{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-convergence-of-stochastic-differential","title":"The convergence of stochastic differential equations to their linearisation in small noise limits","arxiv_id":"2309.16334","date":"2023-09-28","proceeding":null,"authors":["Liam Blake","John Maclean","Sanjeeva Balasuriya"],"abstract":"Prediction via deterministic continuous-time models will always be subject to model error, for example due to unexplainable phenomena, uncertainties in any data driving the model, or discretisation/resolution issues. In this paper, we build upon previous small-noise studies to provide an explicit bound for the error between a general class of stochastic differential equations and corresponding computable linearisations written in terms of a deterministic system. Our framework accounts for non-autonomous coefficients, multiplicative noise, and uncertain initial conditions. We demonstrate the predictive power of our bound on diverse numerical case studies. We confirm that our bound is sharp, in that it accurately predicts the error scaling in the moments of the linearised approximation as both the uncertainty in the initial condition and the magnitude of the noise in the differential equation are altered. This paper also provides an extension of stochastic sensitivity, a recently introduced tool for quantifying uncertainty in dynamical systems, to arbitrary dimensions and establishes the link to our characterisation of stochastic differential equation linearisations.","url_abs":"https://arxiv.org/abs/2309.16334v2","url_pdf":"https://arxiv.org/pdf/2309.16334v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-convergence-of-stochastic-differential","repo_url":"https://github.com/liamblake/explicit-characterisation-sde-linearisation","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}