451 0894-9840/03/0400-0451/0 © 2003 Plenum Publishing Corporation Journal of Theoretical Probability, Vol. 16, No. 2, April 2003 (© 2003) Besov Regularity of Stochastic Integrals with Respect to the Fractional Brownian Motion with Parameter H > 1/2 David Nualart 1 and Youssef Ouknine 2 1 Facultat de Matemàtiques, Universitat de Barcelona, Gran Via Corts Catalanes 585, 08007 Barcelona, Spain. E-mail: nualart@mat.ub.es 2 Faculté des Sciences Semlalia, Département de Mathématiques, Université Cadi Ayyad, BP 2390, Marrakech, Morocco. Received November 8, 2001; revised May 3, 2002 Let {B t ,t ¥ [0, 1]} be a fractional Brownian motion with Hurst parameter H> 1 2 . Using the techniques of the Malliavin calculus we show that the trajectories of the indefinite divergence integral > t 0 u s dB s belong to the Besov space B a p, q for all q \ 1, 1 p < a <H, provided the integrand u belongs to the space L p, 1 . Moreover, if u is bounded and belongs to L d,2 for some even integer p \ 2 and for some d large enough, then the trajectories of the indefinite divergence integral > t 0 u s dB s belong to the Besov space B H p, . . KEY WORDS: Fractional Brownian motion; stochastic integrals; Malliavin calculus. 1. INTRODUCTION Let B={B t ,t ¥ [0, 1]} be a fractional Brownian motion with Hurst parameter H. If H ] 1 2 , the process B is not a semimartingale and we cannot apply the stochastic calculus developed by Ito ˆ in order to define stochastic integrals with respect to B. Different approaches have been used in order to define stochastic integrals > t 0 u s dB s . From E |B t -B s | 2 =|t - s| 2H it follows that B has a-Hölder continuous paths for all a <H. As a conse- quence, if the process u has b-Hölder continuous paths with a+b >1, then we can define the pathwise integral > t 0 u s dB s using Young’s approach (see Ref. 24).