The Nonlinear Filtering Problem

In Chapter 4, we were concerned with the problem of approximations for the singularly perturbed system (4.1.1), (4.1.2), and with the associated problem of control approximations. In this chapter, we will be concerned with approximations for the nonlinear filtering problem. Suppose that we observe the noise corrupted data yϵ(•) defined by \(d{y^\varepsilon } = g({x^\varepsilon },{z^\varepsilon })dt + d{w_0}\), where w0(•) is a standard vector-valued Wiener process. Owing to the complexity and high dimension of the original system (4.1.1), (4.1.2), the construction of the optimal filter or even the direct construction of an acceptable approximation can be a very hard task. The possibility of using some sort of averaging method, such as in Chapter 4, to get a simpler filter (say, one for an averaged system) is quite appealing. One would use the filter for the averaged system, but the input would be the true physical observations yϵ(•).