Non-decreasing Lévy processes with weird branding

October 14, 2019 — February 24, 2022

point processes
stochastic processes

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

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

\[ \mathbb{P}(\rv{g}(t)-\rv{g}(s)< 0)=0, \,\forall t \geq s. \]

That is, it is an a.s. non-decreasing Markov process with homogeneous independent increments.

Figure 1: Superordinators

Tutorial introductions to these creatures are in (Kyprianou 2014; Bertoin 1996, 2000; Sato 1999).

The platonic ideal of a subordinator is probably the Lévy-Gamma process. However there are many more; There is no requirement that the support of a subordinator’s paths is dense in \(\mathbb{R}_+\); for example, counting such Poisson processes are subordinators.

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 implicit in 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 \(\mathbb{R}^d,\) although such a process no longer has an intuitive interpretation as the rate-of-time-passing, since time is usually one-dimensional in common experience.

1 Properties

Figure 2: Pure jump processes

Let us see what Dominic Yeo says, channelling Bertoin (1996):

In general, we describe Lévy processes by their characteristic exponent. As a subordinator takes values in \([0,\infty),\) we can use the Laplace exponent instead: \[\mathbb{E}\exp(-\lambda X_t)=:\exp(-t\Phi(\lambda)).\] We can refine the Levy-Khintchine formula; \[\Phi(\lambda)=k+d\lambda+\int_{[0,\infty)}(1-e^{-\lambda x})\Pi(dx),\] where k is the kill rate (in the non-strict case). Because the process is increasing, it must have bounded variation, and so the quadratic part vanishes, and we have a stronger condition on the Levy measure: \(\int(1\wedge x)\Pi(dx)<\infty.\) The expression \(\bar{\Pi}(x):=k+\Pi((x,\infty))\) for the tail of the Levy measure is often more useful in this setting. We can think of this decomposition as the sum of a drift, and a PPP with characteristic measure \(\Pi+k\delta_\infty.\) … we do not want to consider the case that \(X\) is a step process, so either \(d>0\) or \(\Pi((0,\infty))=\infty\) is enough to ensure this.

2 Gamma processes

See Gamma processes.

3 Beta processes

The strictly-increasing flavour, as seen in See Gamma processes.

4 Poisson processes

See Poisson processes.

5 as measure priors

See Measure priors.

6 Compound Poisson processes with non-negative increments


7 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. Dependency structure might be interesting.


8 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.

9 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.


10 Subordination of 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?


11 Generalized Gamma Convolutions

A construction (Bondesson 1979, 2012; Ole E. Barndorff-Nielsen, Maejima, and Sato 2006; James, Roynette, and Yor 2008; Steutel and van Harn 2003) that shows how to represent some startling (to me) processes using subordinators, including Pareto (Thorin 1977a) and Lognormal (Thorin 1977b) ones.

12 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 handy.

13 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? I care about this especially for multivariate Gamma processes.

14 References

Applebaum. 2004. Lévy Processes — from Probability to Finance and Quantum Groups.” Notices of the AMS.
———. 2009. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 116.
Asmussen, and Glynn. 2007. Stochastic Simulation: Algorithms and Analysis.
Aurzada, and Dereich. 2009. Small Deviations of General Lévy Processes.” The Annals of Probability.
Avramidis, L’Ecuyer, and 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. WSC ’03.
Barndorff-Nielsen, Ole E., Maejima, and Sato. 2006. Some Classes of Multivariate Infinitely Divisible Distributions Admitting Stochastic Integral Representations.” Bernoulli.
Barndorff-Nielsen, Ole E., Pedersen, and Sato. 2001. Multivariate Subordination, Self-Decomposability and Stability.” Advances in Applied Probability.
Barndorff-Nielsen, Ole E, and Shephard. 2012. Basics of Lévy Processes.” In Lévy Driven Volatility Models.
Bertoin. 1996. Lévy Processes. Cambridge Tracts in Mathematics 121.
———. 1999. Subordinators: Examples and Applications.” In Lectures on Probability Theory and Statistics: Ecole d’Eté de Probailités de Saint-Flour XXVII - 1997. Lecture Notes in Mathematics.
———. 2000. Subordinators, Lévy Processes with No Negative Jumps, and Branching Processes.
Bondesson. 1979. A General Result on Infinite Divisibility.” The Annals of Probability.
———. 2012. Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics 76.
Borovkov, and Burq. 2001. Kendall’s Identity for the First Crossing Time Revisited.” Electronic Communications in Probability.
Buchmann, Lu, and Madan. 2017. Weak Subordination of Multivariate Lévy Processes and Variance Generalised Gamma Convolutions.” arXiv:1609.04481 [Math].
Burridge, Kwaśnicki, Kuznetsov, et al. 2014. New Families of Subordinators with Explicit Transition Probability Semigroup.” arXiv:1402.1062 [Math].
Doney. 2007. Fluctuation Theory for Lévy Processes: Ecole d’eté de Probabilités de Saint-Flour XXXV, 2005. Lecture Notes in Mathematics 1897.
Gander. 2004. Inference for Stochastic Volatility Models Based on Lévy Processes.”
James, Roynette, and Yor. 2008. Generalized Gamma Convolutions, Dirichlet Means, Thorin Measures, with Explicit Examples.” Probability Surveys.
Kyprianou. 2014. Fluctuations of Lévy Processes with Applications: Introductory Lectures. Universitext.
Leonenko, Meerschaert, Schilling, et al. 2014. Correlation Structure of Time-Changed Lévy Processes.” Communications in Applied and Industrial Mathematics.
Madan, and Seneta. 1990. The Variance Gamma (V.G.) Model for Share Market Returns.” The Journal of Business.
Minami. 2003. A Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship.” Communications in Statistics - Theory and Methods.
———. 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),.
Pérez-Abreu, and Rocha-Arteaga. 2005. Covariance-Parameter Lévy Processes in the Space of Trace-Class Operators.” Infinite Dimensional Analysis, Quantum Probability and Related Topics.
Ranganath, and Blei. 2018. Correlated Random Measures.” Journal of the American Statistical Association.
Rubinstein, and Kroese. 2016. Simulation and the Monte Carlo Method. Wiley series in probability and statistics.
Sato. 1999. Lévy Processes and Infinitely Divisible Distributions.
Seneta. 2007. The Early Years of the Variance-Gamma Process.” In Advances in Mathematical Finance. Applied and Numerical Harmonic Analysis.
Steutel, and van Harn. 2003. Infinite Divisibility of Probability Distributions on the Real Line.
Thorin. 1977a. On the Infinite Divisbility of the Pareto Distribution.” Scandinavian Actuarial Journal.
———. 1977b. On the Infinite Divisibility of the Lognormal Distribution.” Scandinavian Actuarial Journal.
Veillette, and Taqqu. 2010a. Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Lévy Processes.” Statistics & Probability Letters.
———. 2010b. Numerical Computation of First-Passage Times of Increasing Lévy Processes.” Methodology and Computing in Applied Probability.
Wolpert, and Ickstadt. 1998. Poisson/Gamma Random Field Models for Spatial Statistics.” Biometrika.