### 统计代写|金融统计代写financial statistics代考| Stochastic Integrals and Differential Equations

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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

## 统计代写|金融统计代写financial statistics代考|Wiener Process

We begin with a simple symmetric random walk $\left{X_{n} ; n \geq 0\right}$ starting in $0\left(X_{0}=0\right)$. The increments $Z_{n}=X_{n}-X_{n-1}$ are i.i.d. with:
$$\mathrm{P}\left(Z_{n}=1\right)=\mathrm{P}\left(Z_{n}=-1\right)=\frac{1}{2}$$
By shortening the period of time of two successive observations we accelerate the process. Simultaneously, the increments of the process become smaller during the shorter period of time. More precisely, we consider a stochastic process $\left{X_{t}^{\Delta} ; t \geq 0\right}$ in continuous time which increases or decreases in a time step $\Delta t$ with

probability $\frac{1}{2}$ by $\Delta x$. Between these jumps the process is constant (alternatively we could interpolate linearly). At time $t=n \cdot \Delta t$ the process is:
$$X_{l}^{\Delta}=\sum_{k=1}^{n} Z_{k} \cdot \Delta x=X_{n} \cdot \Delta x$$
where the increments $Z_{1} \Delta x, Z_{2} \Delta x, \ldots$ are mutually independent and take the values $\Delta x$ or $-\Delta x$ with probability $\frac{1}{2}$, respectively. From Sect. $4.1$ we know:
$$\mathrm{E}\left[X_{t}^{\Delta}\right]=0, \quad \operatorname{Var}\left(X_{t}^{\Delta}\right)=(\Delta x)^{2} \cdot \operatorname{Var}\left(X_{n}\right)=(\Delta x)^{2} \cdot n=t \cdot \frac{(\Delta x)^{2}}{\Delta t}$$
Now, we let $\Delta t$ and $\Delta x$ become smaller. For the process in the limit to exist in a reasonable sense, $\operatorname{Var}\left(X_{t}^{\Delta}\right)$ must be finite. On the other hand, $\operatorname{Var}\left(X_{l}^{\Delta}\right)$ should not converge to 0 , since the process would no longer be random. Hence, we must choose:
$$\Delta t \rightarrow 0, \Delta x=c \cdot \sqrt{\Delta t}, \text { such that } \operatorname{Var}\left(X_{l}^{\Delta}\right) \rightarrow c^{2} t$$
If $\Delta t$ is small, then $n=t / \Delta t$ is large. Thus, the random variable $X_{n}$ of the ordinary symmetric random walk is approximately $\mathrm{N}(0, n)$ distributed, and therefore for all $t$ (not only for $t$ such that $t=n \Delta t$ ):
$$\mathcal{L}\left(X_{t}^{\Delta}\right) \approx \mathrm{N}\left(0, n(\Delta x)^{2}\right) \approx \mathrm{N}\left(0, c^{2} t\right)$$

## 统计代写|金融统计代写financial statistics代考|Stochastic Integration

In order to introduce a stochastic process as a solution of a stochastic differential equation we introduce the concept of the Itô-integral: a stochastic integral with respect to a Wiener process. Formally the construction of the Itô-integral is similar to the Stieltjes-integral. However, instead of integrating with respect to a deterministic function (Stieltjes-integral), the Itô-integral integrates with respect to a random function, more precisely, the path of a Wiener process. Since the integrant itself can be random, i.e. it can be a path of a stochastic process, one has to analyze the mutual dependencies of the integrant and the Wiener process.

Let $\left{Y_{l} ; t \geq 0\right}$ be the process to integrate and let $\left{W_{l} ; t \geq 0\right}$ be a standard Wiener process. The definition of a stochastic integral assumes that $\left{Y_{l} ; t \geq 0\right}$ is non-anticipating. Intuitively, it means that the process up to time $s$ does not contain any information about future increments $W_{t}-W_{s}, t>s$, of the Wiener process. In particular, $Y_{s}$ is independent of $W_{I}-W_{s}$.

An integral of a function is usually defined as the limit of the sum of the suitably weighted function. Similarly, the Itô integral with respect to a Wiener process is defined as the limit of the sum of the (randomly) weighted (random) function $\left{Y_{l} ; t \geq 0\right}$ :
$$\begin{array}{r} I_{n}=\sum_{k=1}^{n} Y_{(k-1) \Delta t} \cdot\left(W_{k \Delta t}-W_{(k-1) \Delta t}\right) \ \int_{0}^{l} Y_{s} d W_{s}=\lim {n \rightarrow \infty} I{n}, \end{array}$$
where the limit is to be understood as the limit of a random variable in terms of mean squared error, i.e. it holds
$$\mathrm{E}\left{\left[\int_{0}^{l} Y_{s} d W_{s}-I_{n}\right]^{2}\right} \rightarrow 0, \quad n \rightarrow \infty$$
It is important to note that each summand of $I_{n}$ is a product of two independent random variables. More precisely, $Y_{(k-1) \Delta t}$, the process to integrate at the left border of the small interval $[(k-1) \Delta t, k \Delta t]$ is independent of the increment $W_{k \Delta t}-$ $W_{(k-1) \Delta t}$ of the Wiener process in this interval.

It is not hard to be more precise on the non-anticipating property of $\left{Y_{l} ; t \geq 0\right}$.

## 统计代写|金融统计代写financial statistics代考|Stochastic Differential Equations

Since the Wiener process fluctuates around its expectation 0 it can be approximated by means of symmetric random walks. As for random walks we are interested in stochastic processes in continuous time which grow on average, i.e. which have a

trend or drift. Proceeding from a Wiener process with arbitrary $\sigma$ (see Sect. 5.1) we obtain the generalized Wiener process $\left{X_{t} ; t \geq 0\right}$ with drift rate $\mu$ and variance $\sigma^{2}$ :
$$X_{t}=\mu \cdot t+\sigma \cdot W_{t} \quad, \quad t \geq 0 .$$
The general Wiener process $X_{t}$ is at time $t, \mathrm{~N}\left(\mu t, \sigma^{2} t\right)$-distributed. For its increment in a small time interval $\Delta t$ we obtain
$$X_{t+\Delta l}-X_{l}=\mu \cdot \Delta t+\sigma\left(W_{t+\Delta t}-W_{t}\right) .$$
For $\Delta t \rightarrow 0$ use the differential notation:
$$d X_{l}=\mu \cdot d t+\sigma \cdot d W_{t}$$
This is only a different expression for the relationship (5.3) which we can also write in integral form:
$$X_{t}=\int_{0}^{t} \mu d s+\int_{0}^{t} \sigma d W_{s}$$
Note, that from the definition of the stochastic integral it follows directly that $\int_{0}^{t} d W_{s}=W_{l}-W_{0}=W_{l} .$

The differential notation (5.4) proceeds from the assumption that both the local drift rate given by $\mu$ and the local variance given by $\sigma^{2}$ are constant. A considerably larger class of stochastic processes which is more suited to model numerous economic and natural processes is obtained if $\mu$ and $\sigma^{2}$ in (5.4) are allowed to be time and state dependent. Such processes $\left{X_{t} ; t \geq 0\right}$, which we call Itô-processes, are defined as solutions of stochastic differential equations:
$$d X_{t}=\mu\left(X_{t}, t\right) d t+\sigma\left(X_{t}, t\right) d W_{t}$$
Intuitively, this means:
$$X_{t+\Delta t}-X_{t}=\mu\left(X_{t}, t\right) \Delta t+\sigma\left(X_{t}, t\right)\left(W_{t+\Delta t}-W_{t}\right),$$

## 统计代写|金融统计代写financial statistics代考|Wiener Process

XlΔ=∑ķ=1n从ķ⋅ΔX=Xn⋅ΔX

Δ吨→0,ΔX=C⋅Δ吨, 这样 曾是⁡(XlΔ)→C2吨

## 统计代写|金融统计代写financial statistics代考|Stochastic Integration

\mathrm{E}\left{\left[\int_{0}^{l} Y_{s} d W_{s}-I_{n}\right]^{2}\right} \rightarrow 0, \quad n \rightarrow \infty\mathrm{E}\left{\left[\int_{0}^{l} Y_{s} d W_{s}-I_{n}\right]^{2}\right} \rightarrow 0, \quad n \rightarrow \infty

## 统计代写|金融统计代写financial statistics代考|Stochastic Differential Equations

X吨=μ⋅吨+σ⋅在吨,吨≥0.

X吨+Δl−Xl=μ⋅Δ吨+σ(在吨+Δ吨−在吨).

dXl=μ⋅d吨+σ⋅d在吨

X吨=∫0吨μds+∫0吨σd在s

dX吨=μ(X吨,吨)d吨+σ(X吨,吨)d在吨

X吨+Δ吨−X吨=μ(X吨,吨)Δ吨+σ(X吨,吨)(在吨+Δ吨−在吨),

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## MATLAB代写

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