Summary
This paper deals with the following problem, given a two parameter stochastic process, under what conditions is it possible to stop the process at any stopping line? It is shown that the class of stoppable processes is strictly larger than the class of two parameter integrators. Sufficient conditions for a weak martingale to be stoppable are derived and the stopped r.v. is represented as a one parameter optional dual projection.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bakry, D.: Une remarque sur les semimartingales à deux indices. Lect. Notes in Math.850, 671–672 (1980)
Bakry, D.: Semi-martingales à deux indices. Lecture Notes in Math.920, 355–369 (1982)
Cairoli, R., Walsh, J.B.: Stochastic integrals in the plane. Acta Math.134, 111–183 (1975)
Cairoli, R., Walsh, J.B.: Régions d'arrêt, localisations et prolongements de martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb.44, 279–306 (1978)
Clarkson, J.A., Adams, C.R.: On definitions of bounded variation for functions of two variables. Trans. Am. Math. Soc.,35, 824–854 (1933)
Dellacherie, C., Meyer, P.A.: Probabilities and potential, Part B. Amsterdam: North Holland 1982
Meyer, P.A.: Théorie élémentaire des processus à deux indices. Lecture Notes in Math.863, 1–39 (1981)
Walsh, J.B.: Optional increasing paths. Lect. Notes in Math.863, 172–201
Wong, E., Zakai, M.: Weak martingales and stochastic integrals in the plane. Ann. Probab.4, 570–586 (1976)
Author information
Authors and Affiliations
Additional information
Work partially supported by a grant from the Research Authority at Bar-Ilan University
Work supported by the fund for promotion of research at the Technion
Rights and permissions
About this article
Cite this article
Merzbach, E., Zakai, M. Stopping a two parameter weak martingale. Probab. Th. Rel. Fields 76, 499–507 (1987). https://doi.org/10.1007/BF00960070
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00960070