Compound Poisson process

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A compound Poisson process is a continuous-time (random) 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. A compound Poisson process, parameterised by a rate $\lambda >0$ and jump size distribution G, is a process $\{\,Y(t):t\geq 0\,\}$ given by

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

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

When $D_{i}$ are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process which has the feature that two or more events occur in a very short time .

Properties of the compound Poisson process

Using conditional expectation, the expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:

\,E(Y(t))=E(E(Y(t)|N(t)))=E(N(t)E(D))=E(N(t))E(D)=\lambda tE(D).

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

{\begin{aligned}\operatorname {var} (Y(t))&=E(\operatorname {var} (Y(t)|N(t)))+\operatorname {var} (E(Y(t)|N(t)))\\&=E(N(t)\operatorname {var} (D))+\operatorname {var} (N(t)E(D))\\&=\operatorname {var} (D)E(N(t))+E(D)^{2}\operatorname {var} (N(t))\\&=\operatorname {var} (D)\lambda t+E(D)^{2}\lambda t\\&=\lambda t(\operatorname {var} (D)+E(D)^{2})\\&=\lambda tE(D^{2}).\end{aligned}}

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

\,\Pr(Y(t)=i)=\sum _{n}\Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n)

{\begin{aligned}E(e^{sY})&=\sum _{i}e^{si}\Pr(Y(t)=i)\\&=\sum _{i}e^{si}\sum _{n}\Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n)\\&=\sum _{n}\Pr(N(t)=n)\sum _{i}e^{si}\Pr(Y(t)=i|N(t)=n)\\&=\sum _{n}\Pr(N(t)=n)\sum _{i}e^{si}\Pr(D_{1}+D_{2}+\cdots +D_{n}=i)\\&=\sum _{n}\Pr(N(t)=n)M_{D}(s)^{n}\\&=\sum _{n}\Pr(N(t)=n)e^{n\ln(M_{D}(s))}\\&=M_{N(t)}(\ln(M_{D}(s)))\\&=e^{\lambda t\left(M_{D}(s)-1\right)}.\end{aligned}}

Exponentiation of measures

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

\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

\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

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

and

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

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

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Properties of the compound Poisson process

Exponentiation of measures

See also