\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]

Gamma processes provide the classic subordinator models, i.e. non-decreasing Lévy processes. By “gamma process” in fact I mean specifically a Lévy process with gamma increments. Other processes that happen to have gamma marginals, e.g. (Wolpert 2006) are not germane.

Ground zero for these processes appears to be (Ferguson 1974; Ferguson and Klass 1972).

Tutorial introductions to gamma processes can be found in in (Applebaum 2009; Asmussen and Glynn 2007; Rubinstein and Kroese 2016; Kyprianou 2014). Existence proofs etc are deferred to those sources. You could also see Wikipedia.

The marginal distribution of the an increment of duration \(t\) is given by the gamma distribution, which we had better cover first.

## Gamma distribution

Let us take a brief divergence into the gamma distribution which is the increment distribution the gamma process (Indeed, all divisible distributions induce Lévy processes.)

The density \(g(x;t,\alpha, \lambda )\) of the univariate gamma is

\[
g(x; \alpha, \lambda)=
\frac{ \lambda^{\alpha} }{ \Gamma (\alpha) } x^{\alpha\,-\,1}e^{-\lambda x},
x\geq 0.
\]
This is the *shape-rate* parameterisation, with rate \(\lambda\)
and shape \(\alpha,\).
We can think of the gamma distribution as the distribution
at time 1 of a gamma process.

If \(\rv{g}\sim \operatorname{Gamma}(\alpha, \lambda)\) then \(\bb E(\rv{g})=\alpha/\lambda\) and \(\var(\rv{g})=\alpha/\lambda^2.\)

We use various facts about the gamma distribution which quantify its divisibility properties.

- If \(\rv{g}_1\sim \operatorname{Gamma}(\alpha_1, \lambda),\,\rv{g}_2\sim \operatorname{Gamma}(\alpha_2, \lambda),\) and \(\rv{g}_1\perp \rv{g}_2,\) then \(\rv{g}_1+\rv{g}_2\sim \operatorname{Gamma}(\alpha_1+\alpha_2, \lambda)\) (additive rule)
- If \(\rv{g}\sim \operatorname{Gamma}(\alpha, \lambda)\) then \(c \rv{g}\sim \operatorname{Gamma}(\alpha, \lambda/c)\) (multiplicative rule)
- If \(\rv{g}_1\sim \operatorname{Gamma}(\alpha_1, \lambda)\perp \rv{g}_1\sim \operatorname{Gamma}(\alpha_2, \lambda)\) then \(\frac{\rv{g}_1}{\rv{g}_1+\rv{g}_2}\sim \operatorname{Beta}(\alpha_1, \alpha_2)\) independent of \(\rv{g}_1+\rv{g}_2\) (stick-breaking rule)

### Moments

Also note that the moment generating function of the gamma distribution is

\[ \bb{E}[\exp(\rv{g} s)]=\left(1-{\frac {s}{\lambda }}\right)^{-\alpha }{\text{ for }}s<\lambda\] which gives us expressions for all moments fairly easily;

\[\begin{aligned} \bb{E}[\rv{g}]&=\left.\frac{\dd}{\dd s}\bb{E}[\exp(\rv{g} t)]\right|_{s=0}\\ &=\left.\frac{\dd}{\dd s}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha }\right|_{s=0}\\ &=\left.\frac{\alpha}{\lambda}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha -1}\right|_{s=0}\\ &=\alpha/\lambda\\ \bb{E}[\rv{g}^2] &=\left.\frac{\dd}{\dd s}\frac{\alpha}{\lambda}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha -1}\right|_{s=0}\\ &=\left.\frac{\alpha^2+\alpha}{\lambda^2}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha -2}\right|_{s=0}\\ &=\frac{\alpha(\alpha+1)}{\lambda^2}\\ \bb{E}[\rv{g}^3] &=\left.\frac{\dd}{\dd s}\frac{\alpha^2+\alpha}{\lambda^2}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha -2}\right|_{s=0}\\ &=\frac{\alpha(\alpha+1)(\alpha+2)}{\lambda^3}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha -2}\\ &=\frac{\alpha(\alpha+1)(\alpha+2)}{\lambda^3}\\ \bb{E}[\rv{g}^4] &=\frac{\alpha(\alpha+1)(\alpha+2)(\alpha+3)}{\lambda^4}\\ &\dots\\ \bb{E}[\rv{g}^n] &=\frac{\langle \alpha \rangle_{n}}{\lambda^n}\\ \end{aligned}\]

Here \(\langle \alpha \rangle_{n}:=\frac{\Gamma(\alpha+n)}{\Gamma(\alpha)}\) is the rising factorial.

## Multivariate gamma distribution with dependence

I am not sure what general correlations are possible here, but one obvious possibility is to choose a transform matrix \(M\) with non-negative entries. Then the RV \(\{M\rv{g}(t)\}\) is still marginally a gamma variate, but the components of the vector are no longer independent. Is this the most general possible gamma distribution? What is the covariance structure of that process? 🏗

## Gamma superpositions

If we wish to know the distribution of the sum of a set of scaled gamma random variables, we can use moment-generating-function approaches
(Mathai 1982; Moschopoulos 1985).
It does not come out *so* simply as in the Gaussian case, with a recursive coefficient definition.

## The Gamma process

The univariate gamma *process* \(\{\rv{g}(t;\alpha,\lambda)\}_t\)
is an independent-increment process,
with time index \(t\) and parameters by \(\alpha, \lambda.\)
We assume it is started at \(\rv{g}(0)=0\).

The marginal density \(g(x;t,\alpha, \lambda )\) of the process at time \(t\) is a gamma RV, specifically, \[ g(x;t, \alpha, \lambda) =\frac{ \lambda^{\alpha t} } { \Gamma (\alpha t) } x^{\alpha t\,-\,1}e^{-\lambda x}, x\geq 0. \] That is, \(\rv{g}(t) \sim \operatorname{Gamma}(\alpha(t_{i+1}-t_{i}), \lambda)\).

which corresponds to increments per unit time in terms of \(\bb E(\rv{g}(1))=\alpha/\lambda\) and \(\var(\rv{g}(1))=\alpha/\lambda^2.\)

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

This leads to a method for simulating a path of a gamma process at a sequence of increasing times, \(\{t_1, t_2, t_3, t_L\}.\) Given \(\rv{g}(t_1;\alpha, \lambda),\) we know that the increments are distributed as independent variates \(\rv{g}_i:=\rv{g}(t_{i+1})-\rv{g}(t_{i})\sim \operatorname{Gamma}(\alpha(t_{i+1}-t_{i}), \lambda)\). Presuming we may simulate from the Gamma distribution, it follows that

\[\rv{g}(t_i)=\sum_{j \lt i}\left( \rv{g}(t_{i+1})-\rv{g}(t_{i})\right)=\sum_{j \lt i} \rv{g}_j.\]

A standard \(d\)-dimensional gamma process is the concatenation of \(d\) independent univariate gamma processes.

## Gamma bridge

Consider a univariate gamma process, \(\rv{g}(t)\) with \(\rv{g}(0)=0.\) The gamma bridge, analogous to the Brownian bridge, is the gamma process conditional upon attaining a fixed the value \(S=\rv{g}(1)\) at terminal time \(1.\) We write \(\rv{g}_{S}:=\{\rv{g}(t)\mid \rv{g}(1)=S\}_{0\lt t \lt 1}\) for the paths of this process.

We can simulate from the gamma bridge easily. Given the increments of the process are independent, if we have a gamma process \(\rv{g}\) on the index set \([0,1]\) such that \(\rv{g}(1)=S\), then we can simulate from the bridge paths which connect these points at intermediate time \(t,\, 0<t<1\) by recalling that we have known distributions for the increments; in particular \(\rv{g}(t)\sim\operatorname{Gamma}(\alpha, \lambda)\) and \(\rv{g}(1)-\rv{g}(t)\sim\operatorname{Gamma}(\alpha (1-t), \lambda)\) and these increments, as increments over disjoints sets, are themselves independent. Then, by the stick breaking rule,

\[\frac{\rv{g}(t)}{\rv{g}(1)}\sim\operatorname{Beta}(\alpha t, \alpha(1-t))\] independent of \(\rv{g}(1).\) We can therefore sample from a path of the bridge \(\rv{g}_{S}(t)\) for some \(t\lt 1\) by simulating \(\rv{g}_{S}(t)=B S,\) where \(B\sim \operatorname{Beta}(\alpha (t),\alpha (1-t)).\)

## Time-warped gamma process

Çinlar (1980) walks us through the mechanics of (deterministically) time-warping Gamma processes, which ends up being not too unpleasant. Predictable stochastic time-warps look like they should be OK. See (Singpurwalla 1997) for an application. Why bother? Linear superpositions of gamma processes can be hard work, and sometime the generalisation from time-warping can come out nicer. 🏗

## Matrix gamma processes

I am tempted to look at matrix-valued extensions, much as the Wishart distribution is a matrix valued extension of the gamma distribution.
“Wishart processes” are indeed a thing (Pfaffel 2012) but the common definition, unlike the common definition of the gamma process, is not necessarily (element-wise, or in any other sense) monotonic.
That is, it generalises the *square Bessel process*, which
is indeed marginally gamma distributed (specifically \(\chi^2\) distributed)
but also non-monotonic, so it is not a natural extension of what we have here.

## Centred gamma process

Define \(H_t:=\rv{g}_t-\alpha t /\lambda.\) Then we have a mean-zero process, in fact, a martingale, since we have subtracted the compensator from it.

We call the marginal distribution at time \(t=1\) a *centred gamma distribution*, and will recycle the letter \(H=G-\alpha /\lambda\).
As a linear transform of a random variable, the MGF of the centred gamma distribution is (ignoring questions of region of convergence)…

\[\begin{aligned} \bb{E}[\exp(H s)]&=\exp (\alpha s/\lambda)\bb{E}[\exp(G s)]\\ &=\exp (\alpha s/\lambda)\left(1-{\frac {s}{\lambda }}\right)^{-\alpha }\\ &=\exp (\alpha s/\lambda - \alpha \log(1-s/\lambda))\end{aligned}\]

## As a Lévy process

For arguments \(x, t>0\) and parameters \(\alpha, \lambda>0,\) we have the increment density as simply a gamma density:

\[ p_{X}(t, x)=\frac{\lambda^{\alpha t} x^{\alpha t-1} \mathrm{e}^{-x \lambda}}{ \Gamma(\alpha t)}, \]

This gives us a spectrally positive Lévy measure

\[\pi_{X}(x)=\frac{\alpha}{x} \mathrm{e}^{-\lambda x} \]

and Laplace exponent

\[\Phi_{X}(z)=\alpha \ln (1+ z/\lambda), z \geq 0. \]

That is, the Poisson rate, with respect to time \(t\) of jumps whose size is in the range \([x, x+dx)\), is \(\pi(x)dx.\) We think of this as an infinite superposition of Poisson processes driving different sized jumps, where the jumps are mostly tiny. This is how I think about Lévy process theory, at least.

Applebaum, David. 2004. “Lévy Processes—from Probability to Finance and Quantum Groups.” *Notices of the AMS* 51 (11): 12. https://core.ac.uk/download/pdf/50531.pdf.

———. 2009. *Lévy Processes and Stochastic Calculus*. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.

Asmussen, Søren, and Peter W. Glynn. 2007. *Stochastic Simulation: Algorithms and Analysis*. 2007 edition. New York: Springer.

Avramidis, Athanassios N., Pierre L’Ecuyer, and Pierre-Alexandre Tremblay. 2003. “New Simulation Methodology for Finance: Efficient Simulation of Gamma and Variance-Gamma Processes.” In *Proceedings of the 35th Conference on Winter Simulation: Driving Innovation*, 319–26. WSC ’03. New Orleans, Louisiana: Winter Simulation Conference. http://www-perso.iro.umontreal.ca/~lecuyer/myftp/papers/wsc03vg.pdf.

Barndorff-Nielsen, Ole E., Makoto Maejima, and Ken-Iti Sato. 2006. “Some Classes of Multivariate Infinitely Divisible Distributions Admitting Stochastic Integral Representations.” *Bernoulli* 12 (1): 1–33. https://projecteuclid.org/euclid.bj/1141136646.

Barndorff-Nielsen, Ole E., Jan Pedersen, and Ken-Iti Sato. 2001. “Multivariate Subordination, Self-Decomposability and Stability.” *Advances in Applied Probability* 33 (1): 160–87. https://doi.org/10.1017/S0001867800010685.

Bertoin, Jean. 1996. *Lévy Processes*. Cambridge Tracts in Mathematics 121. Cambridge ; New York: Cambridge University Press.

———. 1999. “Subordinators: Examples and Applications.” In *Lectures on Probability Theory and Statistics: Ecole d’Eté de Probailités de Saint-Flour XXVII - 1997*, edited by Jean Bertoin, Fabio Martinelli, Yuval Peres, and Pierre Bernard, 1717:1–91. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-48115-7_1.

———. 2000. *Subordinators, Lévy Processes with No Negative Jumps, and Branching Processes*. University of Aarhus. Centre for Mathematical Physics and Stochastics …. http://www.maphysto.dk/oldpages/events/LevyBranch2000/notes/bertoin.pdf.

Bhattacharya, Rabi N., and Edward C. Waymire. 2009. *Stochastic Processes with Applications*. Society for Industrial and Applied Mathematics. http://epubs.siam.org/doi/abs/10.1137/1.9780898718997.fm.

Bondesson, Lennart. 2012. *Generalized Gamma Convolutions and Related Classes of Distributions and Densities*. Springer Science & Business Media. http://books.google.com?id=sBDlBwAAQBAJ.

Buchmann, Boris, Benjamin Kaehler, Ross Maller, and Alexander Szimayer. 2015. “Multivariate Subordination Using Generalised Gamma Convolutions with Applications to V.G. Processes and Option Pricing,” February. http://arxiv.org/abs/1502.03901.

Connor, Robert J., and James E. Mosimann. 1969. “Concepts of Independence for Proportions with a Generalization of the Dirichlet Distribution.” *Journal of the American Statistical Association* 64 (325): 194–206.

Çinlar, Erhan. 1980. “On a Generalization of Gamma Processes.” *Journal of Applied Probability* 17 (2): 467–80. https://doi.org/10.2307/3213036.

Émery, Michel, and Marc Yor. 2004. “A Parallel Between Brownian Bridges and Gamma Bridges.” *Publications of the Research Institute for Mathematical Sciences* 40 (3): 669–88. https://doi.org/10.2977/prims/1145475488.

Ferguson, Thomas S. 1974. “Prior Distributions on Spaces of Probability Measures.” *The Annals of Statistics* 2 (4): 615–29. https://doi.org/10.1214/aos/1176342752.

Ferguson, Thomas S., and Michael J. Klass. 1972. “A Representation of Independent Increment Processes Without Gaussian Components.” *The Annals of Mathematical Statistics* 43 (5): 1634–43. https://doi.org/10.1214/aoms/1177692395.

Figueroa-López, José E. 2012. “Jump-Diffusion Models Driven by Lévy Processes.” In *Handbook of Computational Finance*, edited by Jin-Chuan Duan, Wolfgang Karl Härdle, and James E. Gentle, 61–88. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-17254-0_4.

Foti, Nicholas, Joseph Futoma, Daniel Rockmore, and Sinead Williamson. 2013. “A Unifying Representation for a Class of Dependent Random Measures.” In *Artificial Intelligence and Statistics*, 20–28. http://proceedings.mlr.press/v31/foti13a.html.

Gusak, Dmytro, Alexander Kukush, Alexey Kulik, Yuliya Mishura, and Andrey Pilipenko. 2010. *Theory of Stochastic Processes : With Applications to Financial Mathematics and Risk Theory*. Problem Books in Mathematics. New York: Springer New York. http://link.springer.com/book/10.1007%2F978-0-387-87862-1.

Hackmann, Daniel, and Alexey Kuznetsov. 2016. “Approximating Lévy Processes with Completely Monotone Jumps.” *The Annals of Applied Probability* 26 (1): 328–59. https://doi.org/10.1214/14-AAP1093.

Ishwaran, Hemant, and Mahmoud Zarepour. 2002. “Exact and Approximate Sum Representations for the Dirichlet Process.” *Canadian Journal of Statistics* 30 (2): 269–83. https://doi.org/10.2307/3315951.

Itkin, Andrey. 2016. “Efficient Solution of Backward Jump-Diffusion Partial Integro-Differential Equations with Splitting and Matrix Exponentials,” January. https://papers.ssrn.com/abstract=2794898.

James, Lancelot F., Bernard Roynette, and Marc Yor. 2008. “Generalized Gamma Convolutions, Dirichlet Means, Thorin Measures, with Explicit Examples.” *Probability Surveys* 5: 346–415. https://doi.org/10.1214/07-PS118.

Kyprianou, Andreas E. 2014. *Fluctuations of Lévy Processes with Applications: Introductory Lectures*. Second edition. Universitext. Heidelberg: Springer.

Lalley, Steven P. 2007. “Lévy Processes, Stable Processes, and Subordinators.”

Lawrence, Neil D., and Raquel Urtasun. 2009. “Non-Linear Matrix Factorization with Gaussian Processes.” In *Proceedings of the 26th Annual International Conference on Machine Learning*, 601–8. ICML ’09. New York, NY, USA: ACM. https://doi.org/10.1145/1553374.1553452.

Lefebvre, Mario. 2007. *Applied Stochastic Processes*. Universitext. Springer New York. http://link.springer.com/chapter/10.1007/978-0-387-48976-6_2.

Lo, Albert Y., and Chung-Sing Weng. 1989. “On a Class of Bayesian Nonparametric Estimates: II. Hazard Rate Estimates.” *Annals of the Institute of Statistical Mathematics* 41 (2): 227–45. https://doi.org/10.1007/BF00049393.

Mathai, A. M. 1982. “Storage Capacity of a Dam with Gamma Type Inputs.” *Annals of the Institute of Statistical Mathematics* 34 (3): 591–97. https://doi.org/10.1007/BF02481056.

Mathai, A. M., and P. G. Moschopoulos. 1991. “On a Multivariate Gamma.” *Journal of Multivariate Analysis* 39 (1): 135–53. https://doi.org/10.1016/0047-259X(91)90010-Y.

Mathal, A. M., and P. G. Moschopoulos. 1992. “A Form of Multivariate Gamma Distribution.” *Annals of the Institute of Statistical Mathematics* 44 (1): 97–106. https://doi.org/10.1007/BF00048672.

Moschopoulos, P. G. 1985. “The Distribution of the Sum of Independent Gamma Random Variables.” *Annals of the Institute of Statistical Mathematics* 37 (3): 541–44. https://doi.org/10.1007/BF02481123.

Olofsson, Peter. 2005. *Probability, Statistics, and Stochastic Processes*. Hoboken, N.J: Hoboken, N.J. : Wiley-Interscience. http://dx.doi.org/10.1002%2F9780471743064.

Pérez-Abreu, Victor, and Robert Stelzer. 2014. “Infinitely Divisible Multivariate and Matrix Gamma Distributions.” *Journal of Multivariate Analysis* 130 (September): 155–75. https://doi.org/10.1016/j.jmva.2014.04.017.

Pfaffel, Oliver. 2012. “Wishart Processes,” January. http://arxiv.org/abs/1201.3256.

Rubinstein, Reuven Y., and Dirk P. Kroese. 2016. *Simulation and the Monte Carlo Method*. 3 edition. Wiley Series in Probability and Statistics. Hoboken, New Jersey: Wiley.

Sato, Ken-iti, Sato Ken-Iti, and A. Katok. 1999. *Lévy Processes and Infinitely Divisible Distributions*. Cambridge University Press.

Semeraro, Patrizia. 2008. “A Multivariate Variance Gamma Model for Financial Applications.” *International Journal of Theoretical and Applied Finance* 11 (01): 1–18. https://doi.org/10.1142/S0219024908004701.

Shaked, Moshe, and J. George Shanthikumar. 1988. “On the First-Passage Times of Pure Jump Processes.” *Journal of Applied Probability* 25 (3): 501–9. https://doi.org/10.2307/3213979.

Singpurwalla, Nozer. 1997. “Gamma Processes and Their Generalizations: An Overview.” In *Engineering Probabilistic Design and Maintenance for Flood Protection*, edited by Roger Cooke, Max Mendel, and Han Vrijling, 67–75. Boston, MA: Springer US. https://doi.org/10.1007/978-1-4613-3397-5_5.

Singpurwalla, Nozer D., and Mark A. Youngren. 1993. “Multivariate Distributions Induced by Dynamic Environments.” *Scandinavian Journal of Statistics* 20 (3): 251–61. https://www.jstor.org/stable/4616280.

Steutel, Fred W., and Klaas van Harn. 2003. *Infinite Divisibility of Probability Distributions on the Real Line*. CRC Press. http://books.google.com?id=5ddskbtvVjMC.

Tankov, Peter, and Ekaterina Voltchkova. n.d. “Jump-Diffusion Models: A Practitioner’s Guide,” 24.

Veillette, Mark, and Murad S. Taqqu. 2010a. “Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Lévy Processes.” *Statistics & Probability Letters* 80 (7): 697–705. https://doi.org/10.1016/j.spl.2010.01.002.

———. 2010b. “Numerical Computation of First-Passage Times of Increasing Lévy Processes.” *Methodology and Computing in Applied Probability* 12 (4): 695–729. https://doi.org/10.1007/s11009-009-9158-y.

Weide, Hans van der. 1997. “Gamma Processes.” In *Engineering Probabilistic Design and Maintenance for Flood Protection*, edited by Roger Cooke, Max Mendel, and Han Vrijling, 77–83. Boston, MA: Springer US. https://doi.org/10.1007/978-1-4613-3397-5_6.

Wilson, Andrew Gordon, and Zoubin Ghahramani. 2011. “Generalised Wishart Processes.” In *Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence*, 736–44. UAI’11. Barcelona, Spain: AUAI Press. http://dl.acm.org/citation.cfm?id=3020548.3020633.

Wolpert, Robert L. 2006. “Stationary Gamma Processes,” 13.