Stochastic integration with respect to fractional brownian motion

Abstract

For every value of the Hurst index H ∈ (0, 1) we define a stochastic integral with respect to fractional Brownian motion of index H . We do so by approximating fractional Brownian motion by semi-martingales.Then, for H > 1/6, we establish an Itô's change of variables formula, which is more precise than Privault's Ito formula (1998) (established for every H > 0), since it only involves anticipating integrals with respect to a driving Brownian motion.

Large deviations for rough paths of the fractional Brownian motion

“…They are a sort of deterministic counterpart of the stochastic integral with respect to the fractional Brownian motion introduced in [1] (see also [2]). …”

Introduction to Diffusive Representation

“…A complete theoretical statement will be found in Montseny [2005] and various applications to non trivial problems will be found in , Carmona and Coutin [1998a,b], Carmona et al [2003], Casenave andMontseny [2009, 2008], Degerli et al [Ap. 1999], Garcia and Bernussou [1998], Lenczner and Montseny [2005], Lenczner et al [2010], Levadoux and Montseny [2003], Montseny [2007], Mouyon and Imbert [2002], Rumeau et al [2006], Bidan et al [2001], , , Devy-Vareta and , Helie and Matignon [2006], Laudebat et al [2004], Lavernhe et al [2001], Lavernhe and Solhusvik [1998], Leger and Pontier [1999], Lubich and Schädler [2002], Matignon [July 2001], Matignon and Prieur [2005], Mbodge et al [1994], Mbodge and Montseny [1995], Montseny [2002bMontseny [ ,a, 1998], Montseny et al [2000], Nihtila and Tervo [2002], Rouzaud [1998].…”

Stochastic calculus with respect to fractional Brownian motion

“…Thus, we could use the Skorohod integral in formula (2.8), and in that case, the integral T 0 u t dB t coincides with the divergence operator in the Malliavin calculus with respect to the fBm B. The approach of Malliavin calculus to define stochastic integrals with respect to the fBm has been introduced by Decreusefont andÜstünel in [13], and further developed by several authors (Carmona and Coutin [6], Alòs, Mazet and Nualart [3], Alòs and Nualart [4], Alòs, León and Nualart [1], and Hu [18]). …”