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3.4.1 Lebesgue decomposition Let $\lambda, \mu$ be two finite measures on a measurable space $(\Omega, \mathcal{F})$. Then, there exists a non-negative function $f \in$ $L^{1}(\Omega, \mathcal{F}, \mu)$ and a measure $\nu$ singular to $\mu$ (i.e. such that there exists a set $S \in \mathcal{F}$ with $\mu(S)=0$ and $\nu(\Omega \backslash S)=0$ ) such that
$$\lambda(A)=\int_{A} f \mathrm{~d} \mu+\nu(A), \quad \text { for all } A \in \mathcal{F}$$

Proof Consider a linear functional $F x=\int x \mathrm{~d} \lambda$, acting in the space $L^{2}(\Omega, \mathcal{F}, \lambda+\mu)$. The estimate
$$|F x| \leq \sqrt{\lambda(\Omega)} \sqrt{\int_{\Omega}|x|^{2} \mathrm{~d} \lambda} \leq \sqrt{\lambda(\Omega)}|x|_{L^{2}(\Omega, \mathcal{F}, \lambda+\mu)}$$
shows that $F$ is well-defined and bounded. Therefore, there exists a function $y \in L^{2}(\Omega, \mathcal{F}, \lambda+\mu)$ such that $F x=\int_{\Omega} x y \mathrm{~d}(\lambda+\mu)$. Taking $x=1_{A}, A \in \mathcal{F}$, we see that
$$\lambda(A)=\int_{A} y \mathrm{~d} \lambda+\int_{A} y \mathrm{~d} \mu .$$
This in turn proves that $y \geq 0,(\lambda+\mu)$ a.e., and $y \leq 1, \lambda$ a.e. Let $S={\omega \mid y(\omega)=1} \in \mathcal{F}$. By $(3.23), \mu(S)=0$. Rewriting $(3.23)$ in the form $\int_{\Omega}(1-y) 1_{A} \mathrm{~d} \lambda=\int_{\Omega} y 1_{A} \mathrm{~d} \mu$, we see that for any non-negative measurable function $x$ on $\Omega, \int_{\Omega}(1-y) x \mathrm{~d} \lambda=\int_{\Omega} y x \mathrm{~d} \mu$. Define $f(\omega)=\frac{y(\omega)}{1-y(\omega)}$ on $S^{\mathrm{C}}$, and zero on $S$. If $A \in \mathcal{F}$, and $A \subset S^{\complement}$, we may take $x=1_{A} \frac{1}{1-y}$ to see that $\lambda(A)=\int_{A} f \mathrm{~d} \mu$. Also, let $\nu(A)=\lambda(S \cap A)$. Thus, $\nu\left(S^{\mathrm{C}}\right)=0$, i.e. $\mu$ and $\nu$ are singular. Moreover,
$$\lambda(A)=\lambda\left(A \cap S^{\mathbf{C}}\right)+\lambda(A \cap S)=\int_{A \cap S^{\mathrm{c}}} f \mathrm{~d} \mu+\nu(A)=\int_{A} f \mathrm{~d} \mu+\nu(A) .$$
Finally, $f$ belongs to $L^{1}(\Omega, \mathcal{F}, \mu)$, since it is non-negative and $\int_{\Omega} f \mathrm{~d} \mu=$ $\int_{S^{0}} f \mathrm{~d} \mu=\lambda\left(S^{\mathrm{C}}\right)<\infty .$
3.4.2 The Radon-Nikodym Theorem Under assumptions of 3.4.1, suppose additionally that $\mu(A)=0$ for some $A \in \mathcal{F}$ implies that $\lambda(A)=0$. Then $\nu=0$; i.e. $\lambda$ is absolutely continuous with respect to $\mu$.

Proof We know that $\mu(S)=0$, so that $\nu(S)=\lambda(S)=0$. On the other hand, $\nu\left(S^{\mathbf{C}}\right)=0$ so that $\nu=0$.

## 数学代写|随机过程作业代写Stochastic Processes代考|Examples of discrete martingales

3.5.1 Definition Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{F}{n,} n \geq$ 1 , be an increasing sequence of $\sigma$-algebras of measurable sets: $\mathcal{F}{n} \subset$ $\mathcal{F}{n+1} \subset \mathcal{F}$; such a sequence is called a filtration. A sequence $X{n}, n \geq 1$ of random variables $X_{n} \in L^{1}(\Omega, \mathcal{F}, \mathbb{P})$ is termed a martingale if $X_{n}$ is $\mathcal{F}{n}$ measurable and $\mathbb{E}\left(X{n} \mid \mathcal{F}{n-1}\right)=X{n-1}$ for all $n \geq 1$. To be more specific, we should say that $X_{n}$ is a martingale with respect to $\mathcal{F}_{n}$ and

$\mathbb{P}$. However, $\mathcal{F}{n}$ and $\mathbb{P}$ are often clear from the context and for simplicity we omit the phrase “with respect to $\mathcal{F}{n}$ and $\mathbb{P}^{\prime \prime}$. Similarly, a sequence $X_{n} \in L^{1}(\Omega, \mathcal{F}, \mathbb{P}), n \geq 1$ is termed a submartingale (with respect to $\mathcal{F}{n}$ and $\left.\mathbb{P}\right)$ if $X{n}$ are $\mathcal{F}{n}$ measurable and $\mathbb{E}\left(X{n+1} \mid \mathcal{F}{n}\right) \geq X{n}, n \geq 1$. If $-X_{n}$ is a submartingale, $X_{n}$ is called a supermartingale. Filtrations, martingales, supermartingales and submartingales indexed by a finite ordered set are defined similarly.
3.5.2 Exercise Show that $X_{n}, n \geq 1$, is a submartingale iff there exists a martingale $M_{n}, n \geq 1$, and a previsible sequence $A_{n}, n \geq 1$ (i.e. $A_{1}=0$ and $A_{n+1}, n \geq 1$, is $\mathcal{F}{n}$ measurable) such that $A{n+1} \geq$ $A_{n}$ (a.s.) and $X_{n}=M_{n}+A_{n}$. This decomposition, called the Doob decomposition, is unique in $L^{1}(\Omega, \mathcal{F}, \mathbb{P})$.
3.5.3 Sum of independent random variables If $X_{n}, n \geq 1$ are $(\mathrm{mu}-$ tually) independent random variables, and $E X_{n}=0$ for $n \geq 1$, then $S_{n}=\sum_{i=1}^{n} X_{i}$ is a martingale with respect to $\mathcal{F}{n}=\sigma\left(X{1}, \ldots, X_{n}\right)$. Indeed, by 3.3.1 (h)-(i), $\mathbb{E}\left(S_{n+1} \mid \mathcal{F}{n}\right)=\mathbb{E}\left(X{n+1}+S_{n} \mid \mathcal{F}{n}\right)=\mathbb{E}\left(X{n+1} \mid \mathcal{F}{n}\right)+$ $S{n}=E X_{n+1} 1_{\Omega}+S_{n}=S_{n}$, since $X_{n+1}$ is independent of $\sigma\left(X_{1}, \ldots, X_{n}\right)$.

## 数学代写|随机过程作业代写Stochastic Processes代考|Convergence of self-adjoint operators

3.6.1 Motivation In the previous section we have already encountered examples of theorems concerning convergence of conditional expectations. In Theorem $3.3 .1$ point $(\mathrm{m})$ and in Exercise $3.3 .8$ we saw that if the $\sigma$-algebra $\mathcal{G}$ is fixed, then the conditional expectation with respect to this $\sigma$-algebra behaves very much like an integral. In this section we devote ourselves to a short study of theorems that involve limit behavior

of conditional expectation $E\left(X \mid \mathcal{F}{n}\right)$ where $X$ is fixed and $\mathcal{F}{n}$ is a family of $\sigma$-algebras. This will lead us in a natural way to convergence theorems for martingales presented in Section 3.7.

If $\mathcal{F}{n}$ is a filtration in a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, then $L^{1}\left(\Omega, \mathcal{F}{n}, \mathbb{P}\right)$ is a non-decreasing sequence of subspaces of $L^{1}(\Omega, \mathcal{F}, \mathbb{P})$, and $L^{2}\left(\Omega, \mathcal{F}{n}, \mathbb{P}\right)$ is a non-decreasing sequence of subspaces of $L^{2}(\Omega, \mathcal{F}, \mathbb{P})$. If $X$ is a square integrable random variable, then the sequence $X{n}=E\left(X \mid \mathcal{F}{n}\right)$ of conditional expectations of $X$ is simply the sequence of projections of $X$ onto this sequence of subspaces. Thus, it is worth taking a closer look at asymptotic behavior of a sequence $x{n}=P_{n} x$, where $x$ is a member of an abstract Hilbert space $\mathbb{H}$ and $P_{n}$ are projections on a non-decreasing sequence of subspaces $\mathbb{H}{n}$ of this space. In view of Theorem 3.1.18, the assumption that $\mathbb{H}{n}$ is a non-decreasing sequence may be conveniently expressed as $\left(P_{\mathrm{n}} x, x\right) \leq\left(P_{\mathrm{n}+1} x, x\right) \leq(x, x)$.

As an aid in our study we will use the fact that projections are selfadjoint operators (see 3.1.19). Self-adjoint operators are especially important in quantum mechanics, and were extensively studied for decades. Below, we will prove a well-known theorem on convergence of self-adjoint operators and then use it to our case of projections. Before we do that, however, we need to introduce the notion of a non-negative operator and establish a lemma.

## 随机过程代写

3.4.1 Lebesgue 分解 Letλ,μ是可测空间上的两个有限测度(Ω,F). 那么，存在一个非负函数F∈ 大号1(Ω,F,μ)和一个措施ν单数μ（即存在一个集合小号∈F和μ(小号)=0和ν(Ω∖小号)=0) 使得
λ(一种)=∫一种F dμ+ν(一种), 对全部 一种∈F

|FX|≤λ(Ω)∫Ω|X|2 dλ≤λ(Ω)|X|大号2(Ω,F,λ+μ)

λ(一种)=∫一种是 dλ+∫一种是 dμ.

λ(一种)=λ(一种∩小号C)+λ(一种∩小号)=∫一种∩小号CF dμ+ν(一种)=∫一种F dμ+ν(一种).

3.4.2 Radon-Nikodym 定理 在 3.4.1 的假设下，另外假设μ(一种)=0对于一些一种∈F暗示λ(一种)=0. 然后ν=0; IEλ是绝对连续的μ.

## 数学代写|随机过程作业代写Stochastic Processes代考|Examples of discrete martingales

3.5.1 定义让(Ω,F,磷)是一个概率空间，让Fn,n≥1 , 是一个递增序列σ- 可测集的代数：Fn⊂ Fn+1⊂F; 这样的序列称为过滤。一个序列Xn,n≥1随机变量Xn∈大号1(Ω,F,磷)被称为鞅，如果Xn是Fn可测量和和(Xn∣Fn−1)=Xn−1对全部n≥1. 更具体地说，我们应该说Xn是关于的鞅Fn和

3.5.2 练习 证明Xn,n≥1, 是亚鞅当且仅当存在鞅米n,n≥1, 和一个可预见的序列一种n,n≥1（IE一种1=0和一种n+1,n≥1， 是Fn可测量的）使得一种n+1≥ 一种n(as) 和Xn=米n+一种n. 这种分解称为 Doob 分解，在大号1(Ω,F,磷).
3.5.3 独立随机变量之和 IfXn,n≥1是(米在−最终）独立随机变量，和和Xn=0为了n≥1， 然后小号n=∑一世=1nX一世是关于的鞅Fn=σ(X1,…,Xn). 事实上，到 3.3.1 (h)-(i)，和(小号n+1∣Fn)=和(Xn+1+小号n∣Fn)=和(Xn+1∣Fn)+ 小号n=和Xn+11Ω+小号n=小号n， 自从Xn+1独立于σ(X1,…,Xn).

## 数学代写|随机过程作业代写Stochastic Processes代考|Convergence of self-adjoint operators

3.6.1 动机 在上一节中，我们已经遇到了有关条件期望收敛的定理示例。定理3.3.1观点(米)并在运动中3.3.8我们看到，如果σ-代数G是固定的，那么关于这个的条件期望σ-代数的行为非常像积分。在本节中，我们致力于对涉及极限行为的定理进行简短的研究

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