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The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps

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  • Mao, Wei
  • Hu, Liangjian
  • Mao, Xuerong

Abstract

In this paper, we consider a class of stochastic pantograph differential equations with Lévy jumps (SPDEwLJs). By using the Burkholder–Davis–Gundy inequality and the Kunita’s inequality, we prove the existence and uniqueness of solutions to SPDEwLJs whose coefficients satisfying the Lipschitz conditions and the local Lipschitz conditions. Meantime, we establish the p-th exponential estimations and almost surely asymptotic estimations of solutions to SPDEwLJs.

Suggested Citation

  • Mao, Wei & Hu, Liangjian & Mao, Xuerong, 2015. "The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 883-896.
    • Handle: RePEc:eee:apmaco:v:268:y:2015:i:c:p:883-896
    DOI: 10.1016/j.amc.2015.06.109
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    References listed on IDEAS

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    1. Nicola Bruti-Liberati & Eckhard Platen, 2007. "Approximation of jump diffusions in finance and economics," Computational Economics, Springer;Society for Computational Economics, vol. 29(3), pages 283-312, May.
    2. Mao, Xuerong, 1996. "Razumikhin-type theorems on exponential stability of stochastic functional differential equations," Stochastic Processes and their Applications, Elsevier, vol. 65(2), pages 233-250, December.
    3. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    4. Küchler, Uwe & Platen, Eckhard, 2000. "Strong discrete time approximation of stochastic differential equations with time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 54(1), pages 189-205.
    5. Philippe Jorion, 1988. "On Jump Processes in the Foreign Exchange and Stock Markets," The Review of Financial Studies, Society for Financial Studies, vol. 1(4), pages 427-445.
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    Cited by:

    1. Dongdong Gao & Jianli Li, 2024. "Existence and Hyers–Ulam Stability for Random Impulsive Stochastic Pantograph Equations with the Caputo Fractional Derivative," Mathematics, MDPI, vol. 12(8), pages 1-16, April.
    2. Wan, Fangzhe & Hu, Po & Chen, Huabin, 2020. "Stability analysis of neutral stochastic differential delay equations driven by Lévy noises," Applied Mathematics and Computation, Elsevier, vol. 375(C).
    3. Amr Abosenna & Ghada AlNemer & Boping Tian, 2024. "Convergence and Almost Sure Polynomial Stability of Partially Truncated Split-Step Theta Method for Stochastic Pantograph Models with Lévy Jumps," Mathematics, MDPI, vol. 12(13), pages 1-16, June.

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