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Stochastic differential equations (SDEs) are popular tools to analyse timeseries data in many areas, such as mathematical finance, physics, and biology.They provide a mechanistic description of the phenomeon of interest, and theirparameters often have a clear interpretation. These advantages come at the costof requiring a relatively simple model specification. We propose a flexiblemodel for SDEs with time-varying dynamics where the parameters of the processare non-parametric functions of covariates, similar to generalized additivemodels. Combining the SDEs and non-parametric approaches allows for the SDE tocapture more detailed, non-stationary, features of the data-generating process.We present a computationally efficient method of approximate inference, wherethe SDE parameters can vary according to fixed covariate effects, randomeffects, or basis-penalty smoothing splines. We demonstrate the versatility andutility of this approach with three applications in ecology, where there isoften a modelling trade-off between interpretability and flexibility.

Varying-coefficient stochastic differential equations with applications in ecology