# Subordinators

## Non-decreasing Lévy processes with weird branding

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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.

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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 $$\{\rv{g}(t)\}$$ 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.

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

Aurzada, Frank, and Steffen Dereich. 2009. “Small Deviations of General Lévy Processes.” The Annals of Probability 37 (5): 2066–92. https://doi.org/10.1214/09-AOP457.

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.

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.

Borovkov, Konstantin, and Zaeem Burq. 2001. “Kendall’s Identity for the First Crossing Time Revisited.” Electronic Communications in Probability 6: 91–94. https://doi.org/10.1214/ECP.v6-1038.

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.

Doney, Ronald A., and Jean Picard, eds. 2007a. “Spectrally Negative Lévy Processes.” In Fluctuation Theory for Lévy Processes: Ecole d’Eté de Probabilités de Saint-Flour XXXV - 2005, 95–113. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-540-48511-7_9.

———, 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.

Gander, Matthew Peter Sandford. 2004. “Inference for Stochastic Volatility Models Based on Lévy Processes.” https://core.ac.uk/download/pdf/1591714.pdf.

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.

Leonenko, Nikolai N, Mark M Meerschaert, René L Schilling, and Alla Sikorskii. 2014. “Correlation Structure of Time-Changed Lévy Processes.” Communications in Applied and Industrial Mathematics 6 (1). https://doi.org/10.1685/journal.caim.483.

Madan, Dilip B., and Eugene Seneta. 1990. “The Variance Gamma (V.G.) Model for Share Market Returns.” The Journal of Business 63 (4): 511–24. http://finance.martinsewell.com/stylized-facts/distribution/MadanSeneta1990.pdf.

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

———. 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.

Pérez-Abreu, Víctor, and Alfonso Rocha-Arteaga. 2005. “Covariance-Parameter Lévy Processes in the Space of Trace-Class Operators.” Infinite Dimensional Analysis, Quantum Probability and Related Topics 08 (01): 33–54. 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. 1999. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.

Seneta, Eugene. 2007. “The Early Years of the Variance-Gamma Process.” In Advances in Mathematical Finance, edited by Michael C. Fu, Robert A. Jarrow, Ju-Yi J. Yen, and Robert J. Elliott, 3–19. Applied and Numerical Harmonic Analysis. Boston, MA: Birkhäuser. https://doi.org/10.1007/978-0-8176-4545-8_1.

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. “On the Infinite Divisbility of the Pareto Distribution.” Scandinavian Actuarial Journal 1977 (1): 31–40. https://doi.org/10.1080/03461238.1977.10405623.

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

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, 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, 4): 695–729. https://doi.org/10.1007/s11009-009-9158-y.