A regular perturbation expansion in nonlinear filtering

We study solutions of nonlinear filtering problems dxt = b(xt)dt + dwt dyt = hτ(xt)dt + dvt where hε = h0 + εf and ε is a small parameter. We look in particular at the case f(x) a polynomial. A "mixed" multiplicative additive expansion in ε for the unnormalized conditional density qε(t,x) is determined and is of the form e-s0 (l + εο1 + ε2τ2 +...). Then a more abstract problem is discussed, that of regular perturbations to a diffusion which has a generator of the form Lε = L0 + εM + ε2M1, where ε 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.