Gamma process

The gamma process is often abbreviated as ${\displaystyle X(t)\equiv \Gamma (t;\gamma ,\lambda )}$ where ${\displaystyle t}$ represents the time from 0. The shape parameter ${\displaystyle \gamma }$ (inversely) controls the jump size, and the rate parameter ${\displaystyle \lambda }$ controls the rate of jump arrivals, analogously with the gamma distribution.^[4] Both ${\displaystyle \gamma }$ and ${\displaystyle \lambda }$ must be greater than 0. We use the gamma function and gamma distribution in this article, so the reader should distinguish between ${\displaystyle \Gamma (\cdot )}$ (the gamma function), ${\displaystyle \Gamma (\gamma ,\lambda )}$ (the gamma distribution), and ${\displaystyle \Gamma (t;\gamma ,\lambda )}$ (the gamma process).

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

The process can also be defined as a stochastic process ${\displaystyle X(t),t\leq 0}$ with ${\displaystyle X(0)=0}$ and independent increments, whose marginal distribution ${\displaystyle f}$ of the random variable ${\displaystyle X(t)-X(s)}$ for an increment ${\displaystyle l=t-s\geq 0}$ is given by^[4] $${\displaystyle f(x;l,\gamma ,\lambda )=\Gamma (\gamma l,\lambda )={\frac {\lambda ^{\gamma l}}{\Gamma (\gamma l)}}x^{\gamma l-1}e^{-\lambda x}.}$$

It is also possible to allow the shape parameter ${\displaystyle \gamma }$ to vary as a function of time, ${\displaystyle \gamma (t)}$.^[4]

This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources in this section. Unsourced material may be challenged and removed. Find sources: "Gamma process" – news · newspapers · books · scholar · JSTOR (June 2023) (Learn how and when to remove this message)

Because the value at each time ${\displaystyle t}$ has mean ${\displaystyle \gamma t/\lambda }$ and variance ${\displaystyle \gamma t/\lambda ^{2},}$^[5] the gamma process is sometimes also parameterised in terms of the mean (${\displaystyle \mu }$) and variance (${\displaystyle v}$) of the increase per unit time. These satisfy ${\displaystyle \gamma =\mu ^{2}/v}$ and ${\displaystyle \lambda =\mu /v}$.

Multiplication of a gamma process by a scalar constant ${\displaystyle \alpha }$ is again a gamma process with different mean increase rate. $${\displaystyle \alpha \Gamma (t;\gamma ,\lambda )\simeq \Gamma (t;\gamma ,\lambda /\alpha )}$$

The sum of two independent gamma processes is again a gamma process. $${\displaystyle \Gamma (t;\gamma _{1},\lambda )+\Gamma (t;\gamma _{2},\lambda )\simeq \Gamma (t;\gamma _{1}+\gamma _{2},\lambda )}$$

The moment function helps mathematicians find expected values, variances, skewness, and kurtosis. $${\displaystyle \operatorname {E} (X_{t}^{n})=\lambda ^{-n}\cdot {\frac {\Gamma (\gamma t+n)}{\Gamma (\gamma t)}},\ \quad n\geq 0,}$$ where ${\displaystyle \Gamma (z)}$ is the Gamma function.

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

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

The gamma process is used as the distribution for random time change in the variance gamma process. Specifically, combining Brownian motion with a gamma process produces a variance gamma process,^[6] and a variance gamma process can be written as the difference of two gamma processes.^[3]

• Moran process