Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions

Joachim Lebovits, Jacques Lévy Véhel and Erick Herbin

Stochastic Processes and their Applications, 2014, vol. 124, issue 1, 678-708

Abstract: Stochastic integration w.r.t. fractional Brownian motion (fBm) has raised strong interest in recent years, motivated in particular by applications in finance and Internet traffic modelling. Since fBm is not a semi-martingale, stochastic integration requires specific developments. Multifractional Brownian motion (mBm) generalizes fBm by letting the local Hölder exponent vary in time. This is useful in various areas, including financial modelling and biomedicine. The aim of this work is twofold: first, we prove that an mBm may be approximated in law by a sequence of “tangent” fBms. Second, using this approximation, we show how to construct stochastic integrals w.r.t. mBm by “transporting” corresponding integrals w.r.t. fBm. We illustrate our method on examples such as the Wick–Itô, Skorohod and pathwise integrals.

Keywords: Fractional and multifractional Brownian motions; Gaussian processes; Convergence in law; White noise theory; Wick–Itô integral; Skorohod integral; Pathwise integral (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (5)

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DOI: 10.1016/j.spa.2013.09.004

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