Gradinaru, Mihai; Nourdin, Ivan; Russo, Francesco; Vallois, Pierre \(m\)-order integrals and generalized Itô’s formula; the case of a fractional Brownian motion with any Hurst index. (English) Zbl 1083.60045 Ann. Inst. Henri Poincaré, Probab. Stat. 41, No. 4, 781-806 (2005). Suppose \(X\) is a continuous process and \(g: {\mathbb R}\to{\mathbb R}\) is a locally bounded Borel function. Given a positive integer \(m\) and a probability measure \(\nu\) on \([0,1]\), the authors introduce \(m\)-order \(\nu\)-integrals as \[ \int_0^tg(X_u)\,d^{\nu,m}X_u:= \lim_{\varepsilon\downarrow 0} \text{ prob}\, {1\over{\varepsilon}}\int_0^t du\, (X_{u+\varepsilon}-X_u)^m\int_0^1 g(X_u+\alpha(X_{u+\varepsilon}-X_u))\,\nu(d\alpha). \] When \(\nu\) is symmetric, the corresponding integral is an extension of symmetric integrals of Stratonovich type. If \(f\in C^{2n}({\mathbb R})\), \(\nu\) is symmetric, and if \(X\) is a continuous process having a \((2n)\)-variation (i.e. \([X,X,\dots,X]\)), then the Itô formula holds with some extra terms consisting of higher order \(\delta_{1/2}\)-integrals. In the case of the fractional Brownian motion \(B^H\), \(0<H<1\), \(m\)-order \(\nu\)-integral vanishes for all odd indices \(m>1/2H\) and any symmetric \(\nu\). If \(\nu\) is any symmetric probability measure, then the Itô-Stratonovich formula holds for \(H>1/6\), but fails to hold for \(H\leq 1/6\). However, if \(H\leq 1/6\), an Itô formula is still valid provided we proceed through a different regularization of the symmetric integral which involves particular symmetric probability measures. Reviewer: Kestutis Kubilius (Vilnius) Cited in 2 ReviewsCited in 58 Documents MSC: 60H05 Stochastic integrals 60G15 Gaussian processes 60G18 Self-similar stochastic processes PDFBibTeX XMLCite \textit{M. Gradinaru} et al., Ann. Inst. Henri Poincaré, Probab. Stat. 41, No. 4, 781--806 (2005; Zbl 1083.60045) Full Text: DOI Numdam EuDML