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<p>This paper investigates the exponential mean-square stability of both the continuous solution and the split-step <i>θ</i>-Milstein (SSTM) numerical scheme for neutral stochastic delay integro-differential equations (NSDIDEs) with Poisson jumps. Under a set of reasonable conditions, the trivial solution of the NSDIDE is shown to be exponentially mean-square stable. Subsequently, it is demonstrated that this stability property is preserved by the SSTM method under explicit stepsize constraints. Specifically, for <InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><mi>θ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo></math></EquationSource><EquationSource Format="TEX">$\theta \in [0, \tfrac{1}{2})$</EquationSource></InlineEquation>, exponential mean-square stability of the numerical solution is guaranteed provided the stepsize is sufficiently small; for <InlineEquation ID="IEq2"><EquationSource Format="MATHML"><math><mi>θ</mi><mo>∈</mo><mo stretchy="false">(</mo><mfrac><mrow><mn>1</mn></mrow><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></EquationSource><EquationSource Format="TEX">$\theta \in (\tfrac{1}{2}, 1]$</EquationSource></InlineEquation>, the method is stable under a less restrictive stepsize condition, thereby enhancing computational efficiency while maintaining stability. The theoretical findings are further validated through two numerical examples, which confirm the validity of the derived stability conditions.</p>

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A split-step θ-Milstein scheme for NSDIDEs with jumps: exponential stability analysis

  • 期刊论文

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  • 发表日期：

    2026年8月5日

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  • Jianyin Fang,
  • Linna Liu,
  • Shuaibo Zhou

摘要

This paper investigates the exponential mean-square stability of both the continuous solution and the split-step θ-Milstein (SSTM) numerical scheme for neutral stochastic delay integro-differential equations (NSDIDEs) with Poisson jumps. Under a set of reasonable conditions, the trivial solution of the NSDIDE is shown to be exponentially mean-square stable. Subsequently, it is demonstrated that this stability property is preserved by the SSTM method under explicit stepsize constraints. Specifically, for θ∈[0,12)$\theta \in [0, \tfrac{1}{2})$, exponential mean-square stability of the numerical solution is guaranteed provided the stepsize is sufficiently small; for θ∈(12,1]$\theta \in (\tfrac{1}{2}, 1]$, the method is stable under a less restrictive stepsize condition, thereby enhancing computational efficiency while maintaining stability. The theoretical findings are further validated through two numerical examples, which confirm the validity of the derived stability conditions.