# Gamma processes

### Sound more exotic than Brownian motion but really aren't

A simple univariate Lévy process of a particular structure. (monotonic increasing, real-valued.)

Notes to myself on its care and feeding.

We write it as $\Gamma (t;\alpha ,\lambda )$ to facilitate copy-pasting to and from Wikipedia, because I’m editing that article concurrently with this. According to that same source, a Gamma process is a pure jump process with jump intensity

\begin{equation*} \nu (x)=\alpha x^{-1}\exp(-\lambda x). \end{equation*}

That is, the Poisson rate, with respect to “time” $t$, of jumps whose size is in $[x, x+dx)$, is $\nu(x)dx.$ This only makes sense on the first reading if you are already familiar with Lévy processes, or terribly clever.

The marginal distribution of the an increment of duration $t$ is given by the Gamma distribution

\begin{equation*} f(x;t,\alpha ,\lambda )={\frac {\lambda ^{\alpha t}}{\Gamma (\alpha t)}}x^{\alpha t\,-\,1}e^{-\lambda x}. \end{equation*}

This is the shape-rate parameterisation, with rate $\lambda$ and shape $\alpha.$

Note that if $\alpha t=1,$ then $\Gamma (t;\alpha ,\lambda )\sim \operatorname{Exp}(\lambda).$

## Refs

BaPS01
Barndorff-Nielsen, O. E., Pedersen, J., & Sato, K.-I. (2001) Multivariate subordination, self-decomposability and stability. Advances in Applied Probability, 33(1), 160–187. DOI.
BhWa09
Bhattacharya, R. N., & Waymire, E. C.(2009) Stochastic Processes with Applications. . Society for Industrial and Applied Mathematics
Brém72
Brémaud, P. (1972) A martingale approach to point processes. . University of California, Berkeley
Grif76
Griffeath, D. (1976) Introduction to Random Fields. In Denumerable Markov Chains (pp. 425–458). Springer New York
GKKM10
Gusak, D., Kukush, A., Kulik, A., Mishura, Y., & Pilipenko, A. (2010) Theory of stochastic processes : with applications to financial mathematics and risk theory. . New York: Springer New York
Kuto84
Kutoyants, Y. A.(1984) Parameter Estimation for Stochastic Processes. . Berlin: Heldermann Verlag
Lefe07
Lefebvre, M. (2007) Applied Stochastic Processes. . Springer New York
Olof05
Olofsson, P. (2005) Probability, statistics, and stochastic processes. . Hoboken, N.J: Hoboken, N.J.: Wiley-Interscience