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Almost Sure Exponential Stability of Numerical Solutions for Stochastic Pantograph Differential Equations with Poisson Jumps

Author

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  • Amr Abou-Senna

    (Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
    Department of Mathematics, Faculty of Engineering at Shoubra, Benha University, Banha 13511, Egypt
    These authors contributed equally to this work.)

  • Boping Tian

    (Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
    These authors contributed equally to this work.)

Abstract

The stability analysis of the numerical solutions of stochastic models has gained great interest, but there is not much research about the stability of stochastic pantograph differential equations. This paper deals with the almost sure exponential stability of numerical solutions for stochastic pantograph differential equations interspersed with the Poisson jumps by using the discrete semimartingale convergence theorem. It is shown that the Euler–Maruyama method can reproduce the almost sure exponential stability under the linear growth condition. It is also shown that the backward Euler method can reproduce the almost sure exponential stability of the exact solution under the polynomial growth condition and the one-sided Lipschitz condition. Additionally, numerical examples are performed to validate our theoretical result.

Suggested Citation

  • Amr Abou-Senna & Boping Tian, 2022. "Almost Sure Exponential Stability of Numerical Solutions for Stochastic Pantograph Differential Equations with Poisson Jumps," Mathematics, MDPI, vol. 10(17), pages 1-18, September.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:17:p:3137-:d:903890
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    References listed on IDEAS

    as
    1. Shaobo Zhou, 2014. "Almost Surely Exponential Stability of Numerical Solutions for Stochastic Pantograph Equations," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-9, June.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. NicolaBruti-Liberati & Eckhard Platen, 2007. "Strong approximations of stochastic differential equations with jumps," Published Paper Series 2007-7, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    4. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
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