Compound Poisson process: Difference between revisions

Revision as of 17:39, 3 December 2011

Revision as of 16:54, 18 June 2012

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{{Unreferenced|date=December 2009}}

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 <math>\lambda>0\gt0</math> and jump size distribution ''G'', is a process <math>\{\,Y(t) : t \geq 0 \,\}</math> given by

:<math>Y(t) = \sum_{i=1}^{N(t)} D_i</math>

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:<math>

\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}

</math>