Two approaches to consistent estimation of parameters of mixed fractional Brownian motion with trend

We investigate the mixed fractional Brownian motion with trend of the form X-t = theta t + sigma W-t + kappa B-t(H), driven by a standard Brownian motion W and a fractional Brownian motion B-H with Hurst parameter H. We develop and compare two approaches to estimation of four unknown parameters theta, sigma, kappa and H by discrete observations. The first algorithm is more traditional: we estimate sigma, kappa and H using the quadratic variations, while the estimator of theta is obtained as a discretization of a continuous-time estimator of maximum likelihood type. This approach has several limitations, in particular, it assumes that H < 3/4, moreover, some estimators have too low rate of convergence. Therefore, we propose a new method for simultaneous estimation of all four parameters, which is based on the ergodic theorem. Finally, we compare two approaches by Monte Carlo simulations.