The convergence of stochastic differential equations to their linearisation in small noise limits

TL;DR

This paper derives explicit error bounds for approximating stochastic differential equations with their linearizations in small noise limits, accounting for complex factors like non-autonomous coefficients and multiplicative noise, validated through numerical case studies.

Contribution

It extends small-noise analysis to more general SDEs, providing sharp bounds and linking stochastic sensitivity to linearization error in high dimensions.

Findings

Error bounds accurately predict moments' scaling with noise and initial uncertainty.

Bound is sharp and validated through diverse numerical experiments.

Extension of stochastic sensitivity to arbitrary dimensions.

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.