Robust Filtering of L\'evy-driven Stochastic Models

TL;DR

This paper develops robust nonlinear filtering methods for Le9vy-driven stochastic models, ensuring filter stability under jump noise and enabling pathwise convergence from discrete data.

Contribution

It introduces new continuity results for filters with jumps, using rough path theory and flow decompositions, extending robustness analysis beyond continuous semimartingales.

Findings

Proves filter continuity for finitely many jumps in rough p-variation topology.

Establishes filter continuity for infinitely many jumps under separability assumptions.

Provides pathwise convergence guarantees from discrete observations without knowing the probability law.

Abstract

We study robust nonlinear filtering for stochastic models driven by L\'evy processes, where the signal and observation processes are coupled through common Brownian and jump noise. Robustness, defined as the continuous dependence of the filter on the observation path, is essential whenever the observation process deviates from the idealized model, for instance when a path must be reconstructed from discrete-time samples. This question is well understood for continuous semimartingale systems but largely open in the presence of jumps. We construct a version of the filter and establish its continuity in two regimes. For processes with finitely many jumps on compact intervals, we prove continuity in both the rough $p$-variation and $p$-variation topologies on cadlag path space, without requiring a separability condition on the jump coefficients. For processes with infinitely many jumps, we prove continuity in a modified rough $p$-variation topology adapted to cadlag geometric rough paths, under an additional separability assumption. In both cases, our approach relies on Stratonovich and Marcus flow decompositions rather than the It\^o-based methods of recent work. The resulting geometric rough-path lifts yield pathwise convergence guarantees and can be constructed directly from discrete observations without knowledge of the underlying probability law.