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) = γx − 1exp( − λx), for positive x. Thus jumps whose size lies in the interval [x,x + dx] occur as a Poisson process with intensity ν(x)dx.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 γt / λ and variance γt / λ2.

The Gamma process is sometimes also parameterised in terms of the mean (μ) and variance (v) per unit time, which is equivalent to γ = μ2 / 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.


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