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 more computationally tractable or statistically well behaved.
To explore
Lamperti representation for continuous state branching processes,
Ogata's time rescaling: Intensity estimation for point processes uses this as a statistical test. To understand:
 relation to martingale transform.
Subordinator
I'm going to follow Applebaum's presentation (Appl09), which is brusque without being incomprehensible or unmotivated.
A subordinator is just a onedimensional Lévy process which happens to be nondecreasing. i.e. A subordinator is an a.s. nondecreasing stochastic process with state space such that

the increments are stationary

The increments are independent

The process is stochastically continuous

The increments are nonnegative
The first three are standard Lévy process stuff. The last is only for subordinators.
Some definitions additionally require the increment distribution is a.s. positive, rather than nonnegative, or that there are no atoms at zero in the increment distribution.
Curiously, upon giving that definition, many proceed to immediately assert that such a process is a model for a random change of time. This sounds not insane per se, but doesn't have much in the way of narrative flow. TBD: explain why one would bother doing such an arbitrary thing as changing time in such a fashion.
Anyway I hope to use these to get a handle on timechanged residual tests and Lamperti representations. TBC.
The subordinator may be extended to multiple dimensions by requiring that each dimension is a.s. increasing. TBC.
Point process transforms
As used in point process residual goodness of fit tests.
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 , the multivariate point process obtained by rescaling each 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 Brém72, Papa72, AaHo78, Kurt80 and BrNa88. Papangelou (Papa72) gave the following interpretation in the univariate case:
Roughly, moving in 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 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.
Going Multivariate
As seen in BaPS01 and others. How does multivariate time work then?
Refs
 MeNu86: (1986) A Characterization of the Spatial Poisson Process and Changing Time. The Annals of Probability, 14(4), 1380–1390. DOI
 ChSt89: (1989) A goodnessoffit test using Moran’s statistic with estimated parameters. Biometrika, 76(2), 385–392. DOI
 Brém72: (1972) A martingale approach to point processes. University of California, Berkeley
 BrNa88: (1988) A Simple Proof of the Multivariate Random Time Change Theorem for Point Processes. Journal of Applied Probability, 25(1), 210–214. DOI
 Knig70: (1970) An Infinitesimal Decomposition for a Class of Markov Processes. The Annals of Mathematical Statistics, 41(5), 1510–1529. DOI
 CaCh06: (2006) Conditioned Stable Lévy Processes and the Lamperti Representation. Journal of Applied Probability, 43(4), 967–983.
 Lamp67: (1967) Continuousstate branching processes. Bull. Amer. Math. Soc, 73(3), 382–386.
 Meye71: (1971) Demonstration simplifiee d’un theoreme de Knight. In Séminaire de Probabilités V Université de Strasbourg (pp. 191–195). Springer Berlin Heidelberg DOI
 HaPB10: (2010) Discrete Time Rescaling Theorem: Determining Goodness of Fit for Discrete Time Statistical Models of Neural Spiking. Neural Computation, 22(10), 2477–2506. DOI
 Papa72: (1972) Integrability of expected increments of point processes and a related random change of scale. Transactions of the American Mathematical Society, 165, 483–506. DOI
 Appl09: (2009) Lévy processes and stochastic calculus. Cambridge ; New York: Cambridge University Press
 BaPS01: (2001) Multivariate subordination, selfdecomposability and stability. Advances in Applied Probability, 33(1), 160–187. DOI
 RaWu01: (2001) On model selection. In Institute of Mathematical Statistics Lecture Notes  Monograph Series (Vol. 38, pp. 1–57). Beachwood, OH: Institute of Mathematical Statistics
 Scho02: (2002) On Rescaled Poisson Processes and the Brownian Bridge. Annals of the Institute of Statistical Mathematics, 54(2), 445–457. DOI
 Nair90: (1990) Random Space Change for Multiparameter Point Processes. The Annals of Probability, 18(3), 1222–1231. DOI
 AaHo78: (1978) Random time changes for multivariate counting processes. Scandinavian Actuarial Journal, 1978(2), 81–101. DOI
 Kurt80: (1980) Representations of Markov Processes as Multiparameter Time Changes. The Annals of Probability, 8(4), 682–715. DOI
 VeSc04: (2004) Rescaling Marked Point Processes. Australian & New Zealand Journal of Statistics, 46(1), 133–143. DOI
 BTMH05: (2005) Residual analysis for spatial point processes (with discussion). Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 67(5), 617–666. DOI
 GiKM08: (2008) Simulating point processes by intensity projection. In Simulation Conference, 2008. WSC 2008. Winter (pp. 560–568). DOI
 Cox55: (1955) Some Statistical Methods Connected with Series of Events. Journal of the Royal Statistical Society. Series B (Methodological) , 17(2), 129–164.
 ChPR13: (2013) The Lamperti representation of realvalued selfsimilar Markov processes. Bernoulli, 19(5B), 2494–2523. DOI
 BBVK02: (2002) The timerescaling theorem and its application to neural spike train data analysis. Neural Computation, 14(2), 325–346. DOI
 Scho99: (1999) Transforming spatial point processes into Poisson processes. Stochastic Processes and Their Applications, 81(2), 155–164. DOI