Gamma process

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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:^{[citation needed]}

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[]

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

Moment generating function[]

${\displaystyle \operatorname {E} {\Big (}\exp(\theta X_{t}){\Big )}=\left(1-{\frac {\theta }{\lambda }}\right)^{-\gamma t},\ \quad \theta <\lambda }$

Correlation[]

${\displaystyle \operatorname {Corr} (X_{s},X_{t})={\sqrt {\frac {s}{t}}},\ s<t}$, 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.

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