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

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 \(θ ∈ [0, 1/2)\) , exponential mean-square stability of the numerical solution is guaranteed provided the stepsize is sufficiently small; for \(θ ∈ (1/2, 1]\) , 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.