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

• Published in: Numerical algorithms Vol. 98; no. 4; pp. 2005 - 2035
• Main Authors: Cai, Jingjing, Chen, Ziheng, Niu, Yuanling
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2025; Springer Nature B.V
• Subjects: Algebra; Algorithms; Approximation; Computer Science; Convergence; Delay; Differential equations; Diffusion coefficient; Mathematical analysis; Mean square errors; Methods; Neural networks; Numeric Computing; Numerical Analysis; Original Paper; Random variables; Stability; Theory of Computation; 65C30; Generalized monotonicity condition; Mean square convergence rate; Backward Euler method; Neutral stochastic delay differential equations; Mean square exponential stability; AMS subject classification: 60H10; 60H35
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-024-01862-4

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 under generalized monotonicity conditions. These theoretical results are finally supported by several numerical experiments.