Title: State-Dependent Volatility in Latent Dynamical Systems: Estimation with Incomplete Data

Abstract:

While latent state-space models are a staple for analyzing partially observed dynamical systems, conventional approaches typically presuppose that process variability remains constant regardless of the latent state’s position. This assumption often fails in biological, behavioral, and physiological contexts, where variability is frequently tied to the underlying dynamical state, resulting in structured stochasticity that constant-variance models cannot adequately represent. To address this, we propose a state-coupled stochastic volatility framework in which the variance of the latent process is determined by its displacement from a latent equilibrium.

To estimate this relationship when data is partially observed, we introduce a particle expectation-maximization algorithm that integrates bootstrap particle filtering with backward trajectory smoothing. Central to this model is a coupling parameter, $\gamma$, which measures the intensity of the link between the latent state’s position and its process variability. We conducted a large-scale simulation benchmark to assess the model’s ability to recover parameters and detect effects under various conditions, including different levels of coupling strength, observation noise, trajectory duration, and persistence regimes.

The results indicate that our proposed framework consistently lowers recovery bias compared to a heteroskedastic proxy based on observed states, with the most significant gains observed under strong coupling conditions. Furthermore, recovery accuracy improves as latent persistence increases. While detection performance remains robust across a wide array of scenarios, it becomes increasingly superior as observation noise rises. These findings confirm that state-coupled volatility can be successfully identified and estimated under partial observation, provided that the latent-state structure is explicitly accounted for. This framework offers a practical methodological basis for investigating state-dependent variability and determining whether structured stochasticity provides insights into system dynamics that are not available from mean-state trajectories alone.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC