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Pathwise Convergent Approximation for the Fractional SDEs

Author

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  • Kęstutis Kubilius

    (Faculty of Mathematics and Informatics, Vilnius University, Akademijos g. 4, LT-08412 Vilnius, Lithuania
    These authors contributed equally to this work.)

  • Aidas Medžiūnas

    (Faculty of Mathematics and Informatics, Vilnius University, Akademijos g. 4, LT-08412 Vilnius, Lithuania
    These authors contributed equally to this work.)

Abstract

Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional stochastic differential equations (SDEs) driven by stochastic process with Hölder continuous paths of order 1 / 2 < γ < 1 . Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate h 2 γ , where h is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field.

Suggested Citation

  • Kęstutis Kubilius & Aidas Medžiūnas, 2022. "Pathwise Convergent Approximation for the Fractional SDEs," Mathematics, MDPI, vol. 10(4), pages 1-16, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:4:p:669-:d:754460
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    References listed on IDEAS

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    2. Neuenkirch, Andreas, 2008. "Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2294-2333, December.
    3. Mario Abundo & Enrica Pirozzi, 2019. "On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes," Mathematics, MDPI, vol. 7(10), pages 1-12, October.
    4. 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.
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