Parameter estimation for α-fractional bridges

Let α, T > 0. We study the asymptotic properties of a least squares estimator for the parameter α of a fractional bridge defined as dXt = −α Xt T−t dt + dBt, 0 6 t < T , where B is a fractional Brownian motion of Hurst parameter H > 12 . Depending on the value of α, we prove that we may have strong consistency or not as t → T . When we have consistency, we obtain the rate of this convergence as well. Also, we compare our results to the (known) case where B is replaced by a standard Brownian motion W . It is great pleasure for us to dedicate this paper to our friend David Nualart, in celebration of his 60th birthday and with all our admiration.