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

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 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, namely 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 conditioned on the latent state, preserving the separation between continuous dynamics and discontinuous shocks. On synthetic and financial datasets, Deep ZakaiJ improves distributional forecasts while remaining competitive in point accuracy, achieving sharper and better-calibrated predictive intervals than the baselines and recovering interpretable latent structure in synthetic and qualitative case studies.