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Approximate Bayesian Computation (ABC) has become one of the major tools oflikelihood-free statistical inference in complex mathematical models.Simultaneously, stochastic differential equations (SDEs) have developed to anestablished tool for modelling time dependent, real world phenomena withunderlying random effects. When applying ABC to stochastic models, two majordifficulties arise. First, the derivation of effective summary statistics andproper distances is particularly challenging, since simulations from thestochastic process under the same parameter configuration result in differenttrajectories. Second, exact simulation schemes to generate trajectories fromthe stochastic model are rarely available, requiring the derivation of suitablenumerical methods for the synthetic data generation. To obtain summaries thatare less sensitive to the intrinsic stochasticity of the model, we propose tobuild up the statistical method (e.g., the choice of the summary statistics) onthe underlying structural properties of the model. Here, we focus on theexistence of an invariant measure and we map the data to their estimatedinvariant density and invariant spectral density. Then, to ensure that thesemodel properties are kept in the synthetic data generation, we adoptmeasure-preserving numerical splitting schemes. The derived property-based andmeasure-preserving ABC method is illustrated on the broad class of partiallyobserved Hamiltonian type SDEs, both with simulated data and with realelectroencephalography (EEG) data. The proposed ingredients can be incorporatedinto any type of ABC algorithm and directly applied to all SDEs that arecharacterised by an invariant distribution and for which a measure-preservingnumerical method can be derived.

Spectral Density-Based and Measure-Preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs