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 the Brownian motion and $B^H_t$ is independent fractional Brownian motion with the 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.