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

Francisco Delgado-Vences; José Julián Pavón-Español; Arelly Ornelas

stat.MLMar 27, 2026

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

Authors: Francisco Delgado-Vences, José Julián Pavón-Español, Arelly Ornelas

Organizations: Universitat Autónoma de Barcelona, Departament de Matemàtiques, Edifici C, Facultat de Ciències, 080193, Bellaterra, Spain · Facultad de Ciencias - Universidad Nacional Autónoma de México. Ciudad Universitaria, Ciudad de México, Mexico · Secihti - Centro Interdisciplinario de Ciencias Marinas. IPN., BCS, La Paz, México

Abstract

This study addresses the inverse problem of parameter estimation for Stochastic Differential Equations (SDEs) by minimizing a regularized discrepancy functional via Stochastic Gradient Descent (SGD). To achieve computational efficiency, we leverage the Wiener Chaos Expansion (WCE), a spectral decomposition technique that projects the stochastic solution onto an orthogonal basis of Hermite polynomials. This transformation effectively maps the stochastic dynamics into a hierarchical system of deterministic functions, termed the propagator. By reducing the stochastic inference task to a deterministic optimization problem, our framework circumvents the heavy computational burden and sampling requirements of traditional simulation-based methods like MCMC or MLE. The robustness and scalability of the proposed approach are demonstrated through numerical experiments on various non-linear SDEs, including models for individual biological growth. Results show that the WCE-SGD framework provides accurate parameter recovery even from discrete, noisy observations, offering a significant paradigm shift in the efficient modeling of complex stochastic systems.

Figures & tables

Panel of Figure 1 — Figure 1: Convergence analysis of the Gradient Descent algorithm for estimating the drift parameter $\alpha$ in the Geometric Brownian Motion and the OU process. The horizontal line indicates the ground truth value ( $\alpha=0.9$ for the GMB and $\alpha=1.7$ for the OU). Distinct markers represent the evolution of the parameter estimate starting from different 5 initial guesses. As shown, all trajectories converge rapidly toward the true value within approximately 6 iterations, demonstrating the robustness and stability of the Wiener Chaos-based optimization framework.

Figure 2: GBM Parameter Learning Validation on S&P 500 Index Data (2017). Comparison between the daily adjusted closing prices (solid black line) and stochastic simulations based on parameters learned via the proposed WCE-SGD method ( $\hat{\mu}\approx 0.1590$ , $\hat{\sigma}\approx 0.0674$ ). The solid gray line represents the mean of 100 simulations, while the shaded blue area indicates the 95% empirical confidence interval. The close fit demonstrates the successful application of the methodology to noisy empirical financial data.

Figure 3: Stochastic Logistic Parameter Learning on Microbial Growth Data. Comparison between empirical optical density trajectories (gray lines) from a 96-well microtiter plate and stochastic simulations driven by parameters learned via the WCE-SGD framework. The data was pre-standardized to estimate the carrying capacity

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. The methodology accurately captures both the asymmetrical, non-linear drift of the exponential phase and the inherent biological noise, encapsulating the individual trajectories within the 95% confidence bands (shaded area).

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Stochastic differential equations (SDEs) provide a flexible framework for modeling temporal dynamics in partially observed systems. A central task is to calibrate such models from data, which requires inferring latent trajectories and parameters from sparse, noisy observations. Classical smoothing methods for this problem are often limited by path degeneracy and poor scalability. In this work, we developed a novel method based on characterization of the posterior SDE in terms of conditional backward-in-time score defined as the gradient of a function solving a Kolmogorov backward equation with multiplicative updates at observation times. We learn this conditional score using neural networks trained to satisfy both the governing PDE and the observation-induced jump conditions, thereby integrating continuous-time dynamics with discrete Bayesian updates. The resulting score induces a posterior SDE with the same diffusion coefficient but a modified drift, enabling efficient posterior trajectory sampling. We further derive a likelihood-based objective for learning the SDE parameters, yielding an evidence lower bound (ELBO) for joint state smoothing and parameter estimation. This leads to a variational EM-style procedure, where the neural conditional score is optimized to approximate the smoothing distribution, followed by a maximization step over the SDE parameters using samples from the induced posterior. Experiments on nonlinear systems demonstrate accurate and stable inference with a very few observations demonstrating significant improved scalability compared to classical MCMC methods.

Abstract

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