A regular perturbation expansion in nonlinear filtering

We study solutions of nonlinear filtering problems dxt = b(xt)dt + dwt dyt = h??(xt)dt + dvt where h?? = h0 + ??f and ?? is a small parameter. We look in particular at the case f(x) a polynomial. A "mixed" multiplicative additive expansion in ?? for the unnormalized conditional density q??(t,x) is determined and is of the form e-s0 (l + ????1 + ??2??2 +...). Then a more abstract problem is discussed, that of regular perturbations to a diffusion which has a generator of the form L?? = L0 + ??M + ??2M1, where ?? is again a small parameter. A numerical study is then undertaken using the Gauss-Galerkin method and the method of Moving Finite elements to probe the behavior of the asymptotic expansion in some special cases.