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• Streaming data are characterised as evolution of Stochastic Differential Equations. • Gaussian process regression used for nonparametric estimation of SDE coefficients. • Stochastic process theory applied to automatically compare SDE models. • Bootstrapping methods allow for control of the false alarm rate of detector. • Evaluation on datasets demonstrates detector's high discriminative power. This paper characterises streaming data as the evolution of a stochastic differential equation, with the aim of extracting information that can be used to detect anomalies in the stream. Gaussian process regression provides a flexible approach to approximating components of the stochastic differential equation, allowing for complex modelling of underlying data generation dynamics. The proposed algorithm displays superior discriminative power over different time-series anomaly detection methods for both synthetic and NYC taxi datasets, whilst the introduced bootstrapping method for setting the detection threshold provides control over the false alarm rate of the anomaly detector. [ABSTRACT FROM AUTHOR]