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

Abstract

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.

Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient

“…We repeat the outlines of the proof of Theorem 3 in [8]. Please note that under the conditions of the theorem, conditions (C 1 )-(C 3 ) are satisfied.…”

“…Thus, we can only state that Equation (10) has a solution X t = Y −1/(β−1) t until the moment at which Y becomes zero. On the other hand, we do know that the CKLS model driven by a standard Brownian motion (see [6]) with β > 1 and the fractional CKLS model with 1/2 β < 1 (see [8]) have positive solutions.…”

“…For fractional CIR, CKLS, and AS models, the existence of a unique positive solution of Equation (2) was obtained in [7][8][9][10][11][12][13][14]. The proof can be provided in several ways.…”

“…admits a unique positive solution, where h(t, x) is a locally Lipschitz function with respect to the space variable x ∈ (0, ∞). This approach was used in [7][8][9]11,14], where the inverse Lamperti transform was used to obtain conditions under which Equation (1) admits a unique positive solution for fractional CIR, CKLS, and AS models. Unfortunately, we cannot apply the proof of positivity of the solution of (3) given in [14] (e.g., it is not applicable to the AS model).…”

“…A simulation of the fractional CIR process was given in [11] by using the Euler approximation. In [8,14] an almost sure strongly convergent approximation of the considered SDE solution is constructed using the backward Euler scheme, which is positivity preserving.…”