Gamma process

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A gamma process is a random process with independent gamma distributed increments. Often written as \Gamma(t;\gamma,\lambda), it is a pure-jump increasing Lévy process with intensity measure \nu(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 \nu(x)dx. The parameter \gamma controls the rate of jump arrivals and the scaling parameter \lambda inversely controls the jump size. It is assumed that the process starts from a value 0 at t=0.

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

Properties[edit]

Some basic properties of the gamma process are:[citation needed]

marginal distribution

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.

scaling
\alpha\Gamma(t;\gamma,\lambda) = \Gamma(t;\gamma,\lambda/\alpha)\,
adding independent processes
\Gamma(t;\gamma_1,\lambda) + \Gamma(t;\gamma_2,\lambda) = \Gamma(t;\gamma_1+\gamma_2,\lambda)\,
moments
\mathbb{E}(X_t^n) = \lambda^{-n}\Gamma(\gamma t+n)/\Gamma(\gamma t),\ \quad n\geq 0 , where \Gamma(z) is the Gamma function.
moment generating function
\mathbb{E}\Big(\exp(\theta X_t)\Big) = (1-\theta/\lambda)^{-\gamma t},\  \quad \theta<\lambda
correlation
\operatorname{Corr}(X_s, X_t) = \sqrt{s/t},\ s<t, for any gamma process X(t) .

The gamma process is used as the distribution for random time change in the variance gamma process.

References[edit]

  • Lévy Processes and Stochastic Calculus by David Applebaum, CUP 2004, ISBN 0-521-83263-2.