| Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion (2006) | |||||||||||||
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| Exact rate convergence some approximation schemes associated SDEs driven fractional Brownian motion Andreas Neuenkirch Johann Wolfgang Goethe Universit Fachbereich Mathematik Robert Mayer Stra Frankfurt Main Germany neuenkirch math uni frankfurt hal version Nov Ivan Nourdin LPMA Universit Pierre Marie Curie Paris courrier Place Jussieu Paris Cedex France ite nourdin ccr jussieu Abstract this paper derive the exact rate convergence some approximation schemes associated scalar stochastic erential equations driven fractional Brownian motion with Hurst index consider two cases the exact rate convergence the Euler scheme determined show that the error the Euler scheme converges almost surely random variable which particular depends the Malliavin derivative the solution This result extends those contained and When the exact rate convergence the Crank Nicholson scheme determined for particular equation Here show convergence law the error random variable which depends the solution the equation and independent Gaussian random variable Key words Fractional Brownian motion Russo Vallois integrals Doss Sussmann type transformation Stochastic erential equations Euler scheme Crank Nicholson scheme Mixing law Mathematics Subject Classi cation H Introduction Let fractional Brownian motion short fBm with Hurst parameter is continuous centered Gaussian process with covariance function s t t For a standard Brownian motion while for neither semimartingale nor Markov process Moreover holds Bs s a | |||||||||||||
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