Random change of time

Stochastic processes derived by varying the rate of time’s passage, which is more convenient than I imagined

August 5, 2015 — February 10, 2020

functional analysis
time series

🏗 Various notes on a.e. continuous monotonic random changes of index in order build new processes.

In Warping and registration problems you try to align two or more processes; this can sometimes be an alignment problem, but not necessarily.

Figure 1

1 To explore

Lamperti representation for continuous state branching processes,

Ogata’s time rescaling: Intensity estimation for point processes uses this as a statistical test.

Relation to e.g. martingale transforms.

2 Subordinator

A subordinator is a non-decreasing Lévy Process taking values on the reals. AFAICS this is precisely a Gamma process. Curiously, upon giving that definition, many proceed to immediately assert that such a process is a model for a random change of time. This is not insane per se, but doesn’t have much narrative momentum, as a Gamma process can model a bunch of other things than time, and it is a weird specialisation in a field that normally tends to excessive generality.

🏗 I should explain why one would bother doing such an arbitrary thing as changing time though; Basically it is because a time-changed Lévy process is still a Lévy process.

3 Point process rate transform

As used in point process residual goodness of fit tests.

A summary in (Vere-Jones and Schoenberg 2004):

Knight (Knight 1970) 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 (Meyer 1971) extended Knight’s theorem to the case of point processes, showing that given a simple multivariate point process \({N_i ; i = 1, 2, \dots, n}\), 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 (Brown and Nair 1988; Kurtz 1980; Papangelou 1972; Aalen and Hoem 1978; Brémaud 1972) Papangelou (Papangelou 1972) 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 (Nair 1990; Merzbach and Nualart 1986; F. Schoenberg 1999). 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.

4 Going Multivariate

As seen in, e.g. (Ole E. Barndorff-Nielsen, Pedersen, and Sato 2001). How does multivariate time work then?

5 References

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Applebaum. 2009. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 116.
Baddeley, Turner, Møller, et al. 2005. Residual Analysis for Spatial Point Processes (with Discussion).” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Barndorff-Nielsen, Ole E., Pedersen, and Sato. 2001. Multivariate Subordination, Self-Decomposability and Stability.” Advances in Applied Probability.
Barndorff-Nielsen, Ole E, and Shiryaev. 2010. Change of Time and Change of Measure. Advanced Series on Statistical Science & Applied Probability.
Brémaud. 1972. “A Martingale Approach to Point Processes.”
Brown, and Nair. 1988. A Simple Proof of the Multivariate Random Time Change Theorem for Point Processes.” Journal of Applied Probability.
Caballero, and Chaumont. 2006. Conditioned Stable Lévy Processes and the Lamperti Representation.” Journal of Applied Probability.
Chaumont, Pantí, and Rivero. 2013. The Lamperti Representation of Real-Valued Self-Similar Markov Processes.” Bernoulli.
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Çinlar. 1980. On a Generalization of Gamma Processes.” Journal of Applied Probability.
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Lamperti. 1958. On the Isometries of Certain Function-Spaces.” Pacific J. Math.
———. 1967. Continuous-State Branching Processes.” Bull. Amer. Math. Soc.
———. 1972. Semi-Stable Markov Processes. I.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete.
Leonenko, Meerschaert, Schilling, et al. 2014. Correlation Structure of Time-Changed Lévy Processes.” Communications in Applied and Industrial Mathematics.
Madan, and Seneta. 1990. The Variance Gamma (V.G.) Model for Share Market Returns.” The Journal of Business.
Merzbach, and Nualart. 1986. A Characterization of the Spatial Poisson Process and Changing Time.” The Annals of Probability.
Meyer. 1971. Demonstration simplifiee d’un theoreme de Knight.” In Séminaire de Probabilités V Université de Strasbourg. Lecture Notes in Mathematics.
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Papangelou. 1972. Integrability of Expected Increments of Point Processes and a Related Random Change of Scale.” Transactions of the American Mathematical Society.
Rao, and Wu. 2001. On Model Selection.” In Institute of Mathematical Statistics Lecture Notes - Monograph Series.
Schoenberg, Frederic. 1999. Transforming Spatial Point Processes into Poisson Processes.” Stochastic Processes and Their Applications.
Schoenberg, Frederic Paik. 2002. On Rescaled Poisson Processes and the Brownian Bridge.” Annals of the Institute of Statistical Mathematics.
Seneta. 2007. The Early Years of the Variance-Gamma Process.” In Advances in Mathematical Finance. Applied and Numerical Harmonic Analysis.
Vere-Jones, and Schoenberg. 2004. Rescaling Marked Point Processes.” Australian & New Zealand Journal of Statistics.