Gamma process

In mathematics and probability theory, a gamma process, also known as (Moran-)Gamma subordinator,[1] is a random process with independent gamma distributed increments. The gamma distribution has scale parameter $\gamma$ and shape parameter $\lambda$ , often written as $\Gamma (\gamma ,\lambda )$ .[2] Both $\gamma$ and $\lambda$ must be greater than 0. The gamma process, often written as $\Gamma (t;\gamma ,\lambda )$ , 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

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

Some basic properties of the gamma process are:

Marginal distribution

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

That is, its density $f$ is given by

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 $\alpha$ is again a gamma process with different mean increase rate.

\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.

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

Moments

\operatorname {E} (X_{t}^{n})=\lambda ^{-n}\cdot {\frac {\Gamma (\gamma t+n)}{\Gamma (\gamma t)}},\ \quad n\geq 0,

where

\Gamma (z)

is the Gamma function.

Moment generating function

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

Correlation

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

Literature

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

References

Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 536. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
Klenke, Achim, ed. (2008), "The Poisson Point Process", Probability Theory: A Comprehensive Course, London: Springer, pp. 525–542, doi:10.1007/978-1-84800-048-3_24, ISBN 978-1-84800-048-3, retrieved 2023-04-04

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[Klenke536-1] Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 536. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.

[2] Klenke, Achim, ed. (2008), "The Poisson Point Process", Probability Theory: A Comprehensive Course, London: Springer, pp. 525–542, doi:10.1007/978-1-84800-048-3_24, ISBN 978-1-84800-048-3, retrieved 2023-04-04

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