BROWNIAN AND POISSON BRIDGES: APPLICATION TO NONLINEAR FILTERING PROBLEMS AND ERROR ESTIMATES

The nonlinear filtering (NLF) problems described by the stochastic systems with jump diffusive state/observation processes have been attracted more and more attentions. In this paper, we consider the NLF problem modeled by a diffusive state process with the mixed observations and the correlated noises. One of the observation processes is driven by the Brownian motion correlated with the state process, and the other one is an independent Poisson point process. The state’s unnormalized density conditioned on the continuous observation history is described by the Zakai equation. However, in whatever algorithm, the unnormalized density conditioned only on the sub-filtration generated by the discretized observations can be implemented to approximate the solution of the Zakai equation. The main contribution of this paper is that we show under certain conditions the mean square error of this approximation is no more than the order O(√h), where h is the time step, by the technique of Brownian and Poisson bridges. To verify this theoretical convergence rate, we extend the Yau-Yau filtering algorithm originally proposed for the classical NLF problems, to those with the mixed observations and the correlated noises. This algorithm is numerically experimented in the modified cubic sensor problem, which can achieve the error of the order O(√h). Moreover, we compare this algorithm with the sampling importance and resampling (SIR) particle filter to illustrate the superiority of the on-and off-line algorithm in both accuracy and efficiency.