After the general discussion of Levy processes in the previous post, we now discuss a particular class of such processes. The majority of content and notation below is taken from chapters 1-3 of Jean Bertoin’s Saint-Flour notes.
We say is a subordinator if:
- It is a right-continuous adapted stochastic process, started from 0.
- It has stationary, independent increments.
- It is increasing.
Note that the first two conditions are precisely those required for a Levy process. We could also allow the process to take the value , where the hitting time of infinity represents ‘killing’ the subordinator in some sense. If this hitting time is almost surely infinite, we say it is a strict subordinator. There is little to be gained right now from considering anything other than strict subordinators.
- A compound Poisson process, with finite jump measure supported on . Hereafter we exclude this case, as it is better dealt with in other languages.
- A so-called stable Levy process, where , for some . (I’ll define very soon.) Note that checking that the sample paths are increasing requires only that almost surely.
- The hitting time process for Brownian Motion. Note that this does indeed have jumps as we would need. (This has .)
- In general, we describe Levy processes by their characteristic exponent. As a subordinator takes values in , we can use the Laplace exponent instead:
- We can refine the Levy-Khintchine formula;
- 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: .
- The expression 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 . As we said above, we do not want to consider the case that X is a step process, so either d>0 or is enough to ensure this.
We give a snapshot of a couple of observations which make these nice to work with. Define the renewal measure U(dx) by:
If we want to know the distribution function of this U, it will suffice to consider the indicator function in the above.
The reason to exclude step processes specifically is to ensure that X has a continuous inverse:
so is continuous.
In fact, this renewal measure characterises the subordinator uniquely, as we see by taking the Laplace transform:
The Arcsine Law
X is Markov, which induces a so-called regenerative property on the range of X, . Formally, given s, we do not always have (as the process might jump over s), but we can define . Then
In fact, the converse holds as well. Any random set with this regenerative property is the range of some subordinator. Note that is some kind of dual to X, since it is increasing, and the regenerative property induces some Markovian properties.
In particular, we consider the last passage time , in the case of a stable subordinator with . Here, is self-similar with scaling exponent . The distribution of is thus independent of t. In this situation, we can derive the generalised arcsine rule for the distribution of :
The most natural application of this is to the hitting time process of Brownian Motion, which is stable with . Then , in the usual notation for the supremum process. Furthermore, we have equality in distribution of the processes (see previous posts on excursion theory and the short aside which follows):
So gives the time of the last zero of BM before time 1, and the arcsine law shows that its distribution is given by: