- 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 aPoisson 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 aGamma 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.
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