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<p>This paper focuses on the weak convergence of the split-step backward Euler (SSBE) method for stochastic Volterra integral equations (SVIEs). Since SVIEs constitute a class of non-Markovian stochastic differential equations with memory kernels, conventional weak convergence analysis techniques are not directly applicable. To address this, we construct an associated auxiliary equation and demonstrate that applying the Euler-Maruyama (EM) method to this auxiliary equation is equivalent to employing the SSBE method on the original SVIEs. Then, our analysis is based on two existing results: (i) the first-order weak error bound of the EM method for the auxiliary equation is a known result; (ii) there exists a strong convergence relationship of order one between the solution of the auxiliary equation and that of the original equation, as proved in this paper. Combining this equivalence with the above two points, it can be directly concluded that the SSBE method achieves weak convergence of rate 1.</p>

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Weak convergence of the split-step backward Euler method for stochastic Volterra integral equations

  • 期刊论文

  • 发表日期：

    2026年6月20日

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  • Huiting Zhang,
  • Wenqiang Wang

摘要

This paper focuses on the weak convergence of the split-step backward Euler (SSBE) method for stochastic Volterra integral equations (SVIEs). Since SVIEs constitute a class of non-Markovian stochastic differential equations with memory kernels, conventional weak convergence analysis techniques are not directly applicable. To address this, we construct an associated auxiliary equation and demonstrate that applying the Euler-Maruyama (EM) method to this auxiliary equation is equivalent to employing the SSBE method on the original SVIEs. Then, our analysis is based on two existing results: (i) the first-order weak error bound of the EM method for the auxiliary equation is a known result; (ii) there exists a strong convergence relationship of order one between the solution of the auxiliary equation and that of the original equation, as proved in this paper. Combining this equivalence with the above two points, it can be directly concluded that the SSBE method achieves weak convergence of rate 1.