# Subordinators

Non-decreasing Lévy processes with weird branding

October 14, 2019 — February 24, 2022

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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

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

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 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 it 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

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

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## 7 Inverse Gaussian processes

Based on inverse Gaussian increments. 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.

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## 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.

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## 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?

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## 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 1977b) 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

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