Deep ZakaiJ: Structured Filtering for Jump-Diffusion Time Series Forecasting

2026-05-23

AI Summary

This work addresses the challenge of modeling time series driven by unobserved latent states that exhibit both continuous diffusion and unpredictable jumps. The authors propose a novel approach that integrates Zakai nonlinear filtering with neural encoder–decoder architectures. By leveraging Strang splitting, the method enables first-order accurate recursive updates of the belief over latent states and explicitly disentangles continuous and discrete dynamics within the decoder. Notably, this is the first framework to embed Zakai filtering into a differentiable neural architecture, achieving both structural interpretability and high inference accuracy. Empirical evaluations on synthetic, financial, and oceanographic datasets demonstrate substantial improvements in distributional forecasting quality, well-calibrated prediction intervals, and successful recovery of interpretable latent state structures.

Abstract

Time series driven by unobserved latent states frequently exhibit abrupt jump discontinuities whose timing and magnitude cannot be predicted from observed history alone. Classical jump-diffusion models offer a principled mathematical framework but assume rigid parametric forms, while recent neural jump models operate on fully observed trajectories without inferring the hidden states that govern the dynamics. We propose \textit{Deep ZakaiJ}, a latent-state model for partially observed jump-diffusion systems that embeds the Zakai nonlinear filtering equation into a neural encoder--decoder architecture. The encoder recursively updates a belief over the latent state via Strang splitting into three interpretable substeps: prior propagation, diffusion innovation, and jump innovation, yielding a differentiable, first-order-accurate approximation of the exact filtering evolution. The decoder is a structured jump-diffusion model explicitly conditioned on the filtered belief, preserving the separation between continuous dynamics and discontinuous shocks. On synthetic, financial, and oceanographic datasets, \textit{Deep ZakaiJ} improves distributional forecasts while remaining competitive in point accuracy, achieving calibrated predictive intervals and recovering interpretable latent structure in synthetic and qualitative case studies.

Problem

• jump-diffusion
• latent state
• time series forecasting
• nonlinear filtering
• partially observed systems

Innovation

• jump-diffusion
• nonlinear filtering
• latent state inference
• neural encoder-decoder
• Strang splitting