Duncan, Tyrone E.; Hu, Yaozhong; Pasik-Duncan, Bozenna Stochastic calculus for fractional Brownian motion. I: Theory. (English) Zbl 0947.60061 SIAM J. Control Optimization 38, No. 2, 582-612 (2000). 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. Cited in 5 ReviewsCited in 249 Documents MSC: 60H05 Stochastic integrals 60G15 Gaussian processes 60G18 Self-similar stochastic processes 60H30 Applications of stochastic analysis (to PDEs, etc.) Keywords:fractional Brownian motion; stochastic calculus; Itô integral; Stratonovich integral; Itô formula; Wick product; Itô calculus; multiple Itô integrals; multiple Stratonovich integrals PDFBibTeX XMLCite \textit{T. E. Duncan} et al., SIAM J. Control Optim. 38, No. 2, 582--612 (2000; Zbl 0947.60061) Full Text: DOI