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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Kęstutis Kubilius & Aidas Medžiūnas, 2020. "Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient," Mathematics, MDPI, vol. 9(1), pages 1-14, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:18-:d:467115
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    References listed on IDEAS

    as
    1. Hu, Yaozhong & Nualart, David & Song, Xiaoming, 2008. "A singular stochastic differential equation driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2075-2085, October.
    2. Alfonsi, Aurélien, 2013. "Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 602-607.
    3. Kubilius, K. & Skorniakov, V., 2016. "On some estimators of the Hurst index of the solution of SDE driven by a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 159-167.
    4. Kubilius, K., 2020. "CLT for quadratic variation of Gaussian processes and its application to the estimation of the Orey index," Statistics & Probability Letters, Elsevier, vol. 165(C).
    5. Dung, Nguyen Tien, 2016. "Tail probabilities of solutions to a generalized Ait-Sahalia interest rate model," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 98-104.
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