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

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

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

Tutorial introductions to these creatures are in (Kyprianou 2014; Bertoin 1996, 2000; Sato, Ken-Iti, and Katok 1999).

The platonic ideal of a subordinator is the Gamma process, although it is not the simplest process meeting satisfying these criteria.

The terminology is weird. Why â€śsubordinatorâ€ť? I suspect because the study of these objects comes from their use as a model of random rate of passage of time.

The subordination of a one-dimensional LĂ©vy 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.

There is no requirement that the support a subordinatorâ€™s paths is dense in \(\mathbb{R}_+\); for example Poisson processes are supported on the natural numbers.

## Gamma processes

See Gamma processes.

## Poisson processes

See Poisson processes.

## Compound Poisson processes with non-negative increments

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## 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.

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## Positive linear combinations of other subordinators

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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 \(\{\Lambda(t)\}\) with state space \(\mathbb{R}_+^{m}\). Take a transform matrix \(M\in\mathbb{R}^{n\times m}\) with non-negative entries. Then the process \(\{M\Lambda(t)\}\) is still a subordinator (and moreover AFAICS also a LĂ©vy process).

## 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. đźš§

# Refs

Applebaum, David. 2004. â€śLĂ©vy Processesâ€”from Probability to Finance and Quantum Groups.â€ť *Notices of the AMS* 51 (11): 12.

â€”â€”â€”. 2009. *LĂ©vy Processes and Stochastic Calculus*. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.

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

Avramidis, Athanassios N., Pierre Lâ€™Ecuyer, and Pierre-Alexandre Tremblay. 2003. â€śNew Simulation Methodology for Finance: Efficient Simulation of Gamma and Variance-Gamma Processes.â€ť In *Proceedings of the 35th Conference on Winter Simulation: Driving Innovation*, 319â€“26. WSC â€™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. â€śSome Classes of Multivariate Infinitely Divisible Distributions Admitting Stochastic Integral Representations.â€ť *Bernoulli* 12 (1): 1â€“33. https://projecteuclid.org/euclid.bj/1141136646.

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

Barndorff-Nielsen, Ole E, and Neil Shephard. 2012. â€śBasics of LĂ©vy Processes.â€ť In *LĂ©vy Driven Volatility Models*, 70. https://pdfs.semanticscholar.org/fe80/07ba98fafa23ddca98ac5d9b1cf1732f70bd.pdf.

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

â€”â€”â€”. 1999. â€śSubordinators: Examples and Applications.â€ť In *Lectures on Probability Theory and Statistics: Ecole dâ€™EtĂ© de ProbailitĂ©s de Saint-Flour XXVII - 1997*, edited by Jean Bertoin, Fabio Martinelli, Yuval Peres, and Pierre Bernard, 1717:1â€“91. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-48115-7_1.

â€”â€”â€”. 2000. *Subordinators, LĂ©vy Processes with No Negative Jumps, and Branching Processes*. University of Aarhus. Centre for Mathematical Physics and Stochastics â€¦. http://www.maphysto.dk/oldpages/events/LevyBranch2000/notes/bertoin.pdf.

Bondesson, Lennart. 1979. â€śA General Result on Infinite Divisibility.â€ť *The Annals of Probability* 7 (6): 965â€“79. https://www.jstor.org/stable/2243098.

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

Buchmann, Boris, Kevin Lu, and Dilip B. Madan. 2017. â€śWeak Subordination of Multivariate L\â€™evy Processes and Variance Generalised Gamma Convolutions,â€ť November. http://arxiv.org/abs/1609.04481.

Gander, Matthew Peter Sandford. 2004. â€śInference for Stochastic Volatility Models Based on LĂ©vy Processes.â€ť https://core.ac.uk/download/pdf/1591714.pdf.

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

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

Minami, Mihoko. 2003. â€śA Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship.â€ť *Communications in Statistics - Theory and Methods* 32 (12): 2285â€“2304. https://doi.org/10.1081/STA-120025379.

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

PĂ©rez-Abreu, VĂctor, and Alfonso Rocha-Arteaga. 2005. â€śCovariance-Parameter LĂ©vy Processes in the Space of Trace-Class Operators.â€ť *Infinite Dimensional Analysis, Quantum Probability and Related Topics* 08 (01): 33â€“54. https://doi.org/10.1142/S0219025705001846.

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. *LĂ©vy Processes and Infinitely Divisible Distributions*. Cambridge University Press.

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.

Thorin, Olof. 1977a. â€śOn the Infinite Divisbility of the Pareto Distribution.â€ť *Scandinavian Actuarial Journal* 1977 (1): 31â€“40. https://doi.org/10.1080/03461238.1977.10405623.

â€”â€”â€”. 1977b. â€śOn the Infinite Divisibility of the Lognormal Distribution.â€ť *Scandinavian Actuarial Journal* 1977 (3): 121â€“48. https://doi.org/10.1080/03461238.1977.10405635.

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

â€”â€”â€”. 2010b. â€śNumerical Computation of First-Passage Times of Increasing LĂ©vy Processes.â€ť *Methodology and Computing in Applied Probability* 12 (4): 695â€“729. https://doi.org/10.1007/s11009-009-9158-y.