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 $$X_t = \theta t + \sigma W_t + \kappa B^H_t$$ , 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 $$H<\frac{3}{4}$$ , 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.