### 金融代写|鞅论及其在金融中的应用代写Martingale theory代考| Two Properties of Stationary Point Processes

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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

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Two Properties of Stationary Point Processes

Property 1.1.1. Independent stationary point processes have a.s. no common points

Proof: Consider the situation described in Example $1.1 .2$ with $k=2$. It will be shown that $N_{1}$ and $N_{2}$ have almost surely no common point. The proof is given when one of them (say $N_{1}$ ) has a finite intensity, i.e. $E\left[N_{1}(0,1]\right]<\infty$. Then $E\left[N_{1}{t}\right]=0$ for all $t \in \mathbb{R}$, by stationarity. Therefore, by Fubini’s theorem
\begin{aligned} E\left[\int_{\mathbb{R}} N_{1}({s}) N_{2}(d s)\right] &=\int_{M_{1}} \int_{M_{2}} \int_{\mathbb{R}} m_{1}({s}) m_{2}(d s) \mathcal{P}{1}\left(d m{1}\right) \mathcal{P}{2}\left(d m{2}\right) \ &=\int_{M_{2}} \int_{\mathbb{R}}\left[\int_{M_{1}} m_{1}({s}) \mathcal{P}{1}\left(d m{1}\right)\right] m_{2}(d s) \mathcal{P}{2}\left(d m{2}\right) \ &=0 \end{aligned}

and this implies $\int_{\mathbb{R}} N_{1}({s}) N_{2}(d s)=0$, P-a.s.
Property 1.1.2. Let $\left(N, \theta_{t}, P\right)$ be a stationary point process, then
$$P({N(\mathbb{R})=0} \cup{N((0, \infty))=N((-\infty, 0))=+\infty})=1$$
Proof: Let $H_{t}={N((t, \infty))=0} . H_{t}$ increases with $t$ and
$$H=\bigcap_{n=0}^{\infty} H_{-n}={N(\mathbb{R})=0} \subset{N((0, \infty))<\infty}=\bigcup_{n=0}^{\infty} H_{n}=G$$
In addition $\theta_{t} H_{s}=H_{s-t}$ for all $s, t \in \mathbb{R}{\text {, so that }} P\left(H{n}\right)=$ constant, for all $n \in \mathbb{Z}$. Hence $P(G)=\lim {n \rightarrow \infty} P\left(H{n}\right)=\lim {n \rightarrow-\infty} P\left(H{n}\right)=P(H)$, which together with $H \subset G$ imply $P(G-H)=0$. A similar reasoning based on $\widetilde{H}_{s}={N((-\infty, s))=0}$ leads to $P\left(G^{\prime}-H\right)=0$, where $G^{\prime}={N((-\infty, 0))<$ $\infty}$. Therefore $P\left(\left(G \cup G^{\prime}\right)-H\right)=0$.

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Intensity of a Stationary Point Process

Let $\left(N, \theta_{t}, P\right)$ be a stationary point process. The non-negative (possibly infinite) number
$$\lambda=E[N((0,1])]$$
is called the intensity of $\left(N, \theta_{t}, P\right)$. More generally, define for all $C \in \mathcal{B}$
$$\lambda(C)=E[N(C)]$$
From the stationarity of $\left(N, \theta_{t}, P\right), \lambda(C+t)=\lambda(C)$ for all $t \in \mathbb{R}$, and therefore $\lambda$ defines a translation invariant, measure on $(\mathbb{R}, \mathcal{B})$. It is therefore proportional to the Lebesgue measure $l$ so that
$$\lambda(C)=\lambda \times l(C), \quad C \in \mathcal{B} .$$
Hypothesis 1.1.1. In this book we shall assume that the stationary point processes under consideration:

1. have a non-null and finste intensity;
2. are simple, i.e. $\left.P\left(N\left{T_{n}\right}\right)>1\right)=0$ for all $n \in \mathbb{Z}$;
3. are such that $P(N(0, \infty)=N(-\infty, 0)=+\infty)=1$.
These assumptions will generally not be recalled in the text.
Example 1.1.7. Rate of arrivals into a queueing system. In queueing theory $\left{T_{n}\right}$ is the sequence of arrival times of customers into a queueing system; the intensity $\lambda$ of $\left{T_{n}\right}$ is called the arrival rate. .

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|The Campbell Measure

Let $\left((N, Z), \theta_{t}, P\right)$ be a stationary marked point process with marks in $(K, \mathcal{K})$. Define the random counting measure $N_{Z}$ on $(\mathbb{R} \times K, \mathcal{B} \otimes \mathcal{K})$ by
$$N_{Z}(C \times L)=\int_{C} 1_{L}\left(Z_{0} \circ \theta_{s}\right) N(d s)=\sum_{n \in Z} 1_{L}\left(Z_{n}\right) 1_{C}\left(T_{n}\right),$$
where $C \in \mathcal{B}, L \in \mathcal{K}$. For all $t$,
$$N_{Z}\left(\theta_{t} \omega, C \times L\right)=N_{Z}\left(\omega_{1}(C+t) \times L\right) .$$
The intensity measure $\lambda_{Z}$ associated with the stationary marked point process $\left((N, Z), \theta_{t}, P\right)$ is the measure on $(\mathbb{R} \times K, \mathcal{B} \otimes \mathcal{K})$ defined by
$$\lambda_{Z}(C \times L)=E\left[N_{Z}(C \times L)\right],$$
where $C \in \mathcal{B}, L \in \mathcal{K}$. It is called the Campbell measure of $(N, Z)$. Since the intensity $\lambda$ of $\left(N, \theta_{t}, P\right)$ is finite by assumption, $\lambda_{Z}$ is a $\sigma$-finite measure. Indeed, $N_{Z}(C \times L) \leq N(C)$ implies $\lambda_{Z}(C \times L) \leq \lambda(C)$. From the $\theta_{t}$-stationarity of $P$, we have
$$\lambda_{Z}((C+t) \times L)=\lambda_{Z}(C \times L) .$$
Thus, for fixed $L \in \mathcal{K}, \lambda_{Z}(. \times L)$ is a translation invariant, $\sigma$-finite measure on $\mathbb{R}$ and it must therefore be proportional to the Lebesgue measure and necessarily
$$\lambda_{Z}(C \times L)=\lambda_{Z}((0,1] \times L) l(C) .$$
Let $\nu_{Z}$ be the measure on $(K, \mathcal{K})$ defined by
$$\nu_{Z}(L)=\frac{1}{\lambda} \lambda_{Z}((0,1] \times L) .$$
Then, summarizing the above results and definitions
$$\lambda_{Z}(C \times L)=\lambda l(C) \nu_{Z}(L) .$$
The probability measure $\nu_{Z}$ is called the Palm probability distribution of the marks $Z_{n}$.

Note that $\nu_{Z}(L)$ is the average number of points $T_{n}$ in $C \in \mathcal{B}$ such that $Z_{n}$ belongs to $L$, divided by the average number of points $T_{n}$ in $C$. This characterization does not depend upon $C \in \mathcal{B}$ provided $l(C)>0$.

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Two Properties of Stationary Point Processes

H=⋂n=0∞H−n=ñ(R)=0⊂ñ((0,∞))<∞=⋃n=0∞Hn=G

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|Intensity of a Stationary Point Process

λ=和[ñ((0,1])]

λ(C)=和[ñ(C)]

λ(C)=λ×l(C),C∈乙.

1. 具有非零和finste强度；
2. 很简单，即\left.P\left(N\left{T_{n}\right}\right)>1\right)=0\left.P\left(N\left{T_{n}\right}\right)>1\right)=0对全部n∈从;
3. 是这样的磷(ñ(0,∞)=ñ(−∞,0)=+∞)=1.
这些假设通常不会在文本中被召回。
例 1.1.7。进入排队系统的到达率。在排队论中\left{T_{n}\right}\left{T_{n}\right}是顾客进入排队系统的时间序列；强度λ的\left{T_{n}\right}\left{T_{n}\right}称为到达率。.

## 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|The Campbell Measure

ñ从(C×大号)=∫C1大号(从0∘θs)ñ(ds)=∑n∈从1大号(从n)1C(吨n),

ñ从(θ吨ω,C×大号)=ñ从(ω1(C+吨)×大号).

λ从(C×大号)=和[ñ从(C×大号)],

λ从((C+吨)×大号)=λ从(C×大号).

λ从(C×大号)=λ从((0,1]×大号)l(C).

ν从(大号)=1λλ从((0,1]×大号).

λ从(C×大号)=λl(C)ν从(大号).

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