Abstract
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.
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Notes
One needs to put \(\epsilon =1\) at the end for the comparison to the original model.
As before, one needs to put \(\epsilon =1\) at the end for the comparison to the original model.
First order expansion is good enough for short period since the infinitesimal generator contains no 2nd order term.
Note that, one have to resort to discrete Fourier transformation for numerical implementation anyway.
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Acknowledgments
The author thanks Akihiko Takahashi of University of Tokyo for informative discussions that help to understand the connections to the asymptotic expansion in the position space. This research is supported by CARF (Center for Advanced Research in Finance) and the global COE program “The research and training center for new development in mathematics.”
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Fujii, M. Momentum-space approach to asymptotic expansion for stochastic filtering. Ann Inst Stat Math 66, 93–120 (2014). https://doi.org/10.1007/s10463-013-0405-1
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DOI: https://doi.org/10.1007/s10463-013-0405-1