The balanced split step theta approximations of stochastic neutral Hopfield neural networks with time delay and Poisson jumps

Abstract

The balanced numerical approximations of the stochastic neutral Hopfield neural networks (SNHNN) with time delay and Poisson jumps are examined to ascertain the nature of the mathematical model. The numerical approximations of the balanced split-step theta methods for the SNHNN with time delay and Poisson jumps are taken into consideration primarily because they maintain almost surely (a.s.) exponential stability property of numerical methods and produce negligible mean square error when compared to other approaches. Furthermore, in the recent development of numerical approximations for SNHNN with time delay, we note that balanced split-step theta-approximations are a more stable scheme. We showed that the numerical approximations of balanced split-step theta methods of SNHNN with time delay and Poisson jumps have strong convergence order 1/2 and are numerically almost exponentially stable by applying some theoretical significance criteria.Moreover, our main research tools are Lipschitz conditions, linear growth conditions, and the discrete semi martingale convergence theorem. Through numerical experiments, we try to demonstrate the theoretical results obtained in this paper. Finally, we got the confirmation about the theoretical results of the split-step theta-approximations of SNHNN with time delay and Poisson jumps via particular numerical example.