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An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. (English) Zbl 1075.60530
Stochastic calculus for fractional Brownian motion ${B}^{H}\left(t\right)$ with arbitrary Hurst parameter $H\in \left(0,1\right)$ is developed. The fractional Brownian motion and fractional white noise are constructed on the classical white noise space. The fractional Itô integral is defined by means of the Wick product as a Pettis type integral on the space of Hida distributions. White noise techniques are used to prove the generalized Itô formula for functional $F\left({B}^{H}\left(t\right)\right)$ where $F$ is a tempered distribution. As an application the Tanaka formula is proven. Finally, a Clarc-Okone formula for Donsker’s delta function of ${B}^{H}\left(t\right)$ is given and an integral representation of the local time of ${B}^{H}\left(t\right)$ is established.
##### MSC:
 60H05 Stochastic integrals 60H40 White noise theory 60G20 Generalized stochastic processes
##### Keywords:
Tanaka formula; local time