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Numerical Approximation for a Stochastic Fractional Differential Equation Driven by Integrated Multiplicative Noise

Author

Listed:
  • James Hoult

    (Department of Mathematical and Physical Sciences, University of Chester, Chester CH1 4BJ, UK
    These authors contributed equally to this work.)

  • Yubin Yan

    (Department of Mathematical and Physical Sciences, University of Chester, Chester CH1 4BJ, UK
    These authors contributed equally to this work.)

Abstract

We consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order α ∈ ( 0 , 1 ) , and the non-linear terms satisfy the global Lipschitz conditions. We first approximate the noise with the piecewise constant function to obtain the regularized stochastic fractional differential equation. By applying Minkowski’s inequality for double integrals, we establish that the error between the exact solution and the solution of the regularized problem has an order of O ( Δ t α ) in the mean square norm, where Δ t denotes the step size. To validate our theoretical conclusions, numerical examples are presented, demonstrating the consistency of the numerical results with the established theory.

Suggested Citation

  • James Hoult & Yubin Yan, 2024. "Numerical Approximation for a Stochastic Fractional Differential Equation Driven by Integrated Multiplicative Noise," Mathematics, MDPI, vol. 12(3), pages 1-18, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:3:p:365-:d:1324894
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    References listed on IDEAS

    as
    1. Yang, Huizi & Yang, Zhanwen & Ma, Shufang, 2019. "Theoretical and numerical analysis for Volterra integro-differential equations with Itô integral under polynomially growth conditions," Applied Mathematics and Computation, Elsevier, vol. 360(C), pages 70-82.
    2. Anh, P.T. & Doan, T.S. & Huong, P.T., 2019. "A variation of constant formula for Caputo fractional stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 351-358.
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