Parameter Estimation in Stochastic Differential Equations via Wiener Chaos Expansion and Stochastic Gradient Descent

A team of researchers has introduced a novel computational framework that dramatically improves how scientists estimate parameters in Stochastic Differential Equations (SDEs). The method, developed by Francisco Delgado-Vences, José Julián Pavón-Español, and Arelly Ornelas, combines Wiener Chaos Expansion (WCE) with Stochastic Gradient Descent (SGD) to transform what's traditionally been a computationally intensive stochastic problem into a more manageable deterministic optimization task. By projecting the stochastic solution onto an orthogonal basis of Hermite polynomials, the framework creates what they term a 'propagator'—a hierarchical system of deterministic functions that represents the original stochastic dynamics.

This transformation is the key innovation, as it circumvents the heavy sampling requirements and computational burden of traditional methods like Markov Chain Monte Carlo (MCMC) or Maximum Likelihood Estimation (MLE). The researchers demonstrated the framework's robustness through numerical experiments on various non-linear SDEs, including models for individual biological growth. Results show that WCE-SGD provides accurate parameter recovery even from discrete, noisy observations, representing what the authors call 'a significant paradigm shift' in efficiently modeling complex stochastic systems. The 25-page manuscript has been submitted to Applied Mathematical Modelling for publication.