Gamma process

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

The marginal distribution of a Gamma process at time $t$ , is a Gamma distribution with mean $γ t / λ$ and variance $γ t / λ 2$ .

The Gamma process is sometimes also parameterised in terms of the mean ( $μ$ ) and variance ( $v$ ) per unit time, which is equivalent to $γ = μ 2 / v$ and $λ = μ / v$ .

Some basic properties of the Gamma process are:

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

$\Gamma(t;\gamma_1,\lambda) + \Gamma(t;\gamma_2,\lambda) = \Gamma(t;\gamma_1+\gamma_2,\lambda)\,$ (adding independent processes)

$\mathbb{E}(X_t^n) = \lambda^{-n}\Gamma(\gamma t+n)/\Gamma(\gamma t),\ n\geq 0$ (moments), where

Γ(z)

is the Gamma function.

$\mathbb{E}\Big(\exp(\theta X_t)\Big) = (1-\theta/\lambda)^{-\gamma t},\ \theta<\lambda$ (moment generating function)

$Corr(X_s, X_t) = \sqrt{s/t},\ s<t$ , for any Gamma process

X (t)

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.