On deformations, approximations and nonlinear filtering

Abstract

Given a (nonlinear) filtering problem there is associated to it a Lie algebra L(Σ) of differential operators which is the Lie algebra generated by the two operators occurring in the Zakai equation for the unnormalized conditional density of the problem. The representability of L(Σ) or quotients of L(Σ) by means of vectorfields on a finite dimensional manifold is strongly related to the existence of exact finite dimensional recursive filters. However, in many cases, including the cubic sensor problem and the problem dx = dw, dz = (x + ɛx3) dt + dυ, ɛ ≠ 0, the algebra Lɛ(Σ)is isomorphic to the Weyl algebra Wl = ℝ which admits no nonzero homomorphisms into any Lie algebra of vectorfields on a finite dimensional manifold. On the other hand the Lie algebra ‘Lɛ(Σ) mod ɛn’ is finite dimensional for all n which opens up the possibility of the existence of a sequence of converging approximate filters.