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

Joachim Lebovits (1, 2) , Jacques Lévy Véhel (3, 1) , Erick Herbin (1, 4)

1 Regularity - Probabilistic modelling of irregularity and application to uncertainties management
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 MAS - Mathématiques Appliquées aux Systèmes - EA 4037
4 FR3487 - Fédération de Mathématiques de l'Ecole Centrale Paris

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 Hitsuda-Skohorod and Wick-Itô stochastic integrals.

Domaines: Probabilités [math.PR]

Joachim Lebovits, Jacques Lévy Véhel, Erick Herbin. Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions. Stochastic Processes and their Applications, 2014, 124, pp.678-708. ⟨10.1016/j.spa.2013.09.004⟩. ⟨hal-00653808v6⟩