This paper develops an asymptotic expansion technique in momentum space  . 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) as an asymptotic expansion of the conditional distribution appearing in stochastic filtering problems . 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" {\it  for asymptotic expansion in a numerically efficient way. This is found to improve the performance significantly where otherwise the approximation fails badly. The method may be useful for other applications, such as, option pricing in finance as well as measure-valued stochastic dynamics in general .
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 {\it 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 an useful tool for more realistic financial modeling with unobserved parameters, and also for problems involving nonlinear measure-valued processes .
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for stochastic filtering
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as an asymptotic expansion of the conditional distribution appearing in stochastic filtering problems
for the relevant conditional distribution
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"substepping method"
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substepping method
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may be useful for other applications, such as, option pricing in finance as well as
is expected to provide an useful tool for more realistic financial modeling with unobserved parameters, and also for problems involving nonlinear
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stochastic dynamics in general
processes