Strong convergence and stability of the split-step theta method for highly nonlinear neutral stochastic delay integro differential equation

Abstract

This paper deals with the asymptotic boundedness, stability and strong convergence of the split-step theta method for highly nonlinear neutral stochastic delay integro differential equation. Under the generalized coercive condition, we prove that the split-step theta approximation strongly converges to the exact solution of the underlying equation, for θ∈[1/2,1]. Besides, we also prove that when θ∈(1/2,1], the method can preserve the asymptotic boundedness and exponential stability of the original equation without any restrictions on the step-size. Moreover, we also show that for sufficiently small step-size, the asymptotic boundary and decay rate can be reproduced. Finally, the numerical experiments demonstrate the correctness of our theoretical results.