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\(m\)-order integrals and generalized Itô’s formula; the case of a fractional Brownian motion with any Hurst index. (English) Zbl 1083.60045
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.

MSC:
60H05 Stochastic integrals
60G15 Gaussian processes
60G18 Self-similar stochastic processes
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