Estimation of the Hurst index of the solutions of fractional SDE with locally Lipschitz drift

Strongly consistent and asymptotically normal estimates of the Hurst index H are obtained for stochastic differential equations (SDEs) that have a unique positive solution. A strongly convergent approximation of the considered SDE solution is constructed using the backward Euler scheme. Moreover, it is proved that the Hurst estimator preserves its properties, if we replace the solution with its approximation.