### 数学代写|随机过程作业代写Stochastic Processes代考|Construction and basic properties of Brownian motion

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• Statistical Inference 统计推断
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• Foundations of Data Science 数据科学基础
• 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

## 数学代写|随机过程作业代写Stochastic Processes代考|Construction and basic properties of Brownian motion

4.3.1 Construction: first step There exists a process $\left{w_{t}, t \geq 0\right}$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ satisfying (a) -(b) of the definition 4.1.6.
Proof Let $\mathbb{H}=L^{2}\left(\mathbb{R}^{+}\right)$, and let $A$ be the operator described in 4.2.19. Let $w_{t}=A\left(1_{[0, t)}\right)$. Any vector $\left(w_{t_{1}}, \ldots, w_{t_{n}}\right)$ is Gaussian, because for any scalars $\alpha_{i}$, the random variable
$$\sum_{i=1}^{n} \alpha_{i} w_{i}=A\left(\sum_{i=1}^{n} \alpha_{i} 1_{\left(0, t_{i}\right)}\right)$$
is Gaussian. Moreover, $E w_{t}=0$, and $E w_{t} w_{s}=\left(1_{[0, t)}, 1_{[0, s)}\right){L^{2}(\mathbb{R}+)}=$ $\int{0}^{\infty} 1_{[0, t)} 1_{[0, s)} d l e b=s \wedge t .$
4.3.2 Existence of Brownian motion on $[0,1]$ In general it is hard, if possible at all, to check if the process constructed above has continuous paths. We may achieve our goal, however, if we consider a specific orthonormal system (other ways of dealing with this difficulty may be found in $[5,61,100,79] \uparrow)$. We will construct a Brownian motion on $[0,1]$ using the system $x_{n}$ from $4.2 .14$. As in $4.2 .19$, we define
$$A x=\sum_{n=0}^{\infty}\left(x_{n}, x\right) Y_{n}$$
where $Y_{n}$ is a sequence of standard independent random variables. Let
$$w_{t}(\omega)=\left(A 1_{[0, t)}\right)(\omega)=\sum_{n=0}^{\infty}\left(x_{n}, 1_{[0, t)}\right) Y_{n}(\omega)=\sum_{n=0}^{\infty} y_{n}(t) Y_{n}(\omega), \quad t \in[0,1]$$
The argument presented in $4.3 .1$ shows that $w_{t}$ satisfy the first two conditions of the definition of Brownian motion, and it is only the question of continuity of paths that has to be settled.

Note that $y_{n}(t)=\left(x_{n}, 1_{[0, t)}\right)=\int_{0}^{t} x_{n}(s) \mathrm{d} s$ is a continuous function, so that for any $\omega$, the partial sums of the series in (4.6) are (linear combinations of) continuous functions. We will prove that the series converges absolutely and uniformly in $t \in[0,1]$ for almost all $\omega \in \Omega$. To this end write
$$w_{t}(\omega)=\sum_{n=0}^{\infty} \sum_{k=0}^{2^{n}-1} y_{2^{n}+k}(t) Y_{2^{n}+k}(\omega)$$

## 数学代写|随机过程作业代写Stochastic Processes代考|Stochastic integrals

Let us consider the following hazard game related to a Brownian motion $w(t), t \geq 0$. Suppose that at time $t_{0}$ we place an amount $x\left(t_{0}\right)$ as a bet, to have $x\left(t_{0}\right)\left[w\left(t_{0}+h\right)-w\left(t_{0}\right)\right]$ at time $t_{0}+h$. More generally, suppose that we place amounts $x\left(t_{i}\right)$ at times $t_{i}$ to have
$$\sum_{i=1}^{n-1} x\left(t_{i}\right)\left[w\left(t_{i+1}\right)-w\left(t_{i}\right)\right]$$
at time $t_{\mathrm{n}}$ where $0 \leq a=t_{0}<t_{1}<\cdots<t_{n}=b<\infty$. If we imagine that we may change our bets in a continuous manner, we are led to considering the limit of such sums as partitions refine to infinitesimal level. Such a limit, a random variable, would be denoted $\int_{a}^{b} x(t) \mathrm{d} w(t)$. The problem is, however, whether such a limit exists and if it enjoys properties that we are used to associating with integrals. The problem is not a trivial one, for as we know from $4.3 .10, t \rightarrow w(t)$ is not of bounded variation in $[a, b]$, so that this integral may not be a Riemann-Stieltjes integral. This new type of integral was introduced and extensively studied by $\mathrm{K}$. Itô, and is now known as an Itô integral. The first thing the reader must keep in mind to understand this notion is that we will not try to define this integral for every path separately; instead we think of $w(t)$ as an element of the space of square integrable functions, and the integral is to be defined as an element of the same space. But the mere change of the point of view and the space where we are to operate does not suffice. As a function $t \mapsto w(t) \in$ $L^{2}(\Omega, \mathcal{F}, \mathbb{P})$ where $(\Omega, \mathcal{F}, \mathbb{P})$ is the probability space where $w(t), t \geq 0$ are defined, the Brownian motion is not of bounded variation either. To see that consider a uniform partition $t_{i}=a+\frac{i}{n}(b-a), i=0, . ., n$ of $[a, b]$. Since $E\left[w\left(t_{i+1}\right)-w\left(t_{i}\right)\right]^{2}=\frac{1}{n}(b-a)$, the supremum of sums $\sum_{i=1}^{n-1}\left|w\left(t_{i+1}\right)-w\left(t_{i}\right)\right|_{L^{2}(\Omega, \mathcal{F}, P)}$ over partitions of the interval $[a, b]$ is at least $\sum_{i=1}^{n-1} \sqrt{\frac{b-a}{n}}=\sqrt{n} \sqrt{b-a}$, as claimed.

## 数学代写|随机过程作业代写Stochastic Processes代考|Definition

4.4.6 Definition By $4.4 .4$ and $4.4 .5$ there exists a linear isometry between $L_{p}^{2}$ and $L^{2}(\Omega, \mathcal{F}, \mathbb{P})$. This isometry is called the Itô integral and denoted $I(x)=\int_{a}^{b} x \mathrm{~d} w$.
4.4.7 Itô integral as a martingale It is not hard to see that taking $a<b<c$ and $x \in L_{p}^{2}[a, c]$ we have $\int_{a}^{b} x \mathrm{~d} w=\int_{a}^{c} x 1_{[a, b] \times \Omega} \mathrm{d} w$. Hence, by linearity $\int_{a}^{c} x \mathrm{~d} w=\int_{a}^{b} x \mathrm{~d} w+\int_{b}^{c} x \mathrm{~d} w$. Also, we have $E \int_{a}^{c} x \mathrm{~d} w=0$ for simple, and hence all processes $x$ in $L_{p}^{2}[a, c]$, since $E$ is a bounded linear functional on $L^{2}(\Omega, \mathcal{F}, \mathbb{P})$. Finally, $\mathbb{E}\left(\int_{b}^{c} x d w \mid \mathcal{\mathcal { F } _ { b }}\right)=0$ where as before $\mathcal{F}{b}=\sigma(w(s), 0 \leq s \leq b)$. Indeed, if $x$ is a simple process (4.17) (with $b$ replaced by $c$ and $a$ replaced by $b)$, then $\mathbb{E}\left(x{t_{i}} \delta_{i} \mid \mathcal{F}{t{i}}\right)=x_{t_{i}} \mathbb{E}\left(\delta_{i} \mid \mathcal{F}{t{i}}\right)=0$, for all $0 \leq i \leq n-1$, whence by the tower property $\mathbb{E}\left(x_{t_{i}} \delta_{i} \mid \mathcal{F}_{b}\right)=0$ as well, and our formula follows.

Now, assume that $x \in L_{p}^{2}[0, t]$ for all $t>0$. Then we may define $y(t)=$ $\int_{0}^{t} x \mathrm{~d} w$. The process $y(t), t \geq 0$, is a time-continuous martingale with respect to the filtration $\mathcal{F}{t}, t \geq 0$, inherited from the Brownian motion. Indeed, $\mathbb{E}\left(y(t) \mid \mathcal{F}{s}\right)=\mathbb{E}\left(\int_{0}^{s} x \mathrm{~d} w \mid \mathcal{F}{s}\right)+\mathbb{E}\left(\int{s}^{t} x \mathrm{~d} w \mid \mathcal{F}{s}\right)=\int{0}^{s} x \mathrm{~d} w+0=$ $y(s)$, because $\int_{0}^{s} x \mathrm{~d} w$ is $\mathcal{F}{s}$ measurable as a limit of $\mathcal{F}{s}$ measurable functions.
4.4.8 Information about stochastic integrals with respect to square integrable martingales There are a number of ways to generalize the notion of Itô integral. For example, one may relax measurability and integrability conditions and obtain limits of integrals of simple processes in a weaker sense (e.g. in probability and not in $L^{2}$ ). The most important fact, however, seems to be that one may define integrals with respect to processes other than Brownian motion. The most general and yet plausible integrators are so-called continuous local martingales, but in this note we restrict ourselves to continuous square integrable martingales. These are time-continuous martingales $y(t), t \geq 0$ with $E y^{2}(t)<\infty, t \geq 0$, and almost all trajectories continuous. Certainly, Brownian motion is an example of such a process. It may be proven that for any such martingale there exists an adapted, non-decreasing process $a(t)$ such that $y^{2}(t)-a(t)$ is a martingale. A non-decreasing process is one such that almost all paths are non-decreasing. For a Brownian motion, $a(t)$ does not depend on $\omega$ and equals $t$. Now, the point is again that one may prove that the space $L_{p}^{2}[a, b]=L_{p}^{2}[a, b, y]$ of progressively measurable processes $x$ such that $E \int_{a}^{b} x^{2}(s) \mathrm{d} a(s)$ is finite is isometrically isomorphic to $L^{2}(\Omega, \mathcal{F}, \mathbb{P})$. To establish this fact one needs to show that simple processes form a linearly dense set in $L_{p}^{2}[a, b, y]$ and define the Itô integral for simple processes as $I(x)=\sum_{i=0}^{n-1} x_{t_{i}}\left[y\left(t_{i+1}\right)-y\left(t_{i}\right)\right]$. Again, the crucial step is establishing Itô isometry, and the reader now appreciates the way we established it in 4.4.5.

## 数学代写|随机过程作业代写Stochastic Processes代考|Construction and basic properties of Brownian motion

4.3.1 构建：第一步 存在流程\left{w_{t}, t \geq 0\right}\left{w_{t}, t \geq 0\right}在概率空间上(Ω,F,磷)满足定义 4.1.6 的 (a) -(b)。

∑一世=1n一种一世在一世=一种(∑一世=1n一种一世1(0,吨一世))

## 数学代写|随机过程作业代写Stochastic Processes代考|Stochastic integrals

∑一世=1n−1X(吨一世)[在(吨一世+1)−在(吨一世)]

## 数学代写|随机过程作业代写Stochastic Processes代考|Definition

4.4.6 定义依据4.4.4和4.4.5之间存在线性等距大号p2和大号2(Ω,F,磷). 这种等距称为伊藤积分并表示为一世(X)=∫一种bX d在.
4.4.7 作为鞅的伊藤积分 不难看出，取一种<b<C和X∈大号p2[一种,C]我们有∫一种bX d在=∫一种CX1[一种,b]×Ωd在. 因此，通过线性∫一种CX d在=∫一种bX d在+∫bCX d在. 另外，我们有和∫一种CX d在=0简单，因此所有过程X在大号p2[一种,C]， 自从和是一个有界线性泛函大号2(Ω,F,磷). 最后，和(∫bCXd在∣Fb)=0和以前一样Fb=σ(在(s),0≤s≤b). 确实，如果X是一个简单的过程（4.17）（与b取而代之C和一种取而代之b)， 然后和(X吨一世d一世∣F吨一世)=X吨一世和(d一世∣F吨一世)=0， 对全部0≤一世≤n−1, 由塔属性从何而来和(X吨一世d一世∣Fb)=0同样，我们的公式如下。

4.4.8 关于平方可积鞅的随机积分的信息 有多种方法可以推广伊藤积分的概念。例如，人们可以放宽可测性和可积性条件，并在较弱的意义上获得简单过程的积分极限（例如在概率上而不是在大号2）。然而，最重要的事实似乎是人们可以定义与布朗运动以外的过程有关的积分。最一般但最合理的积分器是所谓的连续局部鞅，但在本说明中，我们将自己限制为连续平方可积鞅。这些是时间连续的鞅是(吨),吨≥0和和是2(吨)<∞,吨≥0，并且几乎所有的轨迹都是连续的。当然，布朗运动就是这种过程的一个例子。可以证明，对于任何这样的鞅，都存在一个适应的、非递减的过程一种(吨)这样是2(吨)−一种(吨)是鞅。非递减过程是几乎所有路径都非递减的过程。对于布朗运动，一种(吨)不依赖于ω和等于吨. 现在，重点是人们可以证明空间大号p2[一种,b]=大号p2[一种,b,是]逐渐可测量的过程X这样和∫一种bX2(s)d一种(s)是有限的 是等距同构的大号2(Ω,F,磷). 为了确定这一事实，我们需要证明简单的过程在大号p2[一种,b,是]并将简单过程的 Itô 积分定义为一世(X)=∑一世=0n−1X吨一世[是(吨一世+1)−是(吨一世)]. 同样，关键步骤是建立 Itô 等距，现在读者很欣赏我们在 4.4.5 中建立它的方式。

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