Rates for branching particle approximations of continuous-discrete filters

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 -> mu(t) ((.)) = P{X-t is an element of (.)vertical bar sigma{Y-tk,Y- t(k) <= t}} where tk =k epsilon and Y-tk is a distorted, corrupted, partial observation of Xt(k). Then, one constructs a particle system with observation-dependent branching and n initial particles whose empirical measure at time t,mu(n)(t), closely approximates mu(t). Each part ticle evolves independently of the other particles according to the law of the signal between observation times tk, and branches with small probability at an observation time. For filtering problems where s 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 epsilon. We analyze the algorithm on Levy-stable signals and give rates of convergence for E-1/2 {vertical bar vertical bar mu(n)(t) - mu(t)vertical bar vertical bar(2)(gamma)} where vertical bar vertical bar (.) vertical bar vertical bar gamma is a Sobolev norm, as well as related convergence results.