### 数学代写|随机过程作业代写Stochastic Processes代考|Brownian motion and Hilbert spaces

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• 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|Brownian motion and Hilbert spaces

The Wiener mathematical model of the phenomenon observed by an English botanist Robert Brown in 1828 has been and still is one of the most interesting stochastic processes. Kingman [66] writes that the deepest results in the theory of random processes are concerned with the interplay of the two most fundamental processes: Brownian motion and the Poisson process. Revuz and Yor [100] point out that the Wiener process “is a good topic to center a discussion around because Brownian motion is in the intersection of many fundamental classes of processes. It is a continuous martingale, a Gaussian process, a Markov process or more specifically a process with independent increments”. Moreover, it belongs to the important class of diffusion processes [58]. It is actually quite hard to find a book on probability and stochastic processes that does not describe this process at least in a heuristic way. Not a serious book, anyway.

Historically, Brown noted that pollen grains suspended in water perform a continuous swarming motion. Years (almost a century) later Bachelier and Einstein derived the probability distribution of a position of a particle performing such a motion (the Gaussian distribution) and pointed out its Markovian nature – lack of memory, roughly speaking. But it took another giant, notably Wiener, to provide a rigorous mathematical construction of a process that would satisfy the postulates of Einstein and Bachelier.

It is hard to overestimate the importance of this process. Even outside of mathematics, as Karatzas and Shreve [64] point out “the range of application of Brownian motion (…) goes far beyond a study of microscopic particles in suspension and includes modelling of stock prices, of thermal noise in electric circuits (…) and of random perturbations in a variety of other physical, biological, economic and management systems”. In

mathematics, Wiener’s argument involved a construction of a measure in the infinite-dimensional space of continuous functions on $\mathbb{R}^{+}$, and this construction was given even before establishing firm foundations for the mathematical measure theory.

To mention just the most elementary and yet so amazing properties of this measure let us note that it is concentrated on functions that are not differentiable at any point. Hence, from its perspective, functions that are differentiable at a point form a negligible set in the space of continuous functions. This should be contrasted with quite involved proofs of existence of a single function that is nowhere differentiable and a once quite common belief (even among the greatest mathematicians) that nowhere differentiable functions are not an interesting object for a mathematician to study and, even worse, that all continuous functions should be differentiable somewhere. On the other hand, if the reader expects this process to have only strange and even peculiar properties, he will be surprised to learn that, to the contrary, on the macroscopic level it has strong smoothing properties. For example, if we take any, say bounded, function $x: \mathbb{R} \rightarrow \mathbb{R}$ and for a given $t>0$ consider the function $x_{t}(\tau)=E x(\tau+w(t)), \tau \in \mathbb{R}$ where $w(t)$ is the value of a Wiener process at time $t$, this new function turns out to be infinitely differentiable! Moreover, $x(t, \tau)=x_{t}(\tau)$ is the solution of a famous heat equation $\frac{\partial u}{\partial t}=$ const. $\frac{\partial^{2} u}{\partial \tau^{2}}$ with the initial condition $u(0, \tau)=x(\tau)$. And this fact is just a peak of a huge iceberg of connections between stochastic processes and partial differential equations of second order.

## 数学代写|随机过程作业代写Stochastic Processes代考|Gaussian families & the definition of Brownian motion

4.1.1 Definition A family $\left{X_{t}, t \in \mathbb{T}\right}$ of random variables defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ where $\mathbb{T}$ is an abstract set of indexes is called a stochastic process. The cases $\mathbb{T}=\mathbb{N}, \mathbb{R}, \mathbb{R}^{+},[a, b],[a, b),(a, b]$, $(a, b)$ are very important but do not exhaust all cases of importance. For example, in the theory of point processes, $T$ is a family of measurable sets in a measurable space $[24,66]$. If $\mathbb{T}=\mathbb{N}$ (or $\mathbb{Z}$ ), we say that our process is time-discrete (hence, a time-discrete process is a sequence of random variables). If $\mathbb{T}=\mathbb{R}, \mathbb{R}^{+}$etc. we speak of time-continuous processes. For any $\omega \in \Omega$, the function $t \rightarrow X_{t}(\omega)$ is referred to as realization/sample path/trajectory/path of the process.
4.1.2 Gaussian random variables An n-dimensional random vector
$$X=\left(X_{1}, \ldots, X_{n}\right)$$
is said to be normal or Gaussian iff for any $\alpha=\left(\alpha_{1}, \ldots, \alpha_{n}\right) \in R^{n}$ the random variable $\sum_{j=1}^{n} \alpha_{j} X_{j}$ is normal. It is said to be standard normal if $X_{i}$ are independent and normal $N(0,1)$. A straightforward calculation (see 1.4.7) shows that convolution of normal densities is normal, so that the standard normal vector is indeed normal. In general, however, for a vector to be normal it is not enough for its coordinates to be normal. For instance, let us consider a $0<p<1$ and a vector $\left(X_{1}, X_{2}\right)$ with the density
$$f\left(x_{1}, x_{2}\right)=p \frac{1}{\pi} \mathrm{e}^{-\frac{x^{2}+x_{2}^{2}}{2}}+(1-p) \frac{1}{\pi} \frac{2 \sqrt{3}}{3} \mathrm{e}^{-\frac{4}{3} \frac{x_{1}^{2}+x_{1} x_{2}+x_{2}^{2}}{2}} .$$
Then $X_{1}$ and $X_{2}$ are normal but $\left(X_{1}, X_{2}\right)$ is not. To see that one checks for example that $X_{1}-\frac{1}{2} X_{2}$ is not normal.

## 数学代写|随机过程作业代写Stochastic Processes代考|Complete orthonormal sequences in a Hilbert space

4.2.1 Linear independence Vectors $x_{1}, \ldots, x_{n}$ in a linear space $\mathrm{X}$ are said to be linearly independent iff the relation
where $\alpha_{i} \in \mathbb{R}$ implies $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{n}=0$. In other words, $x_{1}, \ldots, x_{n}$ are independent iff none of them belongs to the subspace spanned by the remaining vectors. In particular, none of them may be zero.

We say that elements of an infinite subset $\mathbb{Z}$ of a linear space are linearly independent if any finite subset of $\mathbb{Z}$ is composed of linearly independent vectors.
4.2.2 Orthogonality and independence A subset $\mathrm{Y}$ of a Hilbert space is said to be composed of orthogonal vectors, or to be an orthogonal set, if for any distinct $x$ and $y$ from $Y,(x, y)=0$, and if $0 \notin \bar{Y}$. If, additionally $|x|=1$ for any $x \in \mathbb{Y}$, the set is said to be composed of orthonormal vectors, or to be an orthonormal set. By a usual abuse of language, we will also say that a sequence is orthonormal (orthogonal) if its values form an orthonormal (orthogonal) set.

Orthogonal vectors are linearly independent, for if (4.2) holds, then $0=\left(\sum_{i=1}^{n} \alpha x_{i}, \sum_{i=1}^{n} \alpha x_{i}\right)=\sum_{i=1}^{n} \alpha^{2}\left|x_{i}\right|^{2}$, which implies $\alpha_{i}=0$, for $i=1, \ldots, n$. On the other hand, if $x_{1}, \ldots, x_{n}$ are linearly independent then one may find a sequence $y_{1}, \ldots, y_{n}$, of orthonormal vectors such that $\operatorname{span}\left{x_{1}, \ldots, x_{n}\right}=\operatorname{span}\left{y_{1}, \ldots, y_{n}\right}$. The proof may be carried by induction. If $n=1$ there is nothing to prove; all we have to do is take $y_{1}=\frac{x_{1}}{\left|x_{1}\right|}$ to make sure that $\left|y_{1}\right|=1$. Suppose now that vectors $x_{1}, \ldots, x_{n+1}$ are linearly independent; certainly $x_{1}, \ldots, x_{n}$ are independent also. Let $y_{1}, \ldots, y_{n}$ be orthonormal vectors such that $\mathrm{Y}:=\operatorname{span}\left{x_{1}, \ldots, x_{n}\right}=\operatorname{span}\left{y_{1}, \ldots, y_{n}\right}$. The vector $x_{n+1}$ does not belong to $\mathrm{Y}$ and so we may take $y_{n+1}=\frac{x_{n+1}-P_{x_{n+1}}}{\left|x_{n+1}-P x_{n+1}\right|}$ where $P$ denotes projection on $\mathbb{Y}$. (The reader will check that $\mathbb{Y}$ is a subspace of $\mathbb{H}$; consult $5.1 .5$ if needed.) Certainly, $y_{1}, \ldots, y_{n+1}$ are orthonormal. Moreover, since $y_{n+1} \in \operatorname{span}\left{x_{1}, \ldots, x_{n+1}\right}$ (since $P x_{n+1} \in \operatorname{span}\left{x_{1}, \ldots, x_{n}\right}$ ), $\operatorname{span}\left{y_{1}, \ldots, y_{n+1}\right} \subset \operatorname{span}\left{x_{1}, \ldots, x_{n+1}\right}$; analogously we prove the converse inclusion. The above procedure of constructing orthonormal vectors from linearly independent vectors is called the Gram-Schmidt orthonormalization procedure. There are a number of examples of sequences of orthogonal polynomials that can be obtained via the GramSchmidt orthonormalization procedure, including (scalar multiples) of Legendre, Hermite and Laguerre polynomials that are of importance both in mathematics and in physics (see [83], [53], [75] Section 40).

## 数学代写|随机过程作业代写Stochastic Processes代考|Gaussian families & the definition of Brownian motion

4.1.1 定义A族\left{X_{t}, t \in \mathbb{T}\right}\left{X_{t}, t \in \mathbb{T}\right}在概率空间上定义的随机变量(Ω,F,磷)在哪里吨是一组抽象的索引，称为随机过程。案例吨=ñ,R,R+,[一种,b],[一种,b),(一种,b], (一种,b)非常重要，但不要穷尽所有重要的案例。例如，在点过程理论中，吨是可测空间中的可测集族[24,66]. 如果吨=ñ（或者从)，我们说我们的过程是时间离散的（因此，时间离散的过程是一系列随机变量）。如果吨=R,R+等等，我们谈到时间连续的过程。对于任何ω∈Ω， 功能吨→X吨(ω)被称为过程的实现/样本路径/轨迹/路径。
4.1.2 高斯随机变量 n 维随机向量
$$X =\left(X_{1}, \ldots, X_{n}\right)$$

## 数学代写|随机过程作业代写Stochastic Processes代考|Complete orthonormal sequences in a Hilbert space

4.2.1 线性独立向量X1,…,Xn在线性空间X据说是线性独立的，当且仅

4.2.2 正交性和独立性子集是的希尔伯特空间被称为由正交向量组成，或者是一个正交集，如果对于任何不同的X和是从是,(X,是)=0， 而如果0∉是¯. 如果，另外|X|=1对于任何X∈是，该集合被称为由正交向量组成，或者是一个正交集合。通过通常的语言滥用，如果序列的值形成正交（正交）集合，我们也会说序列是正交的（正交）。

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