Gamma process

A Gamma process is a Lévy process with independent Gamma increments. Often written as $Gamma(t;gamma,lambda)$, it is a pure-jump increasing Levy process with intensity measure $u(x)=gamma\; x^\{-1\}exp(-lambda\; x)$, for positive $x$. Thus jumps whose size lies in the interval $[x,x+dx]$ occur as a Poisson process with intensity $u(x)dx$.The parameter $gamma$ controls the rate of jump arrivals and the scaling parameter $lambda$ inversely controls the jump size.

The marginal distribution of a Gamma process at time $t$, is a Gamma distribution with mean $gamma\; t/lambda$ and variance $gamma\; t/lambda^2$.

The Gamma process is sometimes also parameterised in terms of the mean ($mu$) and variance ($v$) per unit time, which is equivalent to $gamma\; =\; mu^2/v$ and $lambda\; =\; mu/v$.

Some basic properties of the Gamma process are:

:$alphaGamma(t;gamma,lambda)\; =\; Gamma(t;gamma,lambda/alpha),$ (scaling)

:$Gamma(t;gamma\_1,lambda)\; +\; Gamma(t;gamma\_2,lambda)\; =\; Gamma(t;gamma\_1+gamma\_2,lambda),$ (adding independent processes)

:$mathbb\{E\}(X\_t^n)\; =\; lambda^\{-n\}Gamma(gamma\; t+n)/Gamma(gamma\; t),\; ngeq\; 0$ (moments), where $Gamma(z)$ is the Gamma function.

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A good reference for Levy processes, including the Gamma process, is "Lévy Processes and Stochastic Calculus" by David Applebaum, CUP 2004, ISBN 0-521-83263-2.