This paper provides an optimal filtering methodology in discretely observed continuous-time jump-diffusion models. Although the filtering problem has received little attention, it is useful for estimating latent states, forecasting volatility and returns, computing model diagnostics such as likelihood ratios, and parameter estimation. Our approach combines time-discretization schemes with Monte Carlo methods. It is quite general, applying in nonlinear and multivariate jump-diffusion models and models with nonanalytic observation equations. We provide a detailed analysis of the filter's performance, and analyze four applications: disentangling jumps from stochastic volatility, forecasting volatility, comparing models via likelihood ratios, and filtering using option prices and returns.

This article presents an augmented deep factor model that generates latent factors for cross-sectional asset pricing. The conventional security sorting on firm characteristics for constructing long–short factor portfolio weights is nonlinear modeling, while factors are treated as inputs in linear models. We provide a structural deep-learning framework to generalize the complete mechanism for fitting cross-sectional returns by firm characteristics through generating risk factors (hidden layers). Our model has an economic-guided objective function that minimizes aggregated realized pricing errors. Empirical results on high-dimensional characteristics demonstrate robust asset pricing performance and strong investment improvements by identifying important raw characteristic sources.