The Living Thing / Notebooks : Change of time and probability measure

TBD. Various notes on a.e. continuous monotonic changes of index in order to render a process “simple” in some sense.

In Warping and registration problems you try to align two or more processes. But here, the target is some “null”, basic process. This special case is often a lot more computationally tractable and statistically well behaved.

To explore:

Lamperti representation for continuous state branching processes, Girsanov transformation (may not be a change of measure in the sense that I want it to be; must look up whether I can shoehorn it.) Mass transport problems - do these relate? Maximum spacing estimators. Surely this pops up in copula modelling too? Ogata’s time rescaling: Intensity estimation for point processes uses this as a statistical test.

to understand:

A summary in VeSc04 of the point process flavour:

Knight (Knig70) showed that for any orthogonal sequence of continuous local martingales, by rescaling time for each via its associated predictable process, we form a multivariate sequence of independent standard Brownian motions. Then Meyer (Meye71) extended Knight’s theorem to the case of point processes, showing that given a simple multivariate point process \({N_i ; i = 1, 2, . . . , n}\), the multivariate point process obtained by rescaling each \(N_i\) according to its compensator is a sequence of independent Poisson processes, each having intensity 1. Since then, alternative proofs and variations of this result have been given by Brem72, Papa72, AaHo78, Kurt80 and BrNa88. Papangelou (Papa72) gave the following interpretation in the univariate case:

Roughly, moving in \([0, \infty)\) so as to meet expected future points at a rate of one per time unit (given at each instant complete knowledge of the past), we meet them at the times of a Poisson process.


Generalizations of Meyer’s result to point processes on \(\mathbb{R}^d\) have been established by MeNu86, Nair90 and Scho99. In each case, the method used has been to focus on one dimension of the point process, and rescale each point along that dimension according to the conditional intensity.


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