Multiplicative asymptotic expansions for filtering linear diffusions with small polynomial nonlinearities

When filtering a diffusion process with small polynomial nonlinearities, the conditional density may be expanded in an additive asymptotic expansion about a Kalman filter. Because such an expansion may fail to be a probability density, finite dimensional multiplicative asymptotic expansions are developed here as an alternative approach. Using the robust version of the filtering equation, an existence theorem and upper and lower estimates for the tail behavior of the unnormalized conditional density are obtained, and error bounds are derived using an integral representation of the error and a comparison theorem for partial differential inequalitites.