On compact Itô's formulas for martingales of \(m_ c^ 4\) (Q2638655)

Let \(\{\) \({\mathcal F}_ z\), \(z\in [0,1]^ 2\}\) be an increasing family of sub-\(\sigma\)-fields on a complete probability space (\(\Omega\),\({\mathcal F},P)\), verifying the usual conditions (F1) - (F4) introduced by \textit{R. Cairoli} and \textit{J. B. Walsh} [Acta Math. 134, 111-183 (1975; Zbl 0334.60026)]. The author shows that the class \(m^ 4_ c\) of continuous martingales bounded in \(L^ 4\) is included into the family of semimartingales \(S^{\infty}_ c(L^ 0(P))\) introduced by \textit{M.-F. Allain} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 421-444 (1984; Zbl 0534.60044)]. These semimartingales are characterized by the fact that all the powers \(M^ k_ z\), \(k\geq 1\), define \(L^ 0(P)\)-stochastic measures and the maximal processes \((M^*_ z)^ j\) are integrable with respect to these measures. On the other hand it is shown that the compact Itô's formula obtained by \textit{M. Sanz} [Stochastic Processes Appl. 32, No.1, 69-92 (1989; Zbl 0675.60041)] for martingales of \(m^ 4_ c\) can be regarded as a particular case of the Itô's formula established by M. F. Allain for semimartingales of the class \(S^{\infty}_ c(L^ 0(P))\).