Compound Poisson process

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 $$\lambda > 0$$ 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 the counting variable of 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.

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


 * $$\operatorname E(Y(t)) = \operatorname E(D_1 + \cdots + D_{N(t)}) = \operatorname E(N(t))\operatorname E(D_1) = \operatorname  E(N(t)) \operatorname E(D) = \lambda t \operatorname  E(D).$$

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

\begin{align} \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)\operatorname{var}(D)) + \operatorname{var}(N(t) \operatorname E(D)) \\[5pt] &= \operatorname{var}(D) \operatorname E(N(t)) + \operatorname E(D)^2 \operatorname{var}(N(t)) \\[5pt] &= \operatorname{var}(D)\lambda t + \operatorname E(D)^2\lambda t \\[5pt] &= \lambda t(\operatorname{var}(D) + \operatorname E(D)^2) \\[5pt] &= \lambda t \operatorname E(D^2). \end{align} $$

Lastly, using the law of total probability, the moment generating function can be given as follows:
 * $$\Pr(Y(t)=i) = \sum_n \Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n) $$



\begin{align} \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_n \Pr(N(t)=n) \sum_i e^{si} \Pr(Y(t)=i\mid N(t)=n) \\[5pt] & = \sum_n \Pr(N(t)=n) \sum_i e^{si}\Pr(D_1 + D_2 + \cdots + D_n=i) \\[5pt] & = \sum_n \Pr(N(t)=n) M_D(s)^n \\[5pt] & = \sum_n \Pr(N(t)=n) e^{n\ln(M_D(s))} \\[5pt] & = M_{N(t)}(\ln(M_D(s))) \\[5pt] & = 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.