Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering

24 Pages Posted: 12 Sep 2012 Last revised: 26 Mar 2013

See all articles by Masaaki Fujii

Masaaki Fujii

University of Tokyo - Faculty of Economics

Date Written: March 24, 2013


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.

Keywords: asymptotic expansion, fourier transformation, filtering, zakai equation, measure-valued system

JEL Classification: C10, C13, G10

Suggested Citation

Fujii, Masaaki, Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering (March 24, 2013). Available at SSRN: or

Masaaki Fujii (Contact Author)

University of Tokyo - Faculty of Economics ( email )

7-3-1 Hongo, Bunkyo-ku
Tokyo 113-0033

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