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\begin{document}

\title{Exponential stability of implicit numerical solution for nonlinear
neutral stochastic differential equations with time-varying delay and
Poisson jumps}



\author[1]{Haoyi Mo}%
\author[2]{Linna Liu}%
\author[1]{Mali Xing}%
\author[3]{feiqi Deng}%
\affil[1]{Guangdong University of Technology}%
\affil[2]{Zhongyuan University of Technology}%
\affil[3]{South China University of Technology}%


\vspace{-1em}



  
  \date{June 17, 2024}


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\begin{abstract}
The aim of this work is to investigate the exponential mean-square
stability for neutral stochastic differential equations with
time-varying delay and Poisson jumps. We give some conditions that all
the drift, diffusion and jumps coefficients can be nonlinear, to obtain
the stability of the analytic solution. It is revealed that the implicit
backward Euler-Maruyama numerical solution can reproduce the
corresponding stability of the analytic solution under these nonlinear
conditions. This is different from the explicit Euler-Maruyama numerical
solution whose stability depends on the linear growth condition. With
some requirements related to the delay function and the property of
compensated Poisson process, we deal with time-varying delay and Poisson
jumps. One highly nonlinear example is provided to confirm the
effectiveness of our theory.%
\end{abstract}%



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