A natural generalization of the optional sampling theorem for martingales is given. For discrete valued stopping times the result holds for directed sets; for more general stopping times the result holds for lattices satisfying a type of separability condition. The discrete case improves a lemma of Chow. The general case depends upon a lemma showing that all martingales with respect to $\sigma$-algebras satisfying a "right continuity" condition have a modification which has a regularity property that is similar to, but weaker than, right continuity. A result of Wong and Zakai is obtained as a corollary.
Publié le : 1980-08-14
Classification:
Martingale,
optional sampling,
stopping time,
directed index set,
multidimensional time,
60G45,
60G40,
60G15,
60G05
@article{1176994659,
author = {Kurtz, Thomas G.},
title = {The Optional Sampling Theorem for Martingales Indexed by Directed Sets},
journal = {Ann. Probab.},
volume = {8},
number = {6},
year = {1980},
pages = { 675-681},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994659}
}
Kurtz, Thomas G. The Optional Sampling Theorem for Martingales Indexed by Directed Sets. Ann. Probab., Tome 8 (1980) no. 6, pp. 675-681. http://gdmltest.u-ga.fr/item/1176994659/