Asymptotic expansions and Lie algebras for some nonlinear filtering problems

State estimation problems for systems involving small parameters are treated by both analytical and Lie algebraic approximation techniques. An asymptotic expansion for the unnormalized conditional density corresponding to the case of observations of a Gauss-Markov process through a (weak) polynomial nonlinearity is computed and a convergence result is derived. The expansion is related to certain approximations of the associated estimation Lie algebra. The convergence result is based on arguments used recently to prove existence and uniqueness and to estimate the tail behavior of solutions to nonlinear filtering problems with unbounded coefficients. These arguments are, in turn, adapted from the analytical theory of parabolic PDE's. Simulation results are presented to further assess the performance of the resulting approximate filters.