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Compound Poisson process

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A compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. To be precise, a compound Poisson process, parameterised by a rate {\displaystyle \lambda >0} and jump size distribution G, is a process {\displaystyle \{\,Y(t):t\geq 0\,\ given by

{\displaystyle Y(t)=\sum _{i=1}^{N(t)}D_{i

where, {\displaystyle \{\,N(t):t\geq 0\,\ is the counting variable of a Poisson process with rate {\displaystyle \lambda }, and {\displaystyle \{\,D_{i}:i\geq 1\,\ are independent and identically distributed random variables, with distribution function G, which are also independent of {\displaystyle \{\,N(t):t\geq 0\,\}.\,}[1]

When {\displaystyle D_{i are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process. [citation needed]

Properties of the compound Poisson process#edit

The expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:

{\displaystyle \operatorname {E} (Y(t))=\operatorname {E} (D_{1}+\cdots +D_{N(t)})=\operatorname {E} (N(t))\operatorname {E} (D_{1})=\operatorname {E} (N(t))\operatorname {E} (D)=\lambda t\

Making similar use of the law of total variance, the variance can be calculated as:

{\displaystyle {\begin{aligned}\operatorname {var} (Y(t))&=\operatorname {E} (\operatorname {var} (Y(t)\mid N(t)))+\operatorname {var} (\operatorname {E} (Y(t)\mid N(t)))\\[5pt]&=\operatorname {E} (N(t)\o

Lastly, using the law of total probability, the moment generating function can be given as follows:

{\displaystyle \Pr(Y(t)=i)=\sum _{n}\Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)}
{\displaystyle {\begin{aligned}\operatorname {E} (e^{sY})&=\sum _{i}e^{si}\Pr(Y(t)=i)\\[5pt]&=\sum _{i}e^{si}\sum _{n}\Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)\\[5pt]&=\sum _

Exponentiation of measures#edit

Let N, Y, and D be as above. Let μ be the probability measure according to which D is distributed, i.e.

{\displaystyle \mu (A)=\Pr(D\in A).\,}

Let δ0 be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure

{\displaystyle \exp(\lambda t(\mu -\delta _{0}))\,}

where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by

{\displaystyle \exp(\nu )=\sum _{n=0}^{\infty }{\nu ^{*n} \over n!

and

{\displaystyle \nu ^{*n}=\underbrace {\nu *\cdots *\nu } _{n{\text{ factors

is a convolution of measures, and the series converges weakly.

See also#edit

References#edit

  1. ^ Ross, Sheldon M. (1996). Stochastic processes. Wiley series in probability and statistics (2nd ed.). New York: Wiley. ISBN 978-0-471-12062-9.
Categories Poisson point processesLévy processes

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