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On the transformation of martingales with a two dimensional parameter set by convex functions. (English) Zbl 0558.60042

This paper partially answers a P. A. Meyer’s question: if M is a two- indices martingale and f is a convex function, is f(M) always a martingale? Reducing the problem to the one-dimensional case, it is given as a sufficient condition when M is strong and bounded in a convenient sense that f” is convex too.
Reviewer: B.Prum

MSC:

60G60 Random fields
60G44 Martingales with continuous parameter
60H05 Stochastic integrals
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References:

[1] Cairoli, R.; Walsh, J. R., Stochastic integrals in the plane, Acta Math., 134, 111-183 (1975) · Zbl 0334.60026 · doi:10.1007/BF02392100
[2] Guyon, X.; Prum, B., Semi-martingales à indice dans R^2, Thèse de Doctorat d’Etat (1980), Orsay: Univ. Paris-Sud, Orsay · Zbl 0447.60041
[3] Guyon, X., Deux résultats sur les martingales browniennes à deux indices, C.R. Acad. Sci. Paris, sér. 1, 295, 359-361 (1982) · Zbl 0499.60054
[4] Meyer, P. A., Théorie élémentaire des processus à deux indices (1981), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0461.60072
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