# Subordinators

Non-decreasing Lévy processes with weird branding

October 14, 2019 — February 24, 2022

branching
point processes
probability
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.

Tutorial introductions to these creatures are in .

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 . I know almost nothing about that but I read a little bit of history in . 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 . 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

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.

## 5 as measure priors

See Measure priors.

🏗

## 7 Inverse Gaussian processes

See . 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 that shows how to represent some startling (to me) processes using subordinators, including Pareto and Lognormal ones.

## 12 via Kendall’s identity

leverage a neat result (Kendall’s identity) in 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. 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. The Annals of Probability.
Avramidis, L’Ecuyer, and Tremblay. 2003. In Proceedings of the 35th Conference on Winter Simulation: Driving Innovation. WSC ’03.
Barndorff-Nielsen, Ole E., Maejima, and Sato. 2006. Bernoulli.
Barndorff-Nielsen, Ole E., Pedersen, and Sato. 2001. Advances in Applied Probability.
Barndorff-Nielsen, Ole E, and Shephard. 2012. In Lévy Driven Volatility Models.
Bertoin. 1996. Lévy Processes. Cambridge Tracts in Mathematics 121.
———. 1999. In Lectures on Probability Theory and Statistics: Ecole d’Eté de Probailités de Saint-Flour XXVII - 1997. Lecture Notes in Mathematics.
Bondesson. 1979. The Annals of Probability.
———. 2012. Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics 76.
Borovkov, and Burq. 2001. Electronic Communications in Probability.
Buchmann, Lu, and Madan. 2017. arXiv:1609.04481 [Math].
Burridge, Kwaśnicki, Kuznetsov, et al. 2014. 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.
James, Roynette, and Yor. 2008. Probability Surveys.
Kyprianou. 2014. Fluctuations of Lévy Processes with Applications: Introductory Lectures. Universitext.
Leonenko, Meerschaert, Schilling, et al. 2014. Communications in Applied and Industrial Mathematics.