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Stochastic dynamical systems arise in many scientific fields, such as asset prices in financial markets, neural activity in the brain, or the spread of infectious diseases. Petar Jovanovski's Ph.D. thesis focuses on these systems, which evolve in ways that are partly random. Jovanovski will defend his PhD "Simulation-based parameter inference methods based on data-conditional simulation of stochastic dynamical systems" on September 19.

These processes are described using differential equations. A classic example of randomness in dynamics is Brownian motion: the irregular movement of pollen particles suspended in water, driven by collisions with surrounding molecules. Unlike purely deterministic systems, where the trajectory is smooth and predictable, stochastic processes combine random fluctuations with an underlying trend. To study such phenomena, researchers use stochastic differential equations (SDEs).

"Most of the thesis focuses on continuous-time models, though one paper also considers discrete-time systems. Even though the behavior is continuous in time, we observe it only at discrete points, often corrupted by measurement noise.

"The central aim is parameter inference: estimating the parameters of these models in a Bayesian framework by computing their posterior distribution given the data. However, because the likelihood function is analytically intractable, standard statistical tools cannot be applied directly," says Jovanovski.

Methods that use the observed data to guide the simulations

Approximate Bayesian computation (ABC) is a branch of Bayesian statistics that relies on generating simulated data and then assessing how close the simulations are to the observed data. In the SDE context, the idea is to simulate parameters, generate trajectories under those parameters, and retain the simulations that are sufficiently close to the observed data.

Some trajectories may be rejected simply due to randomness rather than because the parameters are poor. To address this, a data-conditional framework has been developed: instead of simulating trajectories entirely from the SDE, the simulations are conditioned directly on the observed data.

These methods have been applied, for example, to the highly variable Schlögl model, which exhibits noise-induced bistability, and in this challenging case, inference was obtained. In addition, new splitting schemes for chemical reaction networks have been introduced that preserve structural properties which standard numerical methods fail to capture. Overall, these advances reduce simulation rejection rates and enable faster, more reliable inference for stochastic dynamical systems.