### 金融代写|鞅论及其在金融中的应用代写Martingale theory代考|The Palm Calculus of Point Processes

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• Statistical Inference 统计推断
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• 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|Stationary Marked Point Process

Introduction. The input into a queueing system can be viewed as a sequence of required service times together with the times at which these requests arrive, that is, a double sequence $\left{\left(T_{n}, \sigma_{n}\right)\right}$ indexed by the set $Z$ of relative integers, where $\sigma_{n}$ is the amount of service (in time units) needed by customer $n$, who arrives at time $T_{n}$. If there are no batch arrivals, then $T_{n}<T_{n+1}$. Since we are interested in the stationary behavior of the system, the sequence of arrival times $\left{T_{n}\right}$ contains arbitrarily large negative times. By convention, the negative or null times of the arrival sequence will be indexed by negative or null relative integers, and the positive times by positive integers: $\cdots<$ $T_{-2}<T_{-1}<T_{0} \leq 0<T_{1}<T_{2}<\ldots$

The sequence $\left{T_{n}\right}$ is a point process, and the double sequence $\left{\left(T_{n}, \sigma_{\pi}\right)\right}$ is a marked point process, $\sigma_{n}$ being the mark of point $T_{n}$. More complicated mark sequences $\left{Z_{n}\right}$ can be considered, where $Z_{n}$ is an attribute of customer $n$ including, among other possible choices, the amount of service $\sigma_{n}$ he requires and his priority class $U_{n}$. If the queueing system under consideration is a network of queues, the mark $Z_{n}$ will for instance feature the route followed by customer $n$ through the network and the amount of service he requires at each station along his route.

Even in the simplest models, intricate dependencies may exist among the members of the sequence $\left{\left(T_{n}, \sigma_{n}\right)\right}$. Of course, in the so-called elementary theory of queues, the stream of required services is of the $G I / G I$ type, i.e. $\left{T_{n}\right}$ and $\left{\sigma_{n}\right}$ are independent sequences, $\left{\sigma_{n}\right}$ is an $i . i . d$. (independent and identically distributed) sequence, and $\left{T_{n}\right}$ forms a renewal process, say a Poisson process. However, suppose that the customers go through a very simple queueing system with two servers operating at unit speed, with a first come first served queueing discipline. The sequence of times $\left{T_{n}{ }^{\prime}\right}$ at which customers leave the queueing system, with the convention $\cdots<T_{-2}^{\prime}<T_{-1}^{\prime}<$ $T_{0}^{\prime} \leq 0<T_{1}^{\prime}<T_{2}^{\prime}<\ldots$ may be of a complex nature, even in the $G I / G I$ case: customers may overtake one another, the delay incurred by a given customer is a complicated function of the past history of the input stream, etc. Suppose now that the customers, just after leaving, join a second queueing system where they require the same amount of service as in the first system. The input stream into the second system is $\left{\left(T_{n}^{\prime}, \sigma_{n}^{\prime}\right)\right}$ where $\sigma_{n}^{\prime}$ is in general different from $\sigma_{n}$ (due to overtaking). It can be shown that the input stream into the second system will most likely not be a $G I / G I$ stream, except in very special cases (in particular when $\left{T_{n}\right}$ is a homogeneous Poisson process, and $\left{\sigma_{n}\right}$ is i.i.d. with a probability distribution of the exponential type (this is Burke’s theorem)).

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

A counting measure on $\mathbb{R}$ (the real line) is a measure $m$ on $(\mathbb{R}, \mathcal{B})$, where $\mathcal{B}$ denotes the Borel $\sigma$-field of $\mathbb{R}$, such that
(a) $m(C) \in{0,1, \ldots, \infty}$ for all $C \in \mathcal{B}$,
(b) $m([a, b])<\infty$ for all bounded intervals $[a, b]$ of $\mathbb{R}$.
Let $M$ be the set of all counting measures on $\mathbb{R}$. The $\sigma$-field on $M$ generated by the functions $m \rightarrow m(C), C \in \mathcal{B}$, is denoted by $\mathcal{M}$. The measurable space $(M, \mathcal{M})$ is the canonical space of point processes on $\mathrm{R}$.

With each counting measure $m \in M$ (Figure 1.1.1), we can associate a unique sequence $\left{t_{n}\right}, n \in \mathbb{Z}$, of $\overline{\mathbb{R}}$, such that
$$m(.)=\sum_{n \in \mathcal{Z}} \delta_{t_{n}}(.)$$

with $-\infty \leq \cdots \leq t_{-1} \leq t_{0} \leq 0<t_{1} \leq t_{2} \leq \cdots \leq+\infty$ and card $\left{n \in \mathbb{Z} ; t_{n} \in\right.$ $[a, b]}<\infty$, for all $[a, b] \subset \mathbb{R}$. Here $\delta_{x}$ is the Dirac measure at $x \in \overline{\mathbb{R}}$, with the convention $\delta_{\infty}(.)=\delta_{-\infty}(.) \equiv 0$, the measure with no mass.
The time $t_{n}$ is called the $n$-th point of $m$. The mapping $m \rightarrow t_{n}$ is measurable from $\mathcal{M}$ to $\mathcal{B}$.

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

Let $Z=\left{Z_{n}\right}, n \in \mathbb{Z}$, be a sequence of measurable mappings from $(\Omega, \mathcal{F})$ into some measurable space $(K, K)$. It is called a sequence of marks of $\left(N, \theta_{t}\right)$ if for all $n$,
$$Z_{n}(\omega)=Z_{0}\left(\theta_{T_{n} \omega}\right)$$
If moreover $\left(N, \theta_{t}, P\right)$ is a stationary point process, $\left((N, Z), \theta_{t}, P\right)$ is called a stationary marked point process (with marks in $K$ ).

The Shadowing Property. The motivation for definition (1.1.3) is that we want to be sure that the mark $Z_{n}(\omega)$ associated with $T_{n}(\omega)$ follows $T_{n}(\omega)$ when the underlying counting measure $N(\omega)$ is shifted.

To see how this shadowing property works ( $Z_{n}$ is the shadow following $\left.T_{n}\right)$, consider Figure 1.1.4 below, where we have taken $Z_{n}=\sigma_{n}$, the required service of customer $n$, as an example.

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

(a)米(C)∈0,1,…,∞对全部C∈乙,
(b)米([一种,b])<∞对于所有有界区间[一种,b]的R.

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