The maximum likelihood drift estimator for mixed fractional Brownian motion

The paper is concerned with the maximum likelihood estimator (MLE) of the unknown drift parameter $\theta\in\Real$ in the continuous-time regression model $X_t = \theta t + B_t + B^H_t, \quad t\in [0,T]$ where $B_t$ is a Brownian motion and $B^H_t$ is an independent fractional Brownian motion with Hurst parameter $H\in (\frac 1 2,1)$. We derive the exact formula for the MLE in terms of the solution of an integral equation and find the asymptotic distribution of the estimation error. In particular, it turns out that the Brownian part does not contribute to the asymptotic variance of the MLE. Another contribution of this paper is a formula for the Radon--Nikodym derivative of the probability, induced by the mixed fractional Brownian motion $\xi_t = B_t + B^H_t$, $H>3/4$ with respect to the Wiener measure.