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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.
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