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Weak convergence of the split-step backward Euler method for stochastic delay integro-differential equations

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  • Li, Yan
  • Xu, Qiuhong
  • Cao, Wanrong

Abstract

In this paper, our primary objective is to discuss the weak convergence of the split-step backward Euler (SSBE) method, renowned for its exceptional stability when used to solve a class of stochastic delay integro-differential equations (SDIDEs) characterized by global Lipschitz coefficients. Traditional weak convergence analysis techniques are not directly applicable to SDIDEs due to the absence of a Kolmogorov equation. To bridge this gap, we employ modified equations to establish an equivalence between the SSBE method used for solving the original SDIDEs and the Euler–Maruyama method applied to modified equations. By demonstrating first-order strong convergence between the solutions of SDIDEs and the modified equations, we establish the first-order weak convergence of the SSBE method for SDIDEs. Finally, we present numerical simulations to validate our theoretical findings.

Suggested Citation

  • Li, Yan & Xu, Qiuhong & Cao, Wanrong, 2025. "Weak convergence of the split-step backward Euler method for stochastic delay integro-differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 227(C), pages 226-240.
    • Handle: RePEc:eee:matcom:v:227:y:2025:i:c:p:226-240
    DOI: 10.1016/j.matcom.2024.08.005
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    References listed on IDEAS

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    1. Dineshkumar, C. & Udhayakumar, R. & Vijayakumar, V. & Nisar, Kottakkaran Sooppy & Shukla, Anurag, 2021. "A note on the approximate controllability of Sobolev type fractional stochastic integro-differential delay inclusions with order 1," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 1003-1026.
    2. Hu, Peng & Huang, Chengming, 2018. "Delay dependent stability of stochastic split-step θ methods for stochastic delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 663-674.
    3. Liu, Linna & Mo, Haoyi & Deng, Feiqi, 2019. "Split-step theta method for stochastic delay integro-differential equations with mean square exponential stability," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 320-328.
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