A probabilistic approach to solving unknown linear stochastic differential equations with Bayesian uncertainty quantification and inference

Résumé

This work develops a probabilistic framework for solving unknown linear stochastic differential equations (SDEs) by treating the numerical solution as an inference problem. Starting from classical foundations, including the Fokker–Planck and Kushner–Stratonovich formulations, the study derives the evolution of conditional probability densities and presents both the strong and weak forms of the Kushner– Stratonovich equation. Under linear–Gaussian assumptions, the solution remains Gaussian and is fully characterized by its mean and covariance. Deterministic moment equations are derived for state transition and covariance propagation, providing a computationally tractable alternative to full sampling methods. Embracing probabilistic numerics, the framework interprets numerical solvers as inference procedures that return distributions over trajectories, thereby quantifying both process noise and discretization uncertainty. Connections are established with Bayesian probabilistic numerics, and the results demonstrate that moment-based probabilistic solvers deliver calibrated uncertainty for filtering and forecasting tasks. This approach is particularly valuable in engineering, finance, and climate modeling, where reliable uncertainty quantification is critical for decision-making under stochastic dynamics. The paper concludes by recommending robust numerical schemes for the Kushner– Stratonovich equation, comparative studies against Euler–Maruyama and Milstein methods, extensions to partially observed and real-time systems, and validation of the framework using real-world case studies.