Nonlinear filtering with stochastic discontinuities

TL;DR

This paper extends nonlinear filtering theory to include predictable jump times in signals and observations, deriving new equations and demonstrating applications in clinical studies, machine learning, and credit risk.

Contribution

It introduces a framework for filtering with predictable jumps, deriving Kushner-Stratonovich and Zakai equations for this setting, which was not previously addressed.

Findings

Derived Kushner-Stratonovich and Zakai equations for predictable jumps.

Applied the framework to clinical, machine learning, and credit risk models.

Extended classical nonlinear filtering results to predictable discontinuities.

Abstract

Filtering problems with jumps in both the signal and the observation have been extensively studied, typically under the assumption that jump times are totally inaccessible. In many applications, however, jump times are known in advance (i.e., predictable), such as scheduled clinical visits, dividend payment dates, or inspection times in engineering systems. Taking predictable jump times as a starting point, we investigate a filtering problem in which both the signal and the observations can exhibit jumps at predictable times. We derive the corresponding Kushner-Stratonovich and Zakai equations, thereby extending classical nonlinear filtering results to a setting with predictable discontinuities. We illustrate the framework on a Kalman filtering model with predictable jumps and on applications to longitudinal clinical studies, such as spinal muscular atrophy (SMA), as well as to machine learning models (neural jump ODEs) and credit risk.