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

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 \textit{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 (3)

• 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 $K$. 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).

Theorems & Definitions (14)

• remark 1
• definition 1
• remark 2
• remark 3
• Theorem 1
• proof
• remark 4
• remark 5
• remark 6
• remark 7