Summary
Continuous two-parameter strong martingalesM andN with respect to a filtration, which is closely related to the filtration of a Wiener sheet, are constructed such thatM and the continuous increasing process [N] ofN (identical with the quadratic variation) cannot be “boundedly localized”: each stopping domainD for which the processesM or [N] stopped byD, are bounded must be contained in a nontrivial subset of ℝ 2+ which only depends onM orN.
References
Cairoli, R., Walsh, J.: Stochastic integrals in the plane. Acta Mathematica134, 111–183 (1975)
Cairoli, R., Walsh, J.: Régions d'arrêt, localisations et prolongements de martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb.44, 279–306 (1978)
Chevalier, L.: Martingales continues à deux paramètres. Bull. Sc. Math. (2)106, 19–62 (1982)
Merzbach, E.: Stopping for two-dimensional stochastic processes. Stoch. Proc. Appl.10, 49–63 (1980)
Merzbach, E.: Chemins croissants optionnels et théorème de section. Ann. Inst. H. Poincaré19, 223–234 (1983)
Merzbach, E.: Processus stochastiques à indices partiellement ordonnés. Rapport interne55. Ecole Polytechnique, Palaiseau 1979
Nualart, D.: Variations quadratiques et inégalités pour les martingales à deux indices. Preprint, Univ. de Barcelona (1984)
Walsh, J.: Convergence and regularity of multiparameter strong martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb.46, 177–192 (1979)
Wong, E., Zakai, M.: Weak martingales and stochastic integrals in the plane. Ann. Probab.4, 570–587 (1976)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Imkeller, P. A note on the localization of two-parameter processes. Probab. Th. Rel. Fields 73, 119–125 (1986). https://doi.org/10.1007/BF01845995
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01845995
Keywords
- Filtration
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Quadratic Variation