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Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay

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  • Gao, Jianfang
  • Liang, Hui
  • Ma, Shufang

Abstract

This paper is mainly concerned with the strong convergence analysis of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations (SVIEs) with constant delay. The solvability and the boundedness of the numerical solution are established. It is proved that the strong convergence order of the semi-implicit Euler method is 0.5 under Lipschitz conditions. Moreover, the strong superconvergence order is 1.0 if further, the kernel σ of the stochastic term satisfies σ(0)=σ(τ)=0. The theoretical results are illustrated by extensive numerical examples.

Suggested Citation

  • Gao, Jianfang & Liang, Hui & Ma, Shufang, 2019. "Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 385-398.
    • Handle: RePEc:eee:apmaco:v:348:y:2019:i:c:p:385-398
    DOI: 10.1016/j.amc.2018.10.025
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    References listed on IDEAS

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    1. Eckhard Platen, 1999. "An Introduction to Numerical Methods for Stochastic Differential Equations," Research Paper Series 6, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. K. Maleknejad & M. Khodabin & F. Hosseini Shekarabi, 2014. "Modified Block Pulse Functions for Numerical Solution of Stochastic Volterra Integral Equations," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-10, March.
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    Cited by:

    1. Li, Yuping & Yang, Zhanwen & Liang, Hui, 2022. "Analysis of collocation methods for a class of third-kind auto-convolution Volterra integral equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 341-358.
    2. Ahmadinia, M. & Afshariarjmand, H. & Salehi, M., 2023. "Numerical solution of multi-dimensional Itô Volterra integral equations by the second kind Chebyshev wavelets and parallel computing process," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    3. Young Hee Geum & Young Ik Kim, 2020. "Computational Bifurcations Occurring on Red Fixed Components in the λ -Parameter Plane for a Family of Optimal Fourth-Order Multiple-Root Finders under the Möbius Conjugacy Map," Mathematics, MDPI, vol. 8(5), pages 1-17, May.

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