- Properties
- Gamma processes
- Poisson processes
- Compound Poisson processes with non-negative increments
- Inverse Gaussian processes
- An increasing linear function is a subordinator
- Positive linear combinations of other subordinators
- Subordination of other subordinators
- Generalized Gamma Convolutions
- via Kendall’s identity
- Multivariate
- Subordinator-valued stochastic process
- References

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

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). 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.

## Properties

TBD.

## 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 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?

🏗

## Generalized Gamma Convolutions

A construction (Bondesson 1979, 2012; Ole E. 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 using 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.

## 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

*Notices of the AMS*51 (11): 1336–47. https://core.ac.uk/download/pdf/50531.pdf.

*Lévy Processes and Stochastic Calculus*. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.

*Stochastic Simulation: Algorithms and Analysis*. 2007 edition. New York: Springer.

*The Annals of Probability*37 (5): 2066–92. https://doi.org/10.1214/09-AOP457.

*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.

*Bernoulli*12 (1): 1–33. https://projecteuclid.org/euclid.bj/1141136646.

*Advances in Applied Probability*33 (1): 160–87. https://doi.org/10.1017/S0001867800010685.

*Lévy Driven Volatility Models*, 70. https://pdfs.semanticscholar.org/fe80/07ba98fafa23ddca98ac5d9b1cf1732f70bd.pdf.

*Lévy Processes*. Cambridge Tracts in Mathematics 121. Cambridge ; New York: Cambridge University Press.

*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.

*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.

*The Annals of Probability*7 (6): 965–79. https://doi.org/10.1214/aop/1176994890.

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

*Electronic Communications in Probability*6: 91–94. https://doi.org/10.1214/ECP.v6-1038.

*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.

*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.

*Probability Surveys*5: 346–415. https://doi.org/10.1214/07-PS118.

*Fluctuations of Lévy Processes with Applications: Introductory Lectures*. Second edition. Universitext. Heidelberg: Springer.

*Communications in Applied and Industrial Mathematics*6 (1). https://doi.org/10.1685/journal.caim.483.

*The Journal of Business*63 (4): 511–24. http://finance.martinsewell.com/stylized-facts/distribution/MadanSeneta1990.pdf.

*Communications in Statistics - Theory and Methods*32 (12): 2285–2304. https://doi.org/10.1081/STA-120025379.

*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.

*Infinite Dimensional Analysis, Quantum Probability and Related Topics*08 (01): 33–54. https://doi.org/10.1142/S0219025705001846.

*Simulation and the Monte Carlo Method*. 3 edition. Wiley Series in Probability and Statistics. Hoboken, New Jersey: Wiley.

*Lévy Processes and Infinitely Divisible Distributions*. Cambridge University Press.

*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.

*Infinite Divisibility of Probability Distributions on the Real Line*. CRC Press. http://books.google.com?id=5ddskbtvVjMC.

*Scandinavian Actuarial Journal*1977 (1): 31–40. https://doi.org/10.1080/03461238.1977.10405623.

*Scandinavian Actuarial Journal*1977 (3): 121–48. https://doi.org/10.1080/03461238.1977.10405635.

*Statistics & Probability Letters*80 (7, 7): 697–705. https://doi.org/10.1016/j.spl.2010.01.002.

*Methodology and Computing in Applied Probability*12 (4, 4): 695–729. https://doi.org/10.1007/s11009-009-9158-y.