Rigorous convergence bounds for stochastic differential equations with application to uncertainty quantification

• Published in: Physica. D Vol. 481; p. 134742
• Main Authors: Blake, Liam A.A., Maclean, John, Balasuriya, Sanjeeva
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2025
• Subjects: Eddy parameterisation; Gaussian approximations; Multiplicative and time-dependent noise; Stochastic differential equations; Uncertainty quantification; Uncertainty quantification; Multiplicative and time-dependent noise; Stochastic differential equations; Gaussian approximations; Eddy parameterisation
• ISSN: 0167-2789
• DOI: 10.1016/j.physd.2025.134742

Prediction via 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 consider a general class of stochastic differential equations and provide rigorous convergence bounds to an analytically solvable approximation. We provide the explicit convergence rate for all moments of a fully non-autonomous model with both multiplicative noise and uncertain initial conditions. Our second main contribution is to extend stochastic sensitivity, a recently introduced uncertainty quantification tool, to arbitrary dimensions and provide a new calculation method that empowers rapid computation. We demonstrate the power and adaptability of our contributions on a diverse set of numerical examples in 1-, 2-, 3-, and 4-dimensions, including providing stochastic sensitivity calculations for an idealised eddy parameterisation of the Gulf Stream.