Paper: arXiv 2605.12142

Authors: Thorsten Schmidt, Félix B. Tambe-Ndonfack

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical foundation by extending classical nonlinear filtering (Kushner-Stratonovich and Zakai equations) to predictable jump times using advanced stochastic calculus. However, it leans heavily on theory and toy application models, lacking extensive empirical backtesting or data validation on financial time series.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend nonlinear filtering<br>to predictable jump times] --> B[Inputs: Signal & Observation Processes<br>with Predictable Discontinuities]
    B --> C[Derive Filter Equations:<br>Kushner-Stratonovich & Zakai Extensions]
    C --> D[Evaluate Methods & Applications]
    D --> E[Kalman Filter with Jumps]
    D --> F[Real-World Domains:<br>Clinical SMA Data, Neural Jump ODEs, Credit Risk]
    E & F --> G[Outcomes: Validated Framework for Filter<br>Updates at Scheduled/Predictable Events]