r-variations for two-parameter continuous martingales and Itô's formula
type: Item
instance of: scholarly article
title: r-variations for two-parameter continuous martingales and Itô's formula
zbMATH Open document ID: 0675.60041
DOI: 10.1016/0304-4149(89)90054-9
published in: Stochastic Processes and their Applications
publication date: 1989-01-01T00:00:00Z
review text: 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.
reviewed by: David Nualart
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 60G60
zbMATH DE Number: 4105972
zbMATH Keywords: Itô's formula
zbMATH Keywords: two-parameter continuous semimartingales
zbMATH Keywords: two-parameter version of Tanaka's formula
author: Marta Sanz-Solé
MaRDI profile type: Publication
full work available at URL: 0304-4149(89)90054-9
OpenAlex ID: W2017459267
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description: scientific article; zbMATH DE number 4105972