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

Jina Yang, Ky Quan Tran

TL;DR

The paper develops explicit criteria for mean-square exponential stability of neutral stochastic delay differential equations with Markovian switching and arbitrary time-dependent delays, using a contradiction-based comparison approach. It then extends these results to the Euler–Maruyama numerical method, proving stability preservation for sufficiently small steps and the ability to approximate the true exponential decay rate. The main contribution is providing practical, implementable conditions and showing that numerical schemes can faithfully reproduce long-time stability behavior in complex switching-delay environments. Numerical examples illustrate the theoretical findings and the stabilizing effect of switching in NSDDEs.

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.

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

TL;DR

The paper develops explicit criteria for mean-square exponential stability of neutral stochastic delay differential equations with Markovian switching and arbitrary time-dependent delays, using a contradiction-based comparison approach. It then extends these results to the Euler–Maruyama numerical method, proving stability preservation for sufficiently small steps and the ability to approximate the true exponential decay rate. The main contribution is providing practical, implementable conditions and showing that numerical schemes can faithfully reproduce long-time stability behavior in complex switching-delay environments. Numerical examples illustrate the theoretical findings and the stabilizing effect of switching in NSDDEs.

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.

Paper Structure

This paper contains 6 sections, 4 theorems, 98 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation.
  3. Formulation
  4. Stability of the exact solutions
  5. Stability of EM approximations
  6. Concluding remarks

Key Result

Lemma 3.1

Assume that there exist constants $T>0$, $K>0$, and $\lambda>0$ such that Then and for any $(\xi, i_0, t)\in C^{b}_{\mathcal{F}_0}([-r, 0], \mathbb R^d)\times {\cal{M}}\times [0, T).$

Figures (4)

  • Figure 1: Graph of $\{{\mathbb E}|Y^{(1)}_n|^2\}$ for the EM approximation $\{Y^{(1)}_n\}$ of Eq. \ref{['yy.2']}. The expectation is estimated by averaging over 100 simulations.
  • Figure 2: Graph of $\{{\mathbb E}|Y^{(2)}_n|^2\}$ for the EM approximation $\{Y^{(2)}_n\}$ of Eq. \ref{['yy.3']}. The expectation is estimated by averaging over 100 simulations.
  • Figure 3: Graph of $\{{\mathbb E}|Y^{\xi, i_0}_n|^2\}$ for the EM approximation $\{Y^{\xi, i_0}_n\}$ of Eq. \ref{['yy.1']}. The expectation is estimated by averaging over 100 simulations.
  • Figure 4: Graph of $\{ \frac{1}{n\varepsilon}\ln({\mathbb E}|Y^{\xi, i_0}_n|^2)\}$ for the EM approximation $\{Y^{\xi, i_0}_n\}$ of Eq. \ref{['yy.1']}. The expectation is estimated by averaging over 100 simulations.

Theorems & Definitions (12)

  • Definition 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Theorem 4.3
  • proof
  • ...and 2 more