Invited to Present at the 2026 CAA Doctoral Academic Exchange Main Forum

To assess the quality of a probabilistic prediction for stochastic dynamical systems (SDSs), scoring rules assign a numerical score based on the predictive distribution and the measured state. We propose an $\epsilon$-logarithm score that generalizes the logarithm score by considering a neighborhood of radius $\epsilon$. We characterize the probabilistic predictability of an SDS by optimizing the expected score over the space of probability measures, and show how predictability is quantitatively determined by the neighborhood radius, the differential entropies of process noises, and the system dimension. For any predictor, we provide approximations of the expected score with error $O(\epsilon)$. We also analyze the asymptotic behaviors of the score on individual trajectories, proving convergence to the expected score for i.i.d. process noises with speed $O(T^{-1/2})$ in probability. Numerical examples elaborate the results.

Probabilistic prediction of stochastic dynamical systems (SDSs) aims to accurately predict the conditional probability distributions of future states. However, accurate probabilistic predictions hinge on accurate distributional information from a nominal model, which is rarely available in practice. We propose a novel functional-maximin-based distributionally robust probabilistic prediction (DRPP) framework, in which one can design probabilistic predictors with worst-case performance guarantees over a pre-defined ambiguity set of SDSs. Although DRPP requires optimizing over the space of probability measures — generally intractable — we develop a methodology that equivalently transforms the maximin problem from function spaces to Euclidean spaces. Two suboptimal solutions are derived: Noise-DRPP (relaxing constraints on the ambiguity set) and Eig-DRPP (relaxing constraints on the predictor), together with their optimality gaps against the global optimum. Numerical simulations compare the performance of the predictors under different SDSs.