A perturbation theory for Feynman-Kac models and their particle interpretations

Source abstract

We develop a perturbation analysis of Feynman--Kac models, describing how local errors introduced at each step of the Feynman--Kac recursion propagate over time. Under general conditions on the potential functions and the Markov transitions, which are shown to hold for a large class of state-space models, we prove that the global error for any time horizon $T\geq 1$ is bounded by $C\,Δ_T\log(1+Δ_T^{-1})$, where $Δ_T$ is the largest local error; this bound holds both for the prediction and the updated models. These conditions require no Lyapunov function and no stability of the Markov transitions, and allow for non-compact state spaces: confinement is provided by the potential functions. Our perturbation bounds hold in total variation and in the Kolmogorov norm; for the latter, we extend the Hardy--Krause variation and the Koksma--Hlawka inequality to unbounded domains. Our perturbation theory is applied to derive time-uniform bounds for linearized Kalman filters, Euler schemes for partially observed diffusions and nudged state-space models, as well as high-probability error bounds, uniform over whole trajectories, for particle filters and, more generally, for mean field particle methods such as sequential Monte Carlo and diffusion Monte Carlo algorithms.