The Living Thing / Notebooks :

Gamma processes

Sound more exotic than Brownian motion but really aren't

A simple univariate Lévy process of a particular structure. (monotonic increasing)

Notes to myself on its care and feeding.

We write it as \(\Gamma (t;\gamma ,\lambda )\) 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)=\gamma 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.

The marginal distribution of the an increment of duration \(t\) is given by the Gamma distribution

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

This is the shape-rate parameterisation, with rate \(\lambda\) and shape \(\gamma.\)

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

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