- 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

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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 conventionally supported on integers.

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 \(\mathbb{R}^d,\) 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

## References

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