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.
Highlights
The models defined by stochastic differential equations (SDEs) t t Xt = x0 + g(Xs) ds + σ Xsβ dBs,1 2 β < 1, where B is standard Brownian motion, g is continuous function on (0, ∞), x0 > 0 is nonrandom initial value, σ > 0 is a constant, include several well-known models such as Chan–Karolyi–Longstaff–Sanders (CKLS), Cox–Ingersoll–Ross (CIR), Ait-Sahalia and others, which are widely used in many financial applications
Xsβ dBs, 1 2 β < 1, where B is standard Brownian motion, g is continuous function on (0, ∞), x0 > 0 is nonrandom initial value, σ > 0 is a constant, include several well-known models such as Chan–Karolyi–Longstaff–Sanders (CKLS), Cox–Ingersoll–Ross (CIR), Ait-Sahalia and others, which are widely used in many financial applications
Some authors convert the original SDE using the Lamperti transform to SDE with a constant diffusion coefficient. This approach is convenient for considering fractional analogues of the CKLS, CIR, Ait-Sahalia models
Summary
1 2 β < 1, where B is standard Brownian motion, g is continuous function on (0, ∞), x0 > 0 is nonrandom initial value, σ > 0 is a constant, include several well-known models such as Chan–Karolyi–Longstaff–Sanders (CKLS), Cox–Ingersoll–Ross (CIR), Ait-Sahalia and others, which are widely used in many financial applications. Some authors convert the original SDE using the Lamperti transform to SDE with a constant diffusion coefficient This approach is convenient for considering fractional analogues of the CKLS, CIR, Ait-Sahalia models. 1 ,1 , admits a unique positive solution, where f (t, x) is a locally Lipschitz function with respect to the space variable x on x ∈ (0, ∞) This approach was used in [3, 8, 16], where the inverse Lamperti transform was used to obtain conditions under which equation (1) admits a unique positive solution for fractional CIR and CKLS models. Our goal is to construct strongly consistent and asymptotically normal estimator of the Hurst index H for SDE (1), which has a unique positive solution For such processes, we can do this in the same way as done for the diffusion coefficient satisfying the usual Lipschitz conditions (see [6, 7]). In Appendix, we recall same results for fBm and the Love–Young inequality
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