Data-Driven Identification of Stochastic Dynamical Systems

Abstract

This comprehensive review paper examines the state-of-the-art methodologies for data-driven identification of stochastic dynamical systems. We present a systematic analysis of both classical and modern approaches, including maximum likelihood estimation, Bayesian inference, method of moments, and emerging machine learning techniques such as neural networks, Gaussian processes, and sparse identification of nonlinear dynamics (SINDy). The paper provides detailed mathematical foundations, practical implementation guidelines using Python, and comparative analyses of different methodologies. We discuss theoretical properties including consistency, asymptotic normality, and computational complexity. Furthermore, we demonstrate applications across diverse domains including finance, climate science, biology, and engineering. The review identifies current challenges and outlines promising directions for future research, particularly in handling high-dimensional systems, limited data scenarios, and real-time identification requirements. Our analysis reveals that hybrid approaches combining physics-based constraints with data-driven learning offer the most promising pathway for robust system identification in complex stochastic environments.