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 filt er can rarely be constructed, one needs to compute with numerically feasibl e approximations. The signal model can be a jump-diffusion, reflected or no t. 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 inte rval. In a previous paper of the authors [A. Budhiraja and H.J. Kushner, SI AM J. Control Optim., 37 (1999), pp. 1946-1979], it was shown, under quite reasonable conditions on the approximating filter and on the signal and noi se 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 co sts per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter was used. When suitable approximating filters cannot be readily constructed due to excessi ve computational requirements or to problems associated with a high signal dimension, approximations based on random sampling methods (or, perhaps, co mbinations of sampling and analytical methods) become attractive, and are t he subject of a great deal of attention. The work of the previous paper is extended to a wide class of such algorithms. Under quite broad conditions, covering virtually all the cases considered to date, it is shown that the p athwise average errors converge to the same limit that would be obtained if the optimal filter was used, as time goes to infinity and the approximatio n parameter goes to its limit in any way at all. All the extensions (e.g., wide bandwidth observation or system driving noise) in [A. Budhiraja and H. J. Kushner, SIAM J. Control Optim., 37 (1999), pp. 1946-1979] hold for our random sampling algorithms as well.