r-variations for two-parameter continuous martingales and Itô's formula
Let \(M=\{M_ z;z\in [0,1]^ 2\}\) be a two-parameter continuous martingale bounded in \(L^ 4\), and suppose that f is a real-valued function of class \(C^ 4\) such that \(f(0)=0\). The aim of this paper is to establish an Itô's formula of the type \[ f(M_ z)=\sum^{4}_{r=1}(r!)^{-1}\int_{[0,z]}f^{(r)}(M_ u)d\mu^ r_ M(u), \] where the processes \(\mu^ r_ M(z)\) are two-parameter continuous semimartingales called the r-variations of M. In the proof of this result the author makes use of the stochastic integrator properties of the r-variations that have been proved by P. Imkeller. This kind of formula was first introduced by \textit{M.-F. Allain} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 421-444 (1984; Zbl 0534.60044)]. The relation between this Itô formula and the change of variables formulas obtained by \textit{L. Chevalier} [Bull. Sci. Math., II. Ser. 106, 19-62 (1982; Zbl 0493.60055)] and by the reviewer [Ann. Inst. Henri Poincaré, Probab. Stat. 20, 251-275 (1984; Zbl 0543.60062)] is also discussed in the paper. As an application the author obtains a two- parameter version of Tanaka's formula.
two-parameter continuous semimartingales
two-parameter version of Tanaka's formula
scientific article; zbMATH DE number 4105972