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Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

Jingjing Cai, Ziheng Chen, Yuanling Niu

TL;DR

This paper addresses NSDDEs with neutral terms and super-linear drift and diffusion under generalized monotonicity. It analyzes an implicit backward Euler discretization, proving a mean-square convergence rate of $1/2$ and that the numerical scheme inherits the original system's mean-square exponential stability. The authors extend the results to SDDEs, derive an upper bound involving local truncation errors, and corroborate the theory with comprehensive numerical experiments, including stiff and non-globally-monotone cases. These findings support stable, efficient simulation of complex NSDDEs and provide a foundation for developing higher-order methods.

Abstract

This work focuses on the numerical approximations of neutral stochastic delay differential equations with their drift and diffusion coefficients growing super-linearly with respect to both delay variables and state variables. Under generalized monotonicity conditions, we prove that the backward Euler method not only converges strongly in the mean square sense with order $1/2$, but also inherit the mean square exponential stability of the original equations. As a byproduct, we obtain the same results on convergence rate and exponential stability of the backward Euler method for stochastic delay differential equations with generalized monotonicity conditions. These theoretical results are finally supported by several numerical experiments.

Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

TL;DR

This paper addresses NSDDEs with neutral terms and super-linear drift and diffusion under generalized monotonicity. It analyzes an implicit backward Euler discretization, proving a mean-square convergence rate of and that the numerical scheme inherits the original system's mean-square exponential stability. The authors extend the results to SDDEs, derive an upper bound involving local truncation errors, and corroborate the theory with comprehensive numerical experiments, including stiff and non-globally-monotone cases. These findings support stable, efficient simulation of complex NSDDEs and provide a foundation for developing higher-order methods.

Abstract

This work focuses on the numerical approximations of neutral stochastic delay differential equations with their drift and diffusion coefficients growing super-linearly with respect to both delay variables and state variables. Under generalized monotonicity conditions, we prove that the backward Euler method not only converges strongly in the mean square sense with order , but also inherit the mean square exponential stability of the original equations. As a byproduct, we obtain the same results on convergence rate and exponential stability of the backward Euler method for stochastic delay differential equations with generalized monotonicity conditions. These theoretical results are finally supported by several numerical experiments.
Paper Structure (6 sections, 6 theorems, 114 equations, 3 figures)

This paper contains 6 sections, 6 theorems, 114 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and upper mean square error bound
  3. Mean square convergence rate
  4. Mean square exponential stability
  5. Numerical experiments
  6. Conclusion

Key Result

Lemma 2.3

Suppose that Assumption ass:mainasstwo holds with $L\Delta <2$. Then eq:BEmethodintro admits a unique solution $\{X_{n}\}_{n \geq -M}$ with probability one.

Figures (3)

  • Figure 1: Mean square convergence rates of backward Euler method
  • Figure 2: Mean square exponential stability of backward Euler method for \ref{['eq:examplestability']}
  • Figure 3: Convergence order simulations for \ref{['eq:examplestiff']}

Theorems & Definitions (14)

  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Definition 4.1
  • Theorem 4.3
  • ...and 4 more