Gamma process

A gamma process is a random process with independent gamma distributed increments. Often written as ${\displaystyle \Gamma (t;\gamma ,\lambda )}$, it is a pure-jump increasing Lévy process with intensity measure ${\displaystyle \nu (x)=\gamma x^{-1}\exp(-\lambda x),}$ for positive ${\displaystyle x}$. Thus jumps whose size lies in the interval ${\displaystyle [x,x+dx)}$ occur as a Poisson process with intensity ${\displaystyle \nu (x)dx.}$ The parameter ${\displaystyle \gamma }$ controls the rate of jump arrivals and the scaling parameter ${\displaystyle \lambda }$ 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 (${\displaystyle \mu }$) and variance (${\displaystyle v}$) of the increase per unit time, which is equivalent to ${\displaystyle \gamma =\mu ^{2}/v}$ and ${\displaystyle \lambda =\mu /v}$.

Properties

Since we use the Gamma function in these properties, we may write the process at time ${\displaystyle t}$ as ${\displaystyle X_{t}\equiv \Gamma (t;\gamma ,\lambda )}$ to eliminate ambiguity.

Some basic properties of the gamma process are:

Marginal distribution

The marginal distribution of a gamma process at time ${\displaystyle t}$ is a gamma distribution with mean ${\displaystyle \gamma t/\lambda }$ and variance ${\displaystyle \gamma t/\lambda ^{2}.}$

That is, its density ${\displaystyle f}$ is given by

$${\displaystyle f(x;t,\gamma ,\lambda )={\frac {\lambda ^{\gamma t}}{\Gamma (\gamma t)}}x^{\gamma t\,-\,1}e^{-\lambda x}.}$$

Scaling

Multiplication of a gamma process by a scalar constant ${\displaystyle \alpha }$ is again a gamma process with different mean increase rate.

${\displaystyle \alpha \Gamma (t;\gamma ,\lambda )\simeq \Gamma (t;\gamma ,\lambda /\alpha )}$

Adding independent processes

The sum of two independent gamma processes is again a gamma process.

${\displaystyle \Gamma (t;\gamma _{1},\lambda )+\Gamma (t;\gamma _{2},\lambda )\simeq \Gamma (t;\gamma _{1}+\gamma _{2},\lambda )}$

Moments

${\displaystyle \mathbb {E} (X_{t}^{n})=\lambda ^{-n}\Gamma (\gamma t+n)/\Gamma (\gamma t),\ \quad n\geq 0,}$ where ${\displaystyle \Gamma (z)}$ is the Gamma function.

Moment generating function

${\displaystyle \mathbb {E} {\Big (}\exp(\theta X_{t}){\Big )}=(1-\theta /\lambda )^{-\gamma t},\ \quad \theta <\lambda }$

Correlation

${\displaystyle \operatorname {Corr} (X_{s},X_{t})={\sqrt {s/t}},\ s<t}$, for any gamma process ${\displaystyle X(t).}$

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.