# zbMATH — the first resource for mathematics

$$m$$-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
Full Text: