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Stochastic calculus for fractional Brownian motion - I. Theory
Duncan, Tyrone E. Hu, Yaozhong Pasik-Duncan, Bozenna
Duncan, Tyrone E.
Hu, Yaozhong
Pasik-Duncan, Bozenna
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Abstract
In this paper a stochastic calculus is given for the fractional Brownian motions that have the Hurst parameter in (1/2, 1). A stochastic integral of Ito 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 Ito 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.
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Date
2000-02-02
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SIAM PUBLICATIONS
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Keywords
Fractional brownian motion, Multiple stratonovich integrals, Multiple ito integrals, Ito calculus, Stochastic calculus, Ito integral, Stratonovich integra, Ito formula, Wick product
Citation
Duncan, TE; Hu, YZ; Pasik-Duncan, B. Stochastic calculus for fractional Brownian motion - I. Theory. SIAM JOURNAL ON CONTROL AND OPTIMIZATION. Feb 2 2000.38(2):582-612.