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 \(B^H(t)\) with arbitrary Hurst parameter \(H\in (0,1)\) 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 \(F(B^H(t))\) 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 \(B^H(t)\) is given and an integral representation of the local time of \(B^H(t)\) is established.


60H05 Stochastic integrals
60H40 White noise theory
60G20 Generalized stochastic processes
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