Statistical Inference for Stochastic Processes (2005) 8: 71–93 © Springer 2005 71 Statistical Inference with Fractional Brownian Motion ALEXANDER KUKUSH 1 , YULIA MISHURA 1 and ESKO VALKEILA 2 1 Department of Mathematics, Kiev University, Volodimirska Street 64, 01033 Kiev, Ukraine. e-mails: alexander_kukush@univ.kiev.ua; myus@univ.kiev.ua 2 Department of Mathematics, University of Helsinki, P.O. Box 4, FIN-00014, University of Helsinki, Finland. e-mail: esko.valkeila@helsinki.fi Abstract. We give a test between two complex hypothesis; namely we test whether a fractional Brownian motion (fBm) has a linear trend against a certain non-linear trend. We study some related questions, like goodness-of-fit test and volatility estimation in these models. AMS Mathematics Subject Classification: 62M07, 62M09. Key words: fractional Brownian motions, hypothesis testing, goodness-of-fit test, volatility estimation. 1. Introduction 1.1. SETUP A fractional Brownian motion (fBm) Z = Z H is a self-similar Gaussian process with index H ∈ (0, 1): it is a continuous Gaussian process with stationary incre- ments, defined on a probability space (,IF,IP), with the properties • Z 0 = 0. • IEZ t = 0 for every t  0. • IEZ t Z s = 1/2(t 2H + s 2H -|s - t | 2H ) for every s,t  0. FBm is not a semimartingale, if H  = 1/2. There are several approaches to define stochastic integrals w.r.t. fBm and they are briefly mentioned in connection to the linear Equation (1.1) in Section 1.2. How the stochastic integral is defined has an impact on modeling with fBm. Namely, the so-called Riemann–Stieltjes integral contributes to the mean rate of signal, but the so-called Skorohod integral does not contribute to the mean rate (see Duncan et al., 2000, p. 583). Concerning the use of fBm as a model in finance the meaning of the two different integrals is still under discussion (for more details see Hu and Øksendal, 2003; Sottinen and Valkeila, 2003). In the context of the linear Equation (1.1) this means that the difference is a certain non-linear drift. This test is developed in Section 4, where the observations are discrete and the intensity σ of the fBm is unknown. In Sections 2 and 3 we give the necessary background for this test. In addition we give some results on