Mean-square exponential stability of exact and numerical solutions for neutral stochastic delay differential equations with Markovian switching

TL;DR

This paper establishes new criteria for the mean-square exponential stability of neutral stochastic delay differential equations with Markovian switching and shows that numerical solutions can accurately replicate the decay rate of true solutions.

Contribution

It provides novel stability criteria for NSDDEs with Markovian switching and demonstrates the accuracy of Euler-Maruyama numerical solutions in capturing exponential decay rates.

Findings

Derived practical stability criteria for NSDDEs with Markovian switching.

Proved numerical solutions can match true solution decay rates with small step sizes.

Validated results with a numerical example.

Abstract

This paper investigates the mean-square exponential stability of neutral stochastic differential delay equations (NSDDEs) with Markovian switching. The analysis addresses the complexities arising from the interaction between the neutral term, time-varying delays, and structural changes governed by a continuous-time Markov chain. We establish novel and practical criteria for the mean-square exponential stability of both the underlying system and its numerical approximations via the Euler-Maruyama method. Furthermore, we prove that the numerical scheme can reproduce the exponential decay rate of the true solution with arbitrary accuracy, provided the step size is sufficiently small. The theoretical results are supported by a numerical example that illustrates their effectiveness.