Poisson point processes

October 14, 2019 — January 29, 2020

Lévy processes
point processes
stochastic processes

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Poisson processes are possibly the simplest subordinators, i.e. non-decreasing Lévy processes. They pop up everywhere, especially as a representation of point processes, and as the second continuous-time stochastic process anyone learns after the Brownian motion.

1 Basics

A Poisson process \(\{\mathsf{n}\}\sim \operatorname{PoisP}(\lambda)\) is a stochastic process whose inter-occurrence times are identically and independently distributed such that \(\mathsf{t}_i-\mathsf{t}_{i-1}\sim\operatorname{Exp}(\lambda)\) (rate parameterization). By convention, we set all \(\mathsf{t}_i\geq 0\) and \(\mathsf{n}(0)=0\) a.s. The counting process that increases for each event is one convenient representation for such a process. In that case we think of the paths of the process as \(\mathsf{n}: \mathbb{R}\mapsto\mathbb{Z}^+\) such that \(\mathsf{n}(t)\equiv \sum_{i=1}^\mathsf{n}\mathbb{I}_{\{\mathsf{t}_i<t\}}\).

The following is a representation of such a creature based on Matt Motoki’s answer on stackoverflow.

lambda <- 0.3
last_arrival <- 0
arrival_times <- c()
n <- 10
counts = 1:n
for (i in counts) {
  last_arrival <- rexp(1, rate = lambda) + last_arrival
  arrival_times <- c(arrival_times,last_arrival)

plot(arrival_times, counts, pch=16, ylim=c(0, n))
points(arrival_times, c(0, counts[-n]))
  x0 = c(0, arrival_times[-n]),
  y0 = c(0, counts[-n]),
  x1 = arrival_times,
  y1 = c(0, counts[-n])

Poisson paths

Poisson paths

It is easy to show that \(\mathsf{n}(t)\sim\textrm{Pois}(\lambda t)\), where the pmf of a Poisson RV is

\[f(k;\lambda)={\frac {\lambda^{n}}{n!}}e^{-\lambda}.\]

It is a standard result that the increments of such processes over disjoint intervals such process have a Poisson distribution also. For \(t \geq s\):

\[\mathsf{n}(t)-\mathsf{n}(s) \sim \textrm{Pois}\left((t-s)\lambda\right).\]

Note also the standard result that

\[\lambda:=\lim_{h\to 0} \frac{\mathrm E\left(\mathsf{n}(t,t+h)\right)}{h}.\]

We call \(\lambda\) the rate.

2 Poisson distribution

2.1 Moments

Using the moment generating function

\[\Ex[\exp(s\mathsf{n})]= \exp[\lambda (e^{s}-1)]\]

we can find the moments by differentiating,

\[\begin{aligned} \frac{\dd}{\dd s}\Ex[\exp(s\mathsf{n})] &=\frac{\dd}{\dd s} \exp[\lambda (e^{s}-1)]\\ &=\lambda e^{s} \exp[\lambda (e^{s}-1)]\\ \end{aligned}\]

and thus

\[\begin{aligned} \Ex[\mathsf{n}] &= \lambda\\ \Ex[\mathsf{n}^2] &= \lambda+\lambda^2\\ \Ex[\mathsf{n}^3] &= \lambda+\lambda^2 + \lambda^3\\ \Ex[\mathsf{n}^4] &= \lambda+7\lambda^2+6\lambda^3+\lambda^4\\ \dots\\ \Ex[\mathsf{n}^k] &=\sum _{i=0}^{k}\lambda ^{i}\left\{{\begin{matrix}k\\i\end{matrix}}\right\}, \end{aligned}\]

where \(\left\{{\begin{matrix}k\\i\end{matrix}}\right\}\) is a Touchard/Bell polynomial, whatever that is.

3 Poisson bridge

Suppose we are given the value of a Poisson process at time \(0\) and time \(1\) and are concerned with some \(t\in(0,1).\) We wish to know the conditional distribution \(P(\mathsf{n}(t)|\mathsf{n}(1)=S)\), i.e. the Poisson bridge. Using the distributions of the increments and the independence property,

\[\begin{aligned} P[\mathsf{n}(t)=n|\mathsf{n}(1)=S] &=\frac{P[\mathsf{n}(t)=n\cap \mathsf{n}(1)=S]}{P[\mathsf{n}(1)=S]}\\ &=\frac{P[\mathsf{n}(t)=n\cap \tilde{\mathsf{n}}(1-t)=S-n]}{P[\mathsf{n}(1)=S]}\\ &=\frac{ \frac{(\lambda t)^{n}}{n!}e^{-\lambda t} \frac{(\lambda (1-t)^{S-n})}{(S-n)!}e^{-\lambda (1-t)} }{ \frac{\lambda^S }{S!}e^{-\lambda } }\\ &=\frac{(\lambda t)^{n}(\lambda (1-t))^{S-n}}{\lambda^{S}} \frac{e^{-\lambda t}e^{-\lambda (1-t)}}{e^{-\lambda}} \frac{S!}{(S-n)!n!}\\ &=(t)^{n} (1-t)^{S-n} \left(\begin{array}{cc}S\\n\end{array}\right)\\ &=\operatorname{Binom}(n;S, t). \end{aligned}\] So we can simulate a point Poisson bridge at some \(t<1\) by sampling a Binomial random variable.

4 Hitting time

Consider the hitting time of a level \(\ell\in \bb{N}\) for a Poisson process \(\mathsf{n}\).

\[\begin{aligned} \bb{P}\left[\mathsf{n}(t)<\ell\right] &=\bb{P}\left[\inf_s [\mathsf{n}(s) \geq \ell] >t\right]\\ &=\bb{P}\left[ \mathsf{t}_{\ell} >t\right]\\ &=\bb{P}\left[ \left(\sum_{1<k\leq \ell }\mathsf{t}_{k}-\mathsf{t}_{k-1} \right)>t\right]. \end{aligned}\]

But since \(\mathsf{t}_{k}-\mathsf{t}_{k-1} \sim \operatorname{Exp}(\lambda),\) and i.i.d. and the sum of i.i.d. Exponential variates is a Gamma variate, we have that

\[\mathsf{t}_\ell \sim \operatorname{Gamma}(\ell, \lambda)\]

and hence the density of this random variable is the Gamma distribution density,

\[f(t)={\frac{\ell ^{\lambda}}{\Gamma(\lambda )}}x^{\lambda -1}e^{-\ell x}.\]

The Gamma process is a kind of dual to the Poisson; we can think of it as a distribution over specific hitting times of an imaginary latent Poisson process.

5 Taylor expansion

See Schoutens, Leuven, and Studer (2001).

6 References

Giles. 2016. Algorithm 955: Approximation of the Inverse Poisson Cumulative Distribution Function.” ACM Trans. Math. Softw.
Kingman. 1992. Poisson Processes.
Schoutens, Leuven, and Studer. 2001. Stochastic Taylor Expansions for Poisson Processes and Applications Towards Risk Management.”