Gamma process

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title: "Gamma process" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["lévy-processes"] description: "Stochastic process for effort or wear" topic_path: "general/levy-processes" source: "https://en.wikipedia.org/wiki/Gamma_process" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Stochastic process for effort or wear ::

::callout[type=note] the stochastic process ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/6c/Gamma-Process.png" caption="Two different gamma processes from time 0 until time 4. The red process has more occurrences in the timeframe compared to the blue process because its ''shape'' parameter is larger than the blue shape parameter."] ::

A gamma process, also called the Moran-Gamma subordinator, is a two-parameter stochastic process which models the accumulation of effort or wear over time. The gamma process has independent and stationary increments which follow the gamma distribution, hence the name. The gamma process is studied in mathematics, statistics, probability theory, and stochastics, with particular applications in deterioration modeling and mathematical finance.

Notation

The gamma process is often abbreviated as X(t)\equiv\Gamma(t;\gamma, \lambda) where t represents the time from 0. The shape parameter \gamma (inversely) controls the jump size, and the rate parameter \lambda controls the rate of jump arrivals, analogously with the gamma distribution. Both \gamma and \lambda must be greater than 0. We use the gamma function and gamma distribution in this article, so the reader should distinguish between \Gamma(\cdot) (the gamma function), \Gamma(\gamma, \lambda) (the gamma distribution), and \Gamma(t;\gamma, \lambda) (the gamma process).

Definition

The process is a pure-jump increasing Lévy process with intensity measure \nu(x)=\gamma x^{-1} \exp(-\lambda x), for all positive x. It is assumed that the process starts from a value 0 at t=0 meaning X(0)=0. Thus jumps whose size lies in the interval [x,x+dx) occur as a Poisson process with intensity \nu(x),dx.

The process can also be defined as a stochastic process X(t), t \leq 0 with X(0) = 0 and independent increments, whose marginal distribution f of the random variable X(t) - X(s) for an increment l = t-s \geq 0 is given by f(x;l, \gamma, \lambda) = \Gamma(\gamma l, \lambda) = \frac {\lambda^{\gamma l}}{\Gamma (\gamma l)} x^{\gamma l-1}e^{-\lambda x}.

Inhomogenous process

It is also possible to allow the shape parameter \gamma to vary as a function of time, \gamma(t).

Properties

Mean and variance

Because the value at each time t has 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) of the increase per unit time. These satisfy \gamma = \mu^2/v and \lambda = \mu/v.

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

The moment function helps mathematicians find expected values, variances, skewness, and kurtosis. \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

The moment generating function is the expected value of \exp(tX) where X is the random variable. \operatorname E\Big(\exp(\theta X_t)\Big) = \left(1- \frac\theta\lambda\right)^{-\gamma t},\ \quad \theta

Correlation

Correlation displays the statistical relationship between any two gamma processes. \operatorname{Corr}(X_s, X_t) = \sqrt{\frac s t},\ s, for any gamma process X(t) .

Related processes

The gamma process is used as the distribution for random time change in the variance gamma process. Specifically, combining Brownian motion with a gamma process produces a variance gamma process, and a variance gamma process can be written as the difference of two gamma processes.

Notes

References

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::