Rates for branching particle approximations of continuous-discrete filters

Abstract

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that t→X_t is a Markov process and we wish to calculate the measure-valued process t→μ_t(⋅)≐P{X_t∈⋅|σ{Y_tk, t_k≤t}}, where t_k=kɛ and Y_tk is a distorted, corrupted, partial observation of X_tk. Then, one constructs a particle system with observation-dependent branching and n initial particles whose empirical measure at time t, μ_tⁿ, closely approximates μ_t. Each particle evolves independently of the other particles according to the law of the signal between observation times t_k, and branches with small probability at an observation time. For filtering problems where ɛ is very small, using the algorithm considered in this paper requires far fewer computations than other algorithms that branch or interact all particles regardless of the value of ɛ. We analyze the algorithm on Lévy-stable signals and give rates of convergence for E^1/2{‖μⁿ_t−μ_t‖_γ²}, where ‖⋅‖_γ is a Sobolev norm, as well as related convergence results.