Compound Poisson process

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 variable, then this compound Poisson process is named 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:

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

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

\begin{align} \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 t E(D^2). \end{align}

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{align} 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{align}

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.

Fitting a compound Poisson process

The parameters for independent observations of a compound Poisson process can be chosen using a maximum likelihood estimator using Simar's algorithm,[1] which has been shown to converge.[2]