A stochastic calculus for continuous N-parameter strong martingales
Let M be a 4N-integrable, real-valued continuous N-parameter strong martingale with respect to an increasing family of \(\sigma\)-fields satisfying the conditional independence property introduced by \textit{R. Cairoli} and \textit{J. B. Walsh}, Acta Math. 134, 111-183 (1975; Zbl 0334.60026). A stochastic calculus for multiparameter martingales is developed, and several Itô-type stochastic differentiation formulas are established. One of them is in terms of some stochastic measures or ``variations'' associated with M, another is obtained by decomposing these measures and a third one is derived from the second by iterated application of a stochastic version of Green's formula. These results are based on the multiparameter Burkholder L p-inequalities and the Itô formula derived by \textit{M. F. Allain}, Z. Wahrscheinlichkeitstheor. Verw. Gebiete 65, 421-444 (1984; Zbl 0534.60044).
stochastic calculus for multiparameter martingales
multiparameter Burkholder L p-inequalities
scientific article; zbMATH DE number 4064176
scientific article