Statistics and Probability Letters 81 (2011) 243–249 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Drift parameter estimation in fractional diffusions driven by perturbed random walks Karine Bertin a,* , Soledad Torres a , Ciprian A. Tudor b,1 a Departamento de Estadística, CIMFAV Universidad de Valparaíso, Casilla 123-V, 4059 Valparaiso, Chile b Laboratoire Paul Painlevé, Université de Lille 1, F-59655 Villeneuve d’Ascq, France article info Article history: Received 25 January 2010 Received in revised form 13 August 2010 Accepted 6 October 2010 Available online 27 October 2010 MSC: 60G18 62M99 Keywords: Fractional Brownian motion Maximum likelihood estimation Random walk abstract We estimate the drift parameter in a simple linear model driven by fractional Brownian motion. We propose maximum likelihood estimators (MLE) for the drift parameter con- struct by using a random walk approximation of the fractional Brownian motion. © 2010 Elsevier B.V. All rights reserved. 1. Introduction The self-similar processes are of interest for various applications, such as economics, internet traffic or hydrology. The fractional Brownian motion (fBm) is the usual candidate to model phenomena in which the self-similarity property can be observed from the empirical data. Recall that the fractional Brownian motion is a centered Gaussian process with the covariance function R H (t , s) = 1 2 (t 2H + s 2H -|t - s| 2H ), H ∈ (0, 1). The fBm can also be defined as the only Gaussian process which is self-similar with stationary increments (see Embrechts and Maejima, 2002). In the last few years the study, from the stochastic calculus point of view, of such processes has been intensively developed. The most popular self-similar stochastic process that exhibits long-range dependence is of course fractional Brownian motion (fBm). Recently, stochastic integrals of various types with respect to fBm have been constructed and stochastic differential equations driven by fBm have been considered (see e.g. Nualart, 2003). The stochastic analysis of the fractional Brownian motion naturally led to the statistical inference for diffusion processes with fBm as the driving noise. We address in this work the problem of the estimation of the drift parameter in the model dY t = ab(Y t )dt + dB H t , t ∈[0, T ] (1) where (B H t ) t ∈[0,T ] is a fractional Brownian motion with a Hurst index H ∈ (0, 1) and b is a deterministic function satisfying some regularity conditions, and assume that the parameter a ∈ R has to be estimated. Such questions have been treated * Corresponding author. Tel.: +56 322508324. E-mail addresses: karine.bertin@uv.cl, karinebertin@hotmail.com (K. Bertin), soledad.torres@uv.cl (S. Torres), tudor@math.univ-lille1.fr (C.A. Tudor). 1 Associate member: SAMOS, Centre d’Economie de La Sorbonne, Université de Panthéon-Sorbonne Paris 1, 90, rue de Tolbiac, 75634 Paris Cedex 13, France. 0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2010.10.003