Une formule d'Itô pour les martingales continues à deux indices et quelques applications
After having proved the existence of a continuous version of the quadratic variation of a two-parameter continuous martingale [On the quadratic variation of two-parameter continuous martingales. Ann. Probab. 12, 445-457 (1984)], the author provides in the paper under review an Itô differentiation formula for arbitrary two-parameter continuous martingales. Such a formula has already been obtained by \textit{L. Chevalier}, Bull. Sci. Math., II. Ser. 106, 19-62 (1982; Zbl 0493.60055), under the additional assumption that any square integrable martingale has a continuous version; in the presently more general setting the author proves this formula by a more accurate and ingenious application of martingales inequalities. As an application, the author proves the existence, continuity and regularity of the local time of these martingales with respect to a particular random measure (a quadratic variation related to the martingale). Finally a maximal inequality for stochastic integrals in one coordinate is presented.
continuity and regularity of the local time
maximal inequality for stochastic integrals
scientific article; zbMATH DE number 3864227