Poisson process

A Poisson process is a named after Siméon Denis Poisson stochastic process. He is a renewal process whose gains are Poisson distributed.

The rare events described by a Poisson process typically have but a huge risk ( as the product of cost and probability). Therefore, so that, for example, accidents in complex industrial plants, floods, plane crashes, etc. are often modeled in insurance.

Parameter

The distribution of growth has referred a parameter λ, this is called the intensity of the process as per time unit exactly λ jumps are expected ( expected value of the Poisson distribution is also λ ). The height of each jump is one, the times between jumps are exponentially distributed. The Poisson process is thus a discrete process in continuous (ie continuous ) time.

Definition

A stochastic process with càdlàg paths over a probability space is called a ( homogeneous ) Poisson process with intensity and, if the following three conditions are met:

  • (See Almost sure properties).
  • . The Poisson distribution indicated by parameters.
  • Be given for a sequence. Then the family of random variables is stochastically independent.

For the definition of the inhomogeneous Poisson process see Poisson process # The inhomogeneous Poisson process.

Properties

  • A Poisson process is apparently a stochastic process with independent increments.
  • A homogeneous Poisson process is a Markov process.
  • The period between two increases, so is exponentially distributed with parameter.
  • Is a Poisson process, then a Poisson process again. Thus, the growth of homogeneous Poisson processes are stationary.
  • For the expected value and variance.
  • For the quadratic variation, since the continuous martingale part vanishes and all the jumps have a height of 1.
  • Since the path of the process increases monotonically, is a submartingale with respect to its natural filtration.
  • If there is a stochastic process which meets the three defining characteristics, there exists a version of the process càdlàg paths, that is a Poisson process.
  • Ie compensated Poisson process and is a martingale with respect to its natural filtration.
  • A Poisson process is memoryless ( It is therefore necessary that the remaining waiting time to the next jump is independent of the previous waiting time. ( This follows from the exponential distribution ).

Compound Poisson processes

Is a Poisson process with intensity as well as independent, identically distributed random variables independent of, the stochastic process is

Called a compound Poisson process. Like the original Poisson process X is also a jump process independent growth and exponential ( μ ) - distributed intervals between the jumps with jump heights, which are distributed according to Y.

For the expectation value of the formula Forest is ( after the mathematician Abraham Wald )

The inhomogeneous Poisson process

In some cases it may be useful, not as a constant but as a function of time limiting. has to the two conditions

  • For all and

. meet

For an inhomogeneous Poisson process notwithstanding a homogeneous Poisson process:

  • , Again referred to the Poisson distribution with the parameter.
  • For the expectation value.
  • Also applies for the variance.
  • Are and two jump points of the inhomogeneous Poisson process, then an exponential distribution with parameter 1

Cox process

A non-homogeneous Poisson process with stochastic intensity function is called double stochastic Poisson process or after the English mathematician David Cox and Cox process. Consider a particular realization of, a Cox process behaves as an inhomogeneous Poisson process. For the expectation value of

Application Examples

  • General: Count of uniformly distributed events per area, space or time scale (eg number of raindrops on a road, number of stars in a volume V is a three-dimensional Poisson process ).
  • To determine the frequency of rare events such as insurance claims, decay processes, repair jobs or the number of goals in a football match (see the football book of Metin Tolan ).
  • The random number of telephone calls per unit time.
  • The random number to the customer at a switch per unit time.
  • The times in which requirements (people, jobs, phone calls, heap, ...) received by an operator (bank, server, telephone, memory management, ...).
  • The random number of non- germinating seeds from a pack.
  • The locations at which a thread has nubs.
  • Number of pixel defects on a TFT display.
  • Number of potholes on a country road.
  • Number of misprints in a book.
  • Number of accidents per unit time at an intersection.
  • The points in time at which a radioactive substance emitting a particles.
  • Random number of particles emitted from a radioactive substance in a given period.
  • The times of major losses of insurance. In the financial and insurance mathematics, the occurrence of damage to be covered is usually described by a composite Poisson process, in which the individual, occurring independently damage to Y are distributed. Provides you the damage process then still with a deterministic, negative drift ( insurance), one obtains a wealth process of the insurance company. This is followed by questions like: How likely is it that the asset process a certain threshold value x, ie the reserves of the insurance exceeds, and thus suffers a bankruptcy? How much of the negative drift or the rate should be in order to express the probability of bankruptcy under a given threshold?
  • Models for classes of shares, which also jumps are permitted. For this purpose, namely Lévy processes are often used, but as infinite activity is often difficult to measure, and compound Poisson processes are used.
  • Credit risk models help CDS spreads and evaluate other credit derivatives and model.
Martingale (probability theory) Actuarial science Mathematical finance Insolvency
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