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

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.