### 统计代写|随机过程代写stochastic process代考|STAT306

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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|随机过程代写stochastic process代考|Two Basic Methods in This Book

In this section, we shall present two basic methods (via illuminating examples) that will be systematically used throughout this book.

The main method that we employ in this book to deal with the analysis of the structure of stochastic distributed parameter systems is the global Carleman type estimate. This method was introduced by T. Carleman ([47]) in 1939 to prove the uniqueness of solutions to second order elliptic partial differential equations with two variables. The key in [47] is an elementary energy estimate with some exponential weight. This type of weighted energy estimates, now referred to as Carleman estimates, have become one of the major tools in the study of unique continuation property, inverse problems and control problems for many partial differential equations. However, it is only in the last ten plus years that the power of the global Carleman estimate in the context of controllability of stochastic partial differential equations came to be realized. For the readers’ convenience, we explain the main idea of Carleman estimate by the following very simple example:
Example 1.14. Consider the following ordinary differential equation in $\mathbb{R}^n$ :
$$\left{\begin{array}{l} y_t(t)=a(t) y(t) \quad \text { in }[0, T], \ y(0)=y_0 . \end{array}\right.$$
It is well-known that if $a \in L^{\infty}(0, T)$, then there is a constant $\mathcal{C}_T>0$ such that for all solutions of (1.45), it holds that $$\max {t \in[0, T]}|y(t)|{\mathbb{R}^n} \leq \mathcal{C}T\left|y_0\right|{\mathbb{R}^n}, \quad \forall y_0 \in \mathbb{R}^n .$$
Now we give a slightly different proof of this result via Carleman-type estimate:
For any $\lambda \in \mathbb{R}$, it is easy to see that
\begin{aligned} & \frac{d}{d t}\left(e^{-2 \lambda t}|y(t)|{\mathbb{R}^n}^2\right) \ & =-2 \lambda e^{-2 \lambda t}|y(t)|{\mathbb{R}^n}^2+2 e^{-2 \lambda t}\left\langle y_t(t), y(t)\right\rangle_{\mathbb{R}^n}=2(a(t)-\lambda) e^{-2 \lambda t}|y(t)|{\mathbb{R}^n}^2 \end{aligned} Choosing $\lambda=|a|{L^{\infty}(0, T)}$, we find that
$$|y(t)|{\mathbb{R}^n} \leq e^{\lambda T}\left|y_0\right|{\mathbb{R}^n}, \quad t \in[0, T]$$
which proves (1.46).

## 统计代写|随机过程代写stochastic process代考|Integrals and Expectation

In this section, we recall the definitions and some basic results for the Bochner integral and the Pettis integral. We omit the proofs and refer the readers to $[74,143]$. Let us fix a $\sigma$-finite measure space $(\Omega, \mathcal{F}, \mu)$, a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a Banach space $H$.

Let $f(\cdot)$ be an (H-valued) $\mathcal{F}$-simple function in the form (2.2). We call $f(\cdot)$ Bochner integrable if $\mu\left(E_i\right)<\infty$ for each $i=1, \cdots, k$. In this case, for any $E \in \mathcal{F}$, the Bochner integral of $f(\cdot)$ over $E$ is defined by
$$\int_E f(s) d \mu=\sum_{i=1}^k \mu\left(E \cap E_i\right) h_i .$$
In general, we have the following notion.
Definition 2.14. A strongly $\mathcal{F}$-measurable function $f(\cdot): \Omega \rightarrow H$ is said to be Bochner integrable (w.r.t. $\mu$ ) if there exists a sequence of Bochner integrable $\mathcal{F}$-simple functions $\left{f_i(\cdot)\right}_{i=1}^{\infty}$ converging strongly to $f(\cdot), \mu$-a.e. in $\Omega$, so that
$$\lim {i, j \rightarrow \infty} \int{\Omega}\left|f_i(s)-f_j(s)\right|_H d \mu=0 .$$

For any $E \in \mathcal{F}$, the Bochner integral of $f(\cdot)$ over $E$ is defined by
$$\int_E f(s) d \mu=\lim {i \rightarrow \infty} \int{\Omega} \chi_E(s) f_i(s) d \mu(s) \quad \text { in } H .$$
It is easy to verify that the limit in the right hand side of (2.4) exists and its value is independent of the choice of the sequence $\left{f_i(\cdot)\right}_{i=1}^{\infty}$. Clearly, when $H=\mathbb{R}^n$ (for some $n \in \mathbb{N}$ ), the above Bochner integral coincides the usual integral for $\mathbb{R}^n$-valued functions.

Particularly, if $(\Omega, \mathcal{F}, \mathbb{P})$ ia a probability space and $f: \Omega \rightarrow H$ is Bochner integrable (w.r.t. $\mathbb{P}$ ), then we say that $f$ has a mean, denoted by
$$\mathbb{E} f=\int_{\Omega} f d \mathbb{P} \text {. }$$
We also call $\mathbb{E} f$ the (mathematical) expectation of $f$.
The following result reveals the relationship between the Bochner integral (for vector-valued functions) and the usual integral (for scalar functions).

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Two Basic Methods in This Book

$\$ \$$左 {$$
y_t(t)=a(t) y(t) \quad \text { in }[0, T], y(0)=y_0 .
$$正确的。 ## 统计代写|随机过程代写stochastic process代考|Integrals and Expectation 在本节中，我们回顾一下 Bochner 积分和 Pettis 积分的定义和一些基本结果。我们省略了证 明，请读者参考 [74,143]. 让我们修复一个 \sigma-有限测度空间 (\Omega, \mathcal{F}, \mu) ，一个概率空间 (\Omega, \mathcal{F}, \mathbb{P}) 和巴拿赫空间 H. 让 f(\cdot) 是一个 ( \mathrm{H} 值) \mathcal{F} – 形式为 (2.2) 的简单函数。我们称之为 f(\cdot) Bochner 可积若 \mu\left(E_i\right)<\infty 每个 i=1, \cdots, k. 在这种情况下，对于任何 E \in \mathcal{F} ，的博赫纳积分 f(\cdot) 超过 E 由定义$$
\int_E f(s) d \mu=\sum_{i=1}^k \mu\left(E \cap E_i\right) h_i .
$$一般来说，我们有以下概念。 定义 2.14。强烈的 \mathcal{F}-可测量的功能 f(\cdot): \Omega \rightarrow H 据说是 Bochner 可积的 (w r t \mu) 如果存在 \Omega ，以便$$
\lim i, j \rightarrow \infty \int \Omega\left|f_i(s)-f_j(s)\right|H d \mu=0 . $$对于任何 E \in \mathcal{F} ，的博赫纳积分 f(\cdot) 超过 E 由定义$$ \int_E f(s) d \mu=\lim i \rightarrow \infty \int \Omega \chi_E(s) f_i(s) d \mu(s) \quad \text { in } H . $$很容易验证式(2.4)右边的极限存在并且它的值与序列的选择无关 Meft{f_i(lcdot)\right}{i=1}^{linfty}. 显然，当 H=\mathbb{R}^n (对于一些 n \in \mathbb{N} ), 上述 Bochner 积分 与通常的积分一致 \mathbb{R}^n – 值函数。 特别是，如果 (\Omega, \mathcal{F}, \mathbb{P}) \mathrm{ia} 一个概率空间和 f: \Omega \rightarrow H 是 Bochner 可积的 (wrtP), 那么我们 说 f 有一个平均值，表示为$$
\mathbb{E} f=\int_{\Omega} f d \mathbb{P} .


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