Watanabe proved that if $X_t$ is a point process such that $X_t - t$ is a martingale, then $X_t$ is a Poisson process and this result was generalized by Bremaud for doubly stochastic Poisson processes. Here we define two-parameter point processes and extend this property without needing the strong martingale condition. Using this characterization, we study the problem of transforming a two-parameter point process into a two-parameter Poisson process by means of a family of stopping lines as a time change. Nualart and Sanz gave conditions in order to transform a square integrable strong martingale into a Wiener process. Here, we do the same for the Poisson process by a similar method but under more general conditions.
Publié le : 1986-10-14
Classification:
Point process,
Poisson,
two-parameter process,
martingale,
intensity,
changing time,
stopping line,
60G55,
60G48,
60G60,
60G40
@article{1176992378,
author = {Merzbach, Ely and Nualart, David},
title = {A Characterization of the Spatial Poisson Process and Changing Time},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 1380-1390},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992378}
}
Merzbach, Ely; Nualart, David. A Characterization of the Spatial Poisson Process and Changing Time. Ann. Probab., Tome 14 (1986) no. 4, pp. 1380-1390. http://gdmltest.u-ga.fr/item/1176992378/