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# Compound Poisson process

A **compound Poisson process** is a Poisson process where we let the jumps be of random height instead of always one as in the ordinary Poisson process. A compound Poisson process is an
important special case of the more geberal class of Lévy processes. A compound Poisson process can also be reprsented using a discrete time random walk a Poisson process. Let $\{X(k)\}_{k=0,1,\ldots}$ be a discrete time random walk and let $N$ be a Poisson process. We can now obtain a compound Poisson process $Z$ by setting $Z(t)=X(N(t))$.
The characteristic function of compound Poisson proces $X(t)$ is \[\phi_{X(t)}=E[e^{iyX(t)}]=e^{tK(y)},\] where $K(y)=iy\mu+\lambda(\phi_J(y)-1)$, where
$\phi_J$ is the characteristic function for the jump distribution which we
can choose freely.

Questions: Magnus Wiktorsson

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