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 $t\to X_t$ is a Markov process and we wish to calculate the measure-valued process $t\to\mu_t(\cdot)\doteq P\{X_t\in \cdot|\sigma\{Y_{t_k}, t_k\leq t\}\}$, where $t_k=k\epsilon$ and $Y_{t_k}$ is a distorted, corrupted, partial observation of $X_{t_k}$. Then, one constructs a particle system with observation-dependent branching and $n$ initial particles whose empirical measure at time $t$, $\mu_t^n$, closely approximates $\mu_t$. 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 $\epsilon$ 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 L\'{e}vy-stable signals and give rates of convergence for $E^{1/2}\{\|\mu^n_t-\mu_t\|_{\gamma}^2\}$, where $\Vert\cdot\Vert_{\gamma}$ is a Sobolev norm, as well as related convergence results.