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

Kęstutis Kubilius and Aidas Medžiūnas
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Kęstutis Kubilius: Faculty of Mathematics and Informatics, Vilnius University, Akademijos g. 4, 08412 Vilnius, Lithuania
Aidas Medžiūnas: Faculty of Mathematics and Informatics, Vilnius University, Akademijos g. 4, 08412 Vilnius, Lithuania

Mathematics, 2020, vol. 9, issue 1, 1-14

Abstract: We study a class of fractional stochastic differential equations (FSDEs) with coefficients that may not satisfy the linear growth condition and non-Lipschitz diffusion coefficient. Using the Lamperti transform, we obtain conditions for positivity of solutions of such equations. We show that the trajectories of the fractional CKLS model with β > 1 are not necessarily positive. We obtain the almost sure convergence rate of the backward Euler approximation scheme for solutions of the considered SDEs. We also obtain a strongly consistent and asymptotically normal estimator of the Hurst index H > 1 / 2 for positive solutions of FSDEs.

Keywords: fractional Brownian motion; backward Euler approximation; fractional Ait–Sahalia model; fractional CKLS model; Hurst index (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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