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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, Ken-Iti, and Katok 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 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 It is due to Lamperti? I do not have time for that citation rabiit 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
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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): 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.
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://www.jstor.org/stable/2243098.
———. 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, Sato Ken-Iti, and A. Katok. 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): 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.