Poisson Process

Also found in: Wikipedia.

Poisson process

A process given by a discrete random variable which has a Poisson distribution.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Poisson Process


a stochastic process describing the moments at which certain random events occur. In a Poisson process, the number of events occurring within any fixed interval of time has a Poisson distribution, and the numbers of the events occurring in nonoverlapping intervals of time are independent.

Suppose μ(s, t) is the number of events whose moments of occurrence τi satisfy the inequalities 0 ≤ s < τit, and suppose λ (s, t) is the mathematical expectation of μ(s, t). In a Poisson process, for any 0 ≤ s1 < t1 ≤ s2 < t2 ≤ … ≤ sr < tr’ the random variables μ(s1, t1), μ(s2, t2), …, μ(sr, tr) are independent, and the equality μ(s, t) = η has probability

e-λ(s, t)[λ(s, t)]n/n!

In a homogeneous Poisson process, λ (s, t) = a(t – s), where a is the mean number of events in a unit of time and the distances τnn-1 between neighboring moments τn are independent and have an exponential distribution with density ae-at, t ≥ 0.

If there are many independent processes that describe the moments certain rare events occur, the total process yields in the limit a Poisson process under certain conditions.

The Poisson process is a convenient mathematical model that is often used in various applications of probability theory. In particular, it is used to describe a request flow in queuing theory—for example, calls arriving at a telephone exchange or ambulance trips in response to traffic accidents in a large city.

A generalization of the Poisson process is the Poisson random distribution of points in a plane or in space. The number of points here in any fixed region has a Poisson distribution (with mean proportional to the area or volume of the region), and the numbers of points in nonoverlapping regions are independent. This distribution is often used in calculations in such fields as astronomy, physics, ecology, and engineering.


Feller, W. Vvedenie v teoriiu veroiatnostei i ee prilozheniia, vols. 1–2. Moscow, 1967. (Translated from English.)


The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Similar to the situations in [19], and [20], the traffic pattern in the IoT station usually forms a Poisson Process and it is suited to be modeled as a Markov Process.
The researchers found that there was strong evidence indicating a structural change in 1996 under the standard Poisson process model.
For example, Bates [14] studies pricing problem of option where the dynamics of underlying asset follows a jump-diffusion process with stochastic volatility, in which the jump components of asset price follow the compound Poisson process. However, Carr and Wu [15] and Huang and Wu [16] find the fact that the compound Poisson process cannot describe the many small jumps of financial assets.
We also design and implement abnormal behavior detection based on the Poisson process to obtain accurate test results.
At a cell arrival time the rainfall process jumps up by a random amount and at extinction time it jumps down by a random amount, both modeled as Poisson process. Each time the rain intensity changes, an exponential increase occurs either upwards or downwards.
To model the random discharge of pollutants such as raw sewage into this designated portion of the Ganges, we suppose that these discharges occur in accordance with a stationary Poisson process (Tijms, A First Course in Stochastic Models, 2003,) with rate or parameter a > 0.
The photon arrival process N(t) counts the number of arrivals during the time interval [0, t), and most of the previous studies assume that N(t) is a homogeneous Poisson process with rate (photon arrivals per unit time) [1, 10, 14, 15].
Markov Modulated Poisson Process (MMPP) is one of the most used models to capture the typical characteristics of the incoming traffic such as self-similar behavior (correlated traffic), burstiness behavior, and long range dependency, and is simply a Poisson process whose mean value changes according to the evolution of a Markov Chain [11, 12].
The process p is called a homogeneous Poisson process if the values of p over disjoint intervals of time are stochastically independent or equivalently the increments of [xi] over disjoint intervals of time are statistically independent and there exists a nonnegative constant [lambda] such that the probability that there are exactly n events over the time period ([t.sub.1], [t.sub.2]] is given by
It describes the basic math required for derivative pricing and financial engineering, including stochastic differential equation models; Ito's lemma for Brownian motion and Poisson process driven stochastic differential equations; stochastic differential equations that have closed form solutions; the factor model approach to arbitrage pricing; constructing a factor model pricing framework; its application to equity derivatives and interest rate and credit derivatives; approaches to hedging; computational methods used in derivative pricing from the factor model perspective; and the concept of risk neutral pricing.
The optimal control problem with random jumps was first considered by Boel and Varaiya [10]; in this case, the control system is often described by Brownian motion and Poisson processes. On the basis of proving the existence and uniqueness of solutions of a kind of forward backward stochastic differential equationwith Poissonjumps (FBSDEP), Wu and Wang [11] got the explicit form of the optimal control for LQ stochastic control problem where the state variable was described by a stochastic differential equation with a Poisson process (SDEP).