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Numerical Scheme for Stochastic Differential Equations Driven by Fractional Brownian Motion with $$ 1/4

Author

Listed:
  • Héctor Araya

    (Universidad de Valparaíso
    Universidad de Valparaíso)

  • Jorge A. León

    (Cinvestav-IPN)

  • Soledad Torres

    (Universidad de Valparaíso)

Abstract

In this article, we study a numerical scheme for stochastic differential equations driven by fractional Brownian motion with Hurst parameter $$ H \in \left( 1/4, 1/2 \right) $$ H ∈ 1 / 4 , 1 / 2 . Toward this end, we apply Doss–Sussmann representation of the solution and an approximation of this representation using a first-order Taylor expansion. The obtained rate of convergence is $$n^{-2H +\rho }$$ n - 2 H + ρ , for $$\rho $$ ρ small enough.

Suggested Citation

  • Héctor Araya & Jorge A. León & Soledad Torres, 2020. "Numerical Scheme for Stochastic Differential Equations Driven by Fractional Brownian Motion with $$ 1/4," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1211-1237, September.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:3:d:10.1007_s10959-019-00902-3
    DOI: 10.1007/s10959-019-00902-3
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    References listed on IDEAS

    as
    1. Andreas Neuenkirch & Ivan Nourdin, 2007. "Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 20(4), pages 871-899, December.
    2. Elisa Alòs & Jorge León & Josep Vives, 2007. "On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility," Finance and Stochastics, Springer, vol. 11(4), pages 571-589, October.
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