Deep ZakaiJ: Structured Filtering for Jump-Diffusion Time Series Forecasting
For practitioners modeling time series with abrupt jumps (e.g., finance, oceanography), Deep ZakaiJ provides a principled latent-state approach that outperforms existing methods in distributional forecasting while maintaining point accuracy.
Deep ZakaiJ embeds the Zakai filtering equation into a neural encoder-decoder to model partially observed jump-diffusion time series, improving distributional forecasts with calibrated intervals and interpretable latent states.
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.