\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\]

A subordinator is an a.s. non-decreasing Lévy process \(\{\rv{g}(t)\}, t \in \mathbb{R}\) with state space \(\mathbb{R}_+\equiv [0,\infty]\) such that

\[ \mathbb{P}(\rv{g}(t)-\rv{g}(s)\lt 0)=0, \,\forall t \geq s. \]

That is, it is an a.s. non-decreasing Markov process with homogeneous independent increments.

Tutorial introductions to these creatures are in (Kyprianou 2014; Bertoin 1996, 2000; Sato 1999).

The platonic ideal of a subordinator is probably the Gamma process. However it is much more general; There is no requirement that the support of a subordinator’s paths is dense in \(\mathbb{R}_+\); for example Poisson processes are subordinators, and they are supported on the natural numbers.

The terminology is weird. Why “subordinator”? Popular references mention their use as a model of random rate of passage of time in the popular Variance Gamma model (Madan and Seneta 1990). (About which I know almost nothing but I read a little bit of history in (Seneta 2007).) The idea is certainly older than that, though. Surely is due to Lamperti? I do not have time for that citation rabbit hole though.

The subordination of a one-dimensional Lévy process \(\{X(t)\}\) by a one-
dimensional increasing process \(\{T (t)\}\) means introducing
a new process \(\{Y (t)\}\) defined as \(Y (t) = X(T (t))\), where
\(\{X(t)\}\) and \(\{T (t)\}\) are assumed to be independent,
and that one-dimensional increasing process is the *subordinator*.

Subordinators can be generalised to beyond one-dimensional processes; see (Barndorff-Nielsen, Pedersen, and Sato 2001). We can more generally consider subordinators that take values in \(\mathbb{R}^d,\) which is what I usually do, although such processes no longer has a convenient interpretation as the rate-of-time-passing, since time is usually one-dimensional in common experience.

## Gamma processes

See Gamma processes.

## Poisson processes

See Poisson processes.

## Compound Poisson processes with non-negative increments

🏗

## Inverse Gaussian processes

See (Kyprianou 2014). Interesting because (Minami 2003) gives an interpretation of the multivariate Inverse Gaussian process which is suggestive of a natural dependence model via covariance of a dual Gaussian process, and I suspect this makes for a non-trivial dependence structure.

🏗

## Positive linear combinations of other subordinators

🏗

Say we have a collection of \(m\) independent univariate subordinators, not necessarily from the same family or with the same parameters, stacked to form a vector process \(\{\rv{g}(t)\}\) with state space \(\mathbb{R}_+^{m}\). Take a transform matrix \(M\in\mathbb{R}^{n\times m}\) with non-negative entries. Then the process \(\{M\rv{g}(t)\}\) is still a subordinator (and moreover a Lévy process) although the elements are no longer independent. We could think of that the same trick applied to just the increments, which would come down to the same thing.

## Generalized Gamma Convolutions

A construction (Bondesson 1979, 2012; Barndorff-Nielsen, Maejima, and Sato 2006; James, Roynette, and Yor 2008; Steutel and Harn 2003) that shows how to represent some startling (to me) processes as subordinators, including Pareto (Thorin 1977a) and Lognormal (Thorin 1977b) ones.

## via Kendall’s identity

(Burridge et al. 2014) leverage a neat result (Kendall’s identity) in (Borovkov and Burq 2001) to produce several interesting families of subordinators with explicit transition densities. These look nifty.

Applebaum, David. 2004. “Lévy Processes — from Probability to Finance and Quantum Groups.” *Notices of the AMS* 51 (11): 1336–47. 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.

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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.

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Barndorff-Nielsen, Ole E, and Neil Shephard. 2012. “Basics of Lévy Processes.” In *Lévy Driven Volatility Models*, 70. https://pdfs.semanticscholar.org/fe80/07ba98fafa23ddca98ac5d9b1cf1732f70bd.pdf.

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.

Bondesson, Lennart. 1979. “A General Result on Infinite Divisibility.” *The Annals of Probability* 7 (6): 965–79. https://doi.org/10.1214/aop/1176994890.

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

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Buchmann, Boris, Kevin Lu, and Dilip B. Madan. 2017. “Weak Subordination of Multivariate Lévy Processes and Variance Generalised Gamma Convolutions,” November. http://arxiv.org/abs/1609.04481.

Burridge, James, Mateusz Kwaśnicki, Alexey Kuznetsov, and Andreas Kyprianou. 2014. “New Families of Subordinators with Explicit Transition Probability Semigroup,” July. http://arxiv.org/abs/1402.1062.

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———, eds. 2007b. “Subordinators.” In *Fluctuation Theory for Lévy Processes: Ecole d’Eté de Probabilités de Saint-Flour XXXV - 2005*, 9–17. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-540-48511-7_2.

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———. 2007. “Multivariate Inverse Gaussian Distribution as a Limit of Multivariate Waiting Time Distributions.” *Journal of Statistical Planning and Inference*, Special Issue: In Celebration of the Centennial of The Birth of Samarendra Nath Roy (1906-1964), 137 (11): 3626–33. https://doi.org/10.1016/j.jspi.2007.03.038.

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———. 1977b. “On the Infinite Divisibility of the Lognormal Distribution.” *Scandinavian Actuarial Journal* 1977 (3): 121–48. https://doi.org/10.1080/03461238.1977.10405635.

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