Stochastic processes

Stochastic processes have experienced tremendous progress in the last decades. Due to their versatile mathematical aspects, they have become an established and powerful tool for modelling various time-dependent real-world phenomena with underlying random effects. Stochastic processes have applications in most scientific areas, including physics, chemistry, medicine, biology, ecology, finance and many more. Among other, this research project focuses on the class of stochastic processes arising as solutions to stochastic differential equations (SDEs) and their application in the field of neuroscience. We are pursuing two main objectives. First, due to the complexity of the SDEs required to adequately describe the modelled real-world phenomenon of interest, they are typically not solvable exactly, and thus numerical methods are required to approximate and simulate them. Here, we construct reliable and efficient numerical methods for SDEs, which capture their qualitative behaviour and structural properties. Second, based on these structure-preserving numerical schemes, we develop statistical methods to infer different parameters (intrinsic model parameters as well as extrinsic network parameters) of SDEs from observed data, focusing on simulation-based approaches such as approximate Bayesian computation. The developed methods are applied on real datasets, including neural action potential recordings or electroencephalogram (EEG) rhythms. As both objectives aim at developing new algorithms, this research project contributes to the JKU research priority Digital Transformation.