Singular perturbation, state aggregation, and nonlinear filtering

Consider a state process t → xε(t) evolving in Rn whose motion is that of a pure jump process in Rn, in the 0(1) time scale, upon which is super -imposed a continuous motion along the orbits of a gradient-like vector field g in Rn, in the 0(1/ε) time scale i.e. the infinitesimal generator of the state process is of the form L + (1/ε)g. If we consider observations of the form dy = h(xε(t))dt + db(t), t ≥ 0, then for each ϵ > 0 the corresponding nonlinear filter is infinite dimensional. We show however that the projected motion t → x-ϵ(t) onto the equilibrium points of g is, in the limit as ϵ ↓ 0, a finite-state process governed by some explicit L- on the finite state space consisting of the equilibrium points of g. We then show that the corresponding filters converge to a finite-state Wonham filter.