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Stochastic calculus for fractional Brownian motion. I: Theory. (English) Zbl 0947.60061
Summary: A stochastic calculus is given for the fractional Brownian motions that have the Hurst parameter in $(1/2,1)$. A stochastic integral of Itô type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment. This integral uses the Wick product and a derivative in the path space. Some Itô formulae (or change of variables formulae) are given for smooth functions of a fractional Brownian motion or some processes related to a fractional Brownian motion. A stochastic integral of Stratonovich type is defined and the two types of stochastic integrals are explicitly related. A square integrable functional of a fractional Brownian motion is expressed as an infinite series of orthogonal multiple integrals.

60H05Stochastic integrals
60G15Gaussian processes
60G18Self-similar processes
60H30Applications of stochastic analysis
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