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.

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

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