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Exponential stability in mean square of theta approximations for neutral stochastic delay differential equations with Poisson jumps

Author

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  • Pham H. A. Ngoc
  • Bich T. N. Le
  • Ky Q. Tran

Abstract

This paper investigates the exponential stability in mean square of theta numerical solutions for neutral stochastic delay differential equations (NSDDEs) with Poisson jumps and time-dependent delays. Analysing the stability of such equations poses significant challenges due to the combined effects of the neutral term, Poisson jumps, and time-dependent delays. To address a gap in the existing literature, we propose novel criteria for the exponential stability of both exact and numerical solutions derived from the stochastic linear theta method and the split-step theta method. Unlike previous criteria, our approach does not require the differentiability of the time-dependent delay function, allowing us to analyse a wider class of NSDDEs. Furthermore, we demonstrate that for sufficiently small step sizes, the theta approximations can arbitrarily accurately replicate the mean-square exponential decay rate of the exact solutions. We provide two examples to illustrate the effectiveness of our criteria.

Suggested Citation

  • Pham H. A. Ngoc & Bich T. N. Le & Ky Q. Tran, 2025. "Exponential stability in mean square of theta approximations for neutral stochastic delay differential equations with Poisson jumps," International Journal of Systems Science, Taylor & Francis Journals, vol. 56(16), pages 4207-4224, December.
    • Handle: RePEc:taf:tsysxx:v:56:y:2025:i:16:p:4207-4224
    DOI: 10.1080/00207721.2025.2486152
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