Approximate and limit results for nonlinear filters with wide bandwith observation noise

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Abstract
The paper treats a number of problems in nonlinear filtering theory when the signal process is xs:(), a “near” diffusion, and the observation noise is of wide bandwidth (correlated or not with xs()). Natural modifications of the optimal filters for the classical (diffusion) case are described, and robustness and weak convergence results proved. Often the weak convergence is in an infinite dimensional setting, since the basic process is measure valued. These filters are not actually optimal for the “physical” process, and their value depends on the use to which they are put. It is shown that they are actually nearly “optimal” with respect to a wide variety of comparison processors. We also treat the normalized (Césaro) errors (for the “approximate” filters) over an infinite time interval, and show that their distributions are close (for wide bandwidth observation noise) to those obtained if the system were a standard diffusion and the optimal filter used. This is particularly important if the system is in operation for a long time period.