Contractivity of stochastic θ-methods under non-global Lipschitz conditions

• Published in: Applied mathematics and computation Vol. 505; p. 129527
• Main Authors: Biščević, Helena, D'Ambrosio, Raffaele, Di Giovacchino, Stefano
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.11.2025
• Subjects: Exponential mean-square contractivity; Exponential mean-square stability; One-sided Lipschitz continuity; Stochastic differential equations; Exponential mean-square stability; 60H10; Stochastic differential equations; 65C30; 65L07; One-sided Lipschitz continuity; Exponential mean-square contractivity
• ISSN: 0096-3003
• DOI: 10.1016/j.amc.2025.129527

The paper is devoted to address the numerical preservation of the exponential mean-square contractive character of the dynamics of stochastic differential equations (SDEs), whose drift and diffusion coefficients are subject to non-global Lipschitz assumptions. The conservative attitude of stochastic θ-methods is analyzed both for Itô and Stratonovich SDEs. The case of systems with linear drift is also analyzed in terms of spectral properties of the coefficient matrix of the drift. Numerical evidence on selected test problems confirms the effectiveness of the approach. •The paper analyzes numerical preservation of the dissipative character of nonlinear SDEs, under non-global Lipschitz continuity of the coefficients.•The conservative character of theta-Maruyama methods has been analyzed and tested, assessing the robustness of this class of numerical methods.•The preservation of the exponential mean-square character along the discretized dynamics translates into stepsize restrictions.•The research falls in the large scenario of stochastic geometric numerical integration.