Compound Poisson process

Compound Poisson process

A compound Poisson process with rate λ > 0 and jump size distribution G is a continuous-time stochastic 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 λ, 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\,\}.\,

Properties of the compound Poisson process

Using conditional expectation, the expected value of a compound Poisson process can be calculated 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.

See also


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