  This paper develops an asymptotic expansion technique in momentum space for
stochastic filtering. It is shown that Fourier transformation combined with a
polynomial-function approximation of the nonlinear terms gives a closed
recursive system of ordinary differential equations (ODEs) for the relevant
conditional distribution. Thanks to the simplicity of the ODE system, higher
order calculation can be performed easily. Furthermore, solving ODEs
sequentially with small sub-periods with updated initial conditions makes it
possible to implement a substepping method for asymptotic expansion in a
numerically efficient way. This is found to improve the performance
significantly where otherwise the approximation fails badly. The method is
expected to provide a useful tool for more realistic financial modeling with
unobserved parameters, and also for problems involving nonlinear measure-valued
processes.
