### 物理代写|电动力学代写electromagnetism代考|PHYS2213

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## 物理代写|电动力学代写electromagnetism代考|A Basic Stochastic Integral

The following is similar to Example $2 .$
Example 5 Suppose $s=1,2,3, \ldots$ is time, measured in days. Suppose a share, or unit of stock, has value $x(s)$ on day s; suppose $z(s)$ is the number of shares held on day $s$; and suppose $c(s)$ is the change in the value of the shareholding on day $s$ as a result of the change in share value from the previous day so $c(s)=$ $z(s-1)(x(s)-x(s-1))$. Let $w(s)$ be the cumulative change in shareholding value at end of day $s$, so $w(s)=w(s-1)+c(s)$. If share value $x(s)$ and stockholding $z(s)$ are subject to random variability, how is the gain (or loss) from the stockholding to be estimated?

Take initial value (at time $s=0)$ of the share to be $x(0)$ (or $\left.x_{0}\right)$, take the initial shareholding or number of shares owned to be $z(0)$ (or $\left.z_{0}\right)$. Then, at end of day $1(s=1)$,
$$c(1)=z(0) \times(x(1)-x(0)), \quad w(1)=w(0)+c(1)=c(1)$$
At end of day $s$,
$$c(s)=z(s-1) \times(x(s)-x(s-1)), \quad w(s)=w(s-1)+c(s)$$
After $t$ days,
$$w(t)=\sum_{s=1}^{t} z(s-1)(x(s)-x(s-1)) .$$
If the time increments are reduced to arbitrarily small size (so $s$ represents number of “time ticks” -fractions of a second, say), with the meaning of the other variables adjusted accordingly, then
$$w(t)=\sum_{j=1}^{n} z\left(s_{j-1}\right)\left(x\left(s_{j}\right)-x\left(s_{j-1}\right)\right), \quad \text { or } \quad w(t)=\sum z(s) \Delta x(s)$$
The latter expressions are Riemann sum estimates of $\int_{0}^{t} z(s) d x(s)$ (a Stieltjestype integral) whenever the latter exists.
Each of the expressions in (2.4) is sample value of a random variable
$$W(t)=\sum_{j=1}^{n} Z\left(s_{j-1}\right)\left(X\left(s_{j}\right)-X\left(s_{j-1}\right)\right) \text { or } \int_{0}^{t} Z(s) d X(s)$$constructed from the random variables $X, Z$, and $W$. These notations symbolize in a “naive” or “realistic” way-the stochastic integral of the process $Z$ with respect to the process $X$. In chapter 8 of [MTRV], symbols s, or $\mathbf{S}$, or $\mathcal{S}$ are used (in place of the symbol $\int$ ) for various kinds of stochastic integral. In the context described here, S would be the appropriate notation. (See (5.28) below.)

## 物理代写|电动力学代写electromagnetism代考|Choosing a Sample Space

It was mentioned earlier that there are many alternative ways of producing a sample space $\Omega$ (along with the linked probability measure $P$ and family $\mathcal{A}$ of measurable subsets of $\Omega$ ). The set of numbers
$${-5,-4,-3,-2,-1,0,1,2,3,4,5,10}$$
was used as sample space for the random variability in the preceding example of stochastic integration. The measurable space $\mathcal{A}$ was the family of all subsets of $\Omega$, and the example was illustrated by means of two distinct probability measures $P$, one of which was based on Up and Down transitions being equally likely, where for the other measure an Up transition was twice as likely as a Down.
An alternative sample space for this example of random variability is
$$\Omega=\Omega \Omega_{1} \times \Omega \Omega_{2} \times \Omega \Omega_{3} \times \Omega \Omega_{4}$$
where $\Omega_{j}={U, D}$ for $j=1,2,3,4$; so the elements $\omega$ of $\Omega$ consist of sixteen 4-tuples of the form
$$\omega=(\cdot, \cdot \cdot \cdot), \quad \text { such as } \omega=(U, D, D, U) \text { for example. }$$
Let the measurable space $\mathcal{A}$ be the family of all subsets $A$ of $\Omega$; so $\mathcal{A}$ contains $2^{16}$ members, one of which (for example) is
$$A={(D, U, U, D),(U, D, D, U),(D, U, D, U),(U, U, U, U),(D, D, D, D)}$$

with $A$ consisting of five individual four-tuples. Assume that Up transitions and Down transitions are equally likely, and that they are independent events. Then, as before,
$$P({\omega})=\frac{1}{16}$$
for each $\omega \in \Omega$. For $A$ above, $P(A)=\frac{5}{16}$.
To relate this probability structure to the shareholding example, let $\mathbf{R}^{4}=$ $\mathbf{R} \times \mathbf{R} \times \mathbf{R} \times \mathbf{R}$, and let
$$f: \Omega \mapsto \mathbf{R}^{4}, \quad f(\omega)=((x(1), x(2), x(3), x(4)),$$
using Table 2.4; so, for instance,
$$f(\omega)=f((U, D, D, U))=(11,10,9,10)=(x(1), x(2), x(3), x(4)),$$
and so on. Next, let $\mathbf{S}$ denote the stochastic integrals of the preceding section, so for $x=(x(1), x(2), x(3), x(4)) \in \mathbf{R}^{4}$,
$$\mathbf{S}(x)=\int_{0}^{4} z(s) d x(s)=\sum_{s=1}^{4} z(s-1)(x(s)-x(s-1)),$$
so $\mathbf{S}(x)$ gives the values $w(4)$ of Table 2.4. As described in Section $2.3$, the rationale for deducing the probabilities of outcomes $\mathbf{S}(x)$, = $w(4)$, from the probabilities on $\Omega$ is the relationship
$$P(w(4))=P\left(f^{-1}\left(\mathbf{S}^{-1}(w(4))\right)\right) .$$

## 物理代写|电动力学代写electromagnetism代考|More on Basic Stochastic Integral

The constructions in Sections $2.3$ and $2.4$ purported to be about stochastic integration. While a case can be made that (2.6) and (2.7) are actually stochastic integrals, such simple examples are not really what the standard or classical theory of Chapter 1 is all about. The examples and illustrations in Sections $2.3$ and $2.4$ may not really be much help in coming to grips with the standard theory of stochastic integrals outlined in Chapter $1 .$

This is because Chapter 1, on the definition and meaning of classical stochastic integration, involves subtle passages to a limit, whereas (2.6) and (2.7) involve only finite sums and some elementary probability calculations.

From the latter point of view, introducing probability measure spaces and random-variables-as-measurable-functions seems to be an unnecessary complication. So, from such a straightforward starting point, why does the theory become so challenging and “messy”, as portrayed in Chapter $1 ?$

As in Example 2, the illustration in Section $2.3$ involves dividing up the time period (4 days) into 4 sections; leading to sample space $\Omega=\mathbf{R}^{4}$ in (2.15). Why not simply continue in this vein, and subdivide the time into 40 , or 400 , or 4 million steps instead of just 4 ; using sample spaces $\mathbf{R}^{40}$, or $\mathbf{R}^{400}$, or $\mathbf{R}^{4000000}$, respectively? The computations may become lengthier, but no new principle is involved; each of the variables changes in discrete steps at discrete points in time. ${ }^{5}$

Other simplifications can be similarly adopted. For instance, only two kinds of changes are contemplated in Section 2.3: increase (Up) or decrease (Down).

## 物理代写|电动力学代写electromagnetism代考|A Basic Stochastic Integral

C(1)=和(0)×(X(1)−X(0)),在(1)=在(0)+C(1)=C(1)

C(s)=和(s−1)×(X(s)−X(s−1)),在(s)=在(s−1)+C(s)

（2.4）中的每个表达式都是随机变量的样本值

## 物理代写|电动力学代写electromagnetism代考|Choosing a Sample Space

−5,−4,−3,−2,−1,0,1,2,3,4,5,10

Ω=ΩΩ1×ΩΩ2×ΩΩ3×ΩΩ4

ω=(⋅,⋅⋅⋅), 如 ω=(在,D,D,在) 例如。

F:Ω↦R4,F(ω)=((X(1),X(2),X(3),X(4)),

F(ω)=F((在,D,D,在))=(11,10,9,10)=(X(1),X(2),X(3),X(4)),

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