Approximation and Limit Results for Nonlinear Filters Over an Infinite Time Interval: Part II, Random Sampling Algorithms

The paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval. Since the optimal filter can rarely be constructed, one needs to compute with numerically feasible approximations. The signal model can be a jump–diffusion, reflected or not. The observations can be taken either in discrete or continuous time. The cost of interest is the pathwise error per unit time over a long time interval. In a previous paper of the authors [2], it was shown, under quite reasonable conditions on the approximating filter and on the signal and noise processes that (as time, bandwidth, process and filter approximation, etc.) go to their limit in any way at all, the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter were used. When suitable approximating filters cannot be readily constructed due to excessive computational requirements or to problems associated with a high signal dimension, approximations based on random sampling methods (or, perhaps, combinations of sampling and analytical methods) become attractive, and are the subject of a great deal of attention. This is somewhat analogous to the use of monte carlo methods for high dimensional integration problems. Owing to the sampling errors as well as to the other (computational and modeling) approximations that are made, in the filter and signal processes, it is conceivable that the long term pathwise average errors per unit time will be large, even with approximations that would perform well over some bounded time interval. ∗Supported in part by contracts N00014-96-1-0276 and N00014-96-1-0279 from the Office of Naval Research and NSF grant DMI 9812857. †Supported in part by contracts DAAH04-96-1-0075 from the Army Research office and NSF grant ECS 9703895.