}}-X_{s}\right),=X_{s} d X_{s},\left(0 \leq s<s^{\prime} \leq t\right)$. In fact, provided $X_{\mathbf{T}}$ is standand Brownian motion, $\int_{0}^{t} X_{s} d X_{s}$ is $\mathcal{S}{\mathrm{T}}^{g}\left(X{\mathrm{T}}\right), a$

weak stochastic integral which evaluates as $\frac{1}{2} X_{t}^{2}-\frac{1}{2} t . \quad$ (See example 63 , pages $405-406$ of [MTRV].)

Expressed in terms of sample values $x_{s}(0<s<t)$, or in terms of sample path $x_{T}$, this result states that, with $x_{t}=x(t)$ given,
$$\mathcal{S}{T}^{g}\left(x{T}\right)=\int_{0}^{t} x_{s} d x_{s}=\int_{0}^{t} x(s) d x(s)=\frac{1}{2} x_{t}^{2}-\frac{1}{2} t$$
in some weak sense; where $\int_{0}^{t} x(s) d x(s)$ is a Stieltjes-type integral of the pointfunction $x(s)$ with respect to (increments of) the point-function $x(s)$.
(a) Equation (6.4) is a “weak” equation, which can only be valid in some sense of “average value” of one or other side, or both.
(b) Furthermore, the left hand side of (6.4) references infinitely many values $x_{s}$, corresponding to the infinitely many time instants $0<s<t . A s$ in (6.3), this suggests infinitely many sample measurements $x_{s}$. This is counter-intuitive as a method of calculation. It is not practically possible to sample every instant s of time.

In the discussion below, both of these issues are addressed by using (as in (6.4)) a Riemann sum method for the averaging required by (a), so each Riemann sum involves a finite sample consisting of only a finite number of times s.

## 物理代写|电动力学代写electromagnetism代考|Stochastic Sum as Observable

A new type of observable is required:
$$f\left(X_{\mathbf{T}}, \mathcal{N}\right) \simeq f\left(x_{\mathbf{T}}, N\right)\left[\mathbf{R}^{\mathbf{T}}, F_{X_{\mathbf{T}}}\right]$$
where $F_{X_{\mathbf{T}}}$ is a distribution function defined for $I[N] \in \mathcal{I}\left(\mathbf{R}^{\mathbf{T}}\right.$ ) (the set of cells in $\left.\mathbf{R}^{\mathbf{T}}\right)$
$$F_{X_{\mathrm{T}}}: \mathcal{I} \mapsto[0,1], \quad 0 \leq F_{X_{\mathrm{T}}}(I[N]) \leq 1 .$$
In addition to dependence on joint occurrences $\left(x_{s}\right)=x_{\mathbf{T}}$, an observable $f$ is permitted to depend explicitly on partitions $N=\left{s_{1}, \ldots, s_{n-1}, s_{n}\right}$ of $\mathbf{T}$. Likewise, a distribution function $F_{X_{\mathrm{T}}}$ depends on cells $I=I[N]$, and may depend explicitly on the partitions $N=\left{s_{1}, \ldots, s_{n-1}, s_{n}\right}$ of T which (with $s_{n}=t$ ) are the “cylinder labels”, or dimension labels, of the cylindrical intervals $I[N]$ in $\mathbf{R}^{\mathbf{T}}$. For example, with $\left.\left.I_{t}=\right] u_{t}, v_{t}\right](t \in N)$, the incremental Gaussian distribution function $G$ of (5.8) (see page 115 above) depends explicitly on the parameters $u_{t}, v_{t}$, and $t$, for $t \in N$.

A left hand limit (or vertex) $u_{t}$ for a partitioning component cell $\left.I_{t}=\right] u_{t}, v_{t}$ ] $(t \in N)$ is a right hand limit or vertex of an adjoining cell $I_{t}^{\prime}$. Thus, choice of a partition $\mathcal{P}$ of domain $\mathbf{R}^{\mathbf{T}}$ reduces to choice of finite samples $N$ of times, along with choices $\left{u_{t}\right}$ of finite samples of vertices for $t \in N$.

As outlined in Section $6.2$, the fundamental step is to define the expectation $\mathrm{E}\left[f\left(X_{\mathrm{T}}, \mathcal{N}\right)\right]$; that is, to define the integral of $f\left(x_{\mathrm{T}}, N\right)$ with respect to distribution function $F(I[N])$. In particular, when $f\left(x_{\mathbf{T}}, N\right)=\mathcal{R}{\mathbf{T}}^{g}\left(x{\mathbf{T}}, N\right)$, (or $\left.f_{\mathbf{T}}^{g}\left(x_{\mathbf{T}}, N\right)\right)$
\begin{aligned} \mathrm{E}\left[\mathcal{R}{\mathbf{T}}^{g}\left(X{T}, \mathcal{N}\right)\right], &=\int_{\mathbf{R}^{\mathbf{T}}}\left(\mathcal{R}{\mathbf{T}}^{g}\left(x{\mathbf{T}}, N\right)\right) F(I[N]), \ \text { or } \mathrm{E}\left[\oiint_{\mathbf{T}}^{g}\left(X_{\mathbf{T}}, \mathcal{N}\right)\right], &=\int_{\mathbf{R}^{\mathrm{T}}}\left(\oiint_{\mathbf{T}}^{g}\left(x_{\mathbf{T}}, N\right)\right) F(I[N]) \end{aligned}
so $f_{\mathrm{T}}^{g}\left(X_{\mathbf{T}}, \mathcal{N}\right.$ ) is a random variable. Chapter 4 (pages 111-182) of [MTRV] deals with the integration in $\mathbf{R}^{S}$ of integrands of the form $h\left(x_{S}, N, I[N]\right)$, where $S$ is any infinite set (such as intervals of time $\mathbf{T}$ or $T$ ), including integrands $f\left(x_{S}, N\right) F(I[N])$. Briefly, $f\left(x_{S}, N\right) F(I[N])$ is integrable on $\mathbf{R}^{S}$, with integral
$$\int_{\mathbf{R}^{S}} f\left(x_{S}, N\right) F(I[N])=\alpha,$$
if, given $\varepsilon>0$, there exists a gauge $\gamma=\left(L, \delta_{\mathcal{N}}\right)$ such that, for every $\gamma$-fine division $\mathcal{D}$ of $\mathbf{R}^{S}$, the corresponding Riemann sums satisfy
$$\left|\alpha-(\mathcal{D}) \sum f\left(x_{S}, N\right) F(I[N])\right|<\varepsilon$$
Chapter 4 of [MTRV] provides a theory of variation for functions $h(x, N, I[N])$ which is applicable to functions $F(I[N])$ and $f(x, N) F(I[N])$. So, for instance, $F(I[N]$ ) (defined on cells $I[N])$ can be extended to an “outer measure” on arbitrary subsets $A$ of $\mathbf{R}^{S}$. Chapter 4 also provides limit theorems for integrals (such as integrability of limits of integrable functions), and Fubini’s theorem for integrands defined on product domains of the form $\mathbf{R}^{S^{\prime}} \times \mathbf{R}^{S^{\prime \prime}}$.

## 物理代写|电动力学代写electromagnetism代考|Stochastic Sum as Random Variable

This section follows through on the definitions of Section 6.3, using familiar examples to illustrate the theory of stochastic sums, as replacement for both strong and weak stochastic integrals in chapter 8 of [MTRV]. The examples are based on the functions $g_{1}$ to $g_{9}$ of pages 391-392; also listed in (5.31) and (5.32) at the end of Chapter 5 above.

The notation is as set out in section $8.2$ of MTRV (pages $386-390$ ); with

$\mathbf{T}=] 0, t]$ replacing the symbol $\mathcal{T}=] 0, t]$ of $[\mathrm{MTRV}] .$ For any given $x_{\mathbf{T}} \in \mathbf{R}^{\mathbf{T}}$,
\begin{aligned} z_{s} &\left.=] s, s^{\prime}\right], & & 0 \leq s<s^{\prime} \leq t, \ \mathbf{x}\left(z_{s}\right) &=x\left(s^{\prime}\right)-x(s), & & x=x_{\mathrm{T}} \in \mathbf{R}^{\mathrm{T}}, \ g &=g\left(x_{s}, s, \mathbf{x}\left(z_{s}\right), z_{s}\right), & & \text { a stochastic summand (or integrand), } \ \mathbf{X}\left(z_{s}\right) &=X\left(s^{\prime}\right)-X(s), & & X=X_{\mathrm{T}} \text { an observable in sample space } \mathbf{R}^{\mathrm{T}}, \ N &=\left{s_{1}, \ldots, s_{n-1}, s_{n}\right}, & & \text { a partition of } \mathbf{T}, \text { or finite subset of } \mathbf{T}, \ z_{j} &\left.=] s_{j-1}, s_{j}\right], & j=1, \ldots, n, \quad s_{0}=0, s_{n}=t . \end{aligned}
For $g=g_{1}, \ldots, g_{9}$ of $(8.16)$ in page 419 of [MTRV], evaluations of stochastic integrals (strong and weak) have been given in [MTRV]. The idea here is to illustrate stochastic summation by replacing ${ }^{4}$ the stochastic integrals $\mathbf{S}{\mathbf{T}}^{g{j}}\left(X_{\mathbf{T}}\right)$ or $\mathcal{S}{\mathbf{T}}^{g{j}}\left(X_{\mathbf{T}}\right)$ with corresponding stochastic sums of the form
$$\int_{\mathbf{T}}^{g_{j}}\left(X_{\mathbf{T}}, \mathcal{N}\right)=\sum_{j=1}^{n} g_{j}\left(X_{s_{j}}, s_{j}, \mathbf{X}\left(z_{s_{j}}\right), z_{s_{j}}\right)$$

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

(a) 方程（6.4）是一个“弱”方程，它只能在某种意义上的一侧或另一侧或两者的“平均值”上有效。
(b) 此外，(6.4) 的左侧引用了无限多的值Xs, 对应于无数个时刻0<s<吨.一个s在 (6.3) 中，这表明有无限多的样本测量Xs. 作为一种计算方法，这是违反直觉的。实际上不可能对每一个瞬间进行采样。

## 物理代写|电动力学代写electromagnetism代考|Stochastic Sum as Observable

F(X吨,ñ)≃F(X吨,ñ)[R吨,FX吨]

FX吨:我↦[0,1],0≤FX吨(我[ñ])≤1.

∫R小号F(X小号,ñ)F(我[ñ])=一个,

|一个−(D)∑F(X小号,ñ)F(我[ñ])|<e
[MTRV] 的第 4 章提供了函数的变分理论H(X,ñ,我[ñ])适用于函数F(我[ñ])和F(X,ñ)F(我[ñ]). 所以，例如，F(我[ñ]) （在单元格上定义我[ñ])可以扩展到任意子集的“外部度量”一个的R小号. 第 4 章还提供了积分的极限定理（例如可积函数极限的可积性），以及定义在形式乘积域上的被积函数的 Fubini 定理R小号′×R小号′′.

## 物理代写|电动力学代写electromagnetism代考|Stochastic Sum as Random Variable

\begin{aligned} z_{s} &\left.=] s, s^{\prime}\right], & & 0 \leq s<s^{\prime} \leq t, \ \mathbf{x} \left(z_{s}\right) &=x\left(s^{\prime}\right)-x(s), & & x=x_{\mathrm{T}} \in \mathbf{R} ^{\mathrm{T}}, \ g &=g\left(x_{s}, s, \mathbf{x}\left(z_{s}\right), z_{s}\right), & & \text { 一个随机和（或被积函数）， } \ \mathbf{X}\left(z_{s}\right) &=X\left(s^{\prime}\right)-X(s), & & X=X_{\mathrm{T}} \text { 样本空间中的一个观测值 } \mathbf{R}^{\mathrm{T}}, \N &=\left{s_{1}, \ldots, s_ {n-1}, s_{n}\right}, & & \text { } \mathbf{T}, \text { 或 } \mathbf{T}, \z_{j} &\ 的有限子集left.=] s_{j-1}, s_{j}\right], & j=1, \ldots, n, \quad s_{0}=0, s_{n}=t。\end{对齐}\begin{aligned} z_{s} &\left.=] s, s^{\prime}\right], & & 0 \leq s<s^{\prime} \leq t, \ \mathbf{x} \left(z_{s}\right) &=x\left(s^{\prime}\right)-x(s), & & x=x_{\mathrm{T}} \in \mathbf{R} ^{\mathrm{T}}, \ g &=g\left(x_{s}, s, \mathbf{x}\left(z_{s}\right), z_{s}\right), & & \text { 一个随机和（或被积函数）， } \ \mathbf{X}\left(z_{s}\right) &=X\left(s^{\prime}\right)-X(s), & & X=X_{\mathrm{T}} \text { 样本空间中的一个观测值 } \mathbf{R}^{\mathrm{T}}, \N &=\left{s_{1}, \ldots, s_ {n-1}, s_{n}\right}, & & \text { } \mathbf{T}, \text { 或 } \mathbf{T}, \z_{j} &\ 的有限子集left.=] s_{j-1}, s_{j}\right], & j=1, \ldots, n, \quad s_{0}=0, s_{n}=t。\end{对齐}

∫吨Gj(X吨,ñ)=∑j=1nGj(Xsj,sj,X(和sj),和sj)

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