# Stability of Feynman-Kac formulae with path-dependent potentials

## TL;DR

This paper investigates the long-term stability of particle algorithms with path-dependent potentials, providing conditions for stability in models like the mixture Kalman filter and GARCH filter, and extending results to standard particle algorithms.

## Contribution

It offers new theoretical conditions ensuring the asymptotic stability of particle algorithms with path-dependent potentials, including practical criteria for specific filters.

## Key findings

- Derived stability conditions for mixture Kalman and GARCH filters
- Established weaker stability conditions for standard particle algorithms
- Demonstrated asymptotic stability as time approaches infinity

## Abstract

Several particle algorithms admit a Feynman-Kac representation such that the potential function may be expressed as a recursive function which depends on the complete state trajectory. An important example is the mixture Kalman filter, but other models and algorithms of practical interest fall in this category. We study the asymptotic stability of such particle algorithms as time goes to infinity. As a corollary, practical conditions for the stability of the mixture Kalman filter, and a mixture GARCH filter, are derived. Finally, we show that our results can also lead to weaker conditions for the stability of standard particle algorithms, such that the potential function depends on the last state only.