### 数学代写|随机过程统计代写Stochastic process statistics代考|MXB334

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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

## 数学代写|随机过程统计代写Stochastic process statistics代考|Conditional Expectation

In this subsection, we fix a probability measure $\mathbb{P}$ on $(\Omega, \mathcal{F})$, and a function $f \in L_{\mathcal{F}}^{1}(\Omega ; H) \triangleq L^{1}(\Omega, \mathcal{F}, \mathbb{P} ; H)$
Definition 2.49. Let $B \in \mathcal{F}$ with $\mathbb{P}(B)>0$. For any event $A \in \mathcal{F}$, put
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} .$$
Then $\mathbb{P}(\cdot \mid B)$ is a probability measure on $(\Omega, \mathcal{F})$, called the conditional probability given the event $B$, and denoted by $\mathbb{P}{B}(\cdot)$. For any given $A \in \mathcal{F}, \mathbb{P}(A \mid B)$ is called the conditional probability of $A$ given $B$. The conditional expectation of $f$ given the event $B$ is defined by $$\mathbb{E}(f \mid B)=\int{\Omega} f d \mathbb{P}{B}=\frac{1}{\mathbb{P}(B)} \int{B} f d \mathbb{P} .$$
Clearly, the conditional expectation of $f$ given the event $B$ represents the average value of $f$ on $B$.

In many concrete problems, it is not enough to consider the conditional expectation given only one event. Instead, it is quite useful to define the conditional expectation to be a suitable random variable. For example, when consider two conditional expectations $\mathbb{E}(f \mid B)$ and $\mathbb{E}\left(f \mid B^{c}\right)$ simultaneously, we simply define it as a function $\mathbb{E}(f \mid B) \chi_{B}(\omega)+\mathbb{E}\left(f \mid B^{c}\right) \chi_{B^{c}}(\omega)$ rather than regarding it as two numbers. Before considering the general setting, we begin with the following special case.

## 数学代写|随机过程统计代写Stochastic process statistics代考|A Riesz-Type Representation Theorem

In this section, we shall prove a Riesz-type representation theorem, which will play important roles in the study of both controllability and optimal control problems for stochastic evolution equations.

Let $\left(X_{1}, \mathcal{M}{1}, \mu{1}\right)$ and $\left(X_{2}, \mathcal{M}{2}, \mu{2}\right)$ be two finite measure spaces, and let $H$ be a Banach space. Let $\mathcal{M}$ be a sub- $\sigma$-field of $\mathcal{M}{1} \times \mathcal{M}{2}$, and for any $1 \leq p, q<\infty$, let
$L_{\mathcal{M}}^{p}\left(X_{1} ; L^{q}\left(X_{2} ; H\right)\right)=\left{\varphi: X_{1} \times X_{2} \rightarrow H \mid \varphi(\cdot)\right.$ is strongly $\mathcal{M}$-measurable w.r.t. $\mu_{1} \times \mu_{2}$ and $\left.\int_{X_{1}}\left(\int_{X_{2}}\left|\varphi\left(x_{1}, x_{2}\right)\right|{H}^{q} d \mu{2}\right)^{\frac{p}{q}} d \mu_{1}<\infty\right} .$
Likewise, let
$L_{\mathcal{M}}^{\infty}\left(X_{1} ; L^{q}\left(X_{2} ; H\right)\right)=\left{\varphi: X_{1} \times X_{2} \rightarrow H \mid \varphi(\cdot)\right.$ is strongly $\mathcal{M}$-measurable w.r.t. $\mu_{1} \times \mu_{2}$ and ess $\left.\operatorname{up}{x{1} \in X_{1}}\left(\int_{X_{2}}\left|\varphi\left(x_{1}, x_{2}\right)\right|{H}^{q} d \mu{2}\right)^{\frac{1}{q}}<\infty\right}$,
$L_{\mathcal{M}}^{p}\left(X_{1} ; L^{\infty}\left(X_{2} ; H\right)\right)=\left{\varphi: X_{1} \times X_{2} \rightarrow H \mid \varphi(\cdot)\right.$ is strongly $\mathcal{M}$-measurable w.r.t. $\mu_{1} \times \mu_{2}$ and $\left.\int_{X_{1}}\left(\operatorname{ess} \sup {x{2} \in X_{2}}\left|\varphi\left(x_{1}, x_{2}\right)\right|{H}^{p}\right) d \mu{1}<\infty\right}$,
$L_{\mathcal{M}}^{\infty}\left(X_{1} ; L^{\infty}\left(X_{2} ; H\right)\right)=\left{\varphi: X_{1} \times X_{2} \rightarrow H \mid \varphi(\cdot)\right.$ is strongly $\mathcal{M}$-measurable w.r.t. $\mu_{1} \times \mu_{2}$ and ess sup $\left.\left(x_{1}, x_{2}\right) \in X_{1} \times X_{2}\left|\varphi\left(x_{1}, x_{2}\right)\right|_{H}<\infty\right}$.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Conditional Expectation

$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} .$$

$$\mathbb{E}(f \mid B)=\int \Omega f d \mathbb{P} B=\frac{1}{\mathbb{P}(B)} \int B f d \mathbb{P} .$$

$\mathbb{E}(f \mid B) \chi_{B}(\omega)+\mathbb{E}\left(f \mid B^{c}\right) \chi_{B^{c}}(\omega)$ 而不是将其视为两个数字。在考虑一般设置之前，我们先从以下特殊情况 开始。

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