On the quadratic variation of two-parameter continuous martingales
Let \(M=(M_ z\); \(z\in [0,1]^ 2)\) be a square integrable continuous martingale. The main result of this paper is the sample continuity of the quadratic variation \(<M>\), what so far has only been proved for some special kinds of martingales [see the author, Sémin. de Probabilités XVII, Proc. 1981/82, Lect. Notes Math. 986, 398-417 (1983; Zbl 0507.60038)]. The method to show these results is based on a two-parameter Itô formula for \(M^ 2\). The construction of \(<M>\) and a more general Itô formula have been obtained by \textit{L. Chevalier} [Bull. Sci. Math., II. Ser. 106, 19-62 (1982; Zbl 0493.60055)] under the additional assumption that the underlying filtration is such that any square integrable martingale has a continuous version.
scientific article; zbMATH DE number 3856123