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From Stochastic Integration w.r.t. Fractional Brownian Motion to Stochastic Integration wrt Multifractional Brownian Motion

Abstract : Because of numerous applications {\textit{e.g.}} in finance and in Internet traffic modelling, stochastic integration w.r.t. fractional Brownian motion (fBm) became a very popular topic in recent years. However, since fBm is not a semi-martingale the Itô integration can not be used for integration w.r.t. fBm and one then needs specific developments. Multifractional Brownian motion (mBm) is a Gaussian process that generalizes fBm by letting the local Hölder exponent vary in time. In addition to the fields mentioned above, it is useful in many and various areas such as geology and biomedicine. In this work we start from the fact, established in \cite[Thm 2.1.(i)]{Herbin Lebovits Vehel}, that an mBm may be approximated, in law, by a sequence of ''tangent" fBms. We used this result to show how one can define a stochastic integral w.r.t. mBm from the stochastic integral w.r.t. fBm, defined in \cite{Bender 2003}, in the white noise theory sense.
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Submitted on : Friday, April 26, 2013 - 3:44:47 PM
Last modification on : Tuesday, July 30, 2013 - 10:19:20 AM
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Joachim Lebovits. From Stochastic Integration w.r.t. Fractional Brownian Motion to Stochastic Integration wrt Multifractional Brownian Motion. Annals of the University of Bucharest. Mathematical series, București : Editura Universității din București, 2013. ⟨hal-00818321⟩

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