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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).
I know almost nothing about that 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 (Ole E. 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.
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 rich dependence structure.
🏗
An increasing linear function is a subordinator
We get linear functions both as a baked-in affordance of Lévy processes and as a limiting distribution of a Gamma process whose increment variance goes to zero.
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.
🏗
Subordination of other subordinators
Since a subordinator time change of a Lévy process is still a Lévy process, a subordinated subordinator is still a Lévy process and it is still monotone and therefore still a subordinator.
Is this convenient for my purposes?
🏗
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.
Multivariate
How do we construct a multivariate subordinator process, by which I mean a random process whose realisations are coordinate-wise monotone functions
\(\mathbb{R}^D\to\mathbb{R}\)?
One obvious way is to take coordinate-wise iid subordinators as the distributions of each coordinate.
This is by construction what we want.
Questions: How general is such a thing?
How easy is it to conditionally sample paths from it?
What do we want our multivariate subordinator to do?
Subordinator-valued stochastic process
Suppose we want a process evolving along one time axis whose value at any given instant is a subordinator, and at each time the realised subordinator are i.i.d., but the process is stochastically continuous?
This is a useful model as a prior in certain Bayesian models.
One option here seems to be to choose an initial subordinator, which evolves by a stochastically continuous time-change; can this be made to be i.i.d?
What is fix point for stochastic time changes.
A simpler model to to draw a new subordinator finitely often according to some Poisson process.
References
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 10, 2017.
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 3, 2014.
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.