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*

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

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