Gamma process

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

Some basic properties of the gamma process are:

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

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