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Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering

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  • Masaaki Fujii

    (The University of Tokyo)

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.

Suggested Citation

  • Masaaki Fujii, 2012. "Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering," CARF F-Series CARF-F-311, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Mar 2013.
    • Handle: RePEc:cfi:fseres:cf311
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    File URL: https://www.carf.e.u-tokyo.ac.jp/old/pdf/workingpaper/fseries/324.pdf
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    References listed on IDEAS

    as
    1. Akihiko Takahashi & Masashi Toda, 2012. "Note on an Extension of an Asymptotic Expansion Scheme," CARF F-Series CARF-F-286, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Dec 2012.
    2. Akihiko Takahashi & Masashi Toda, 2012. "Note on an Extension of an Asymptotic Expansion Scheme," CIRJE F-Series CIRJE-F-860, CIRJE, Faculty of Economics, University of Tokyo.
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    Cited by:

    1. Akihiko Takahashi & Toshihiro Yamada, 2013. "A Weak Approximation with Asymptotic Expansion and Multidimensional Malliavin Weights," CIRJE F-Series CIRJE-F-909, CIRJE, Faculty of Economics, University of Tokyo.
    2. Akihiko Takahashi & Toshihiro Yamada, 2015. "A weak approximation with asymptotic expansion and multidimensional Malliavin weights (Revised version of CARF-F-335; Forthcoming in Annals of Applied Probability")"," CARF F-Series CARF-F-358, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Apr 2016.
    3. Akihiko Takahashi, 2015. "Asymptotic Expansion Approach in Finance," CARF F-Series CARF-F-356, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2015.

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    1. Masaaki Fujii, 2013. "Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering," CIRJE F-Series CIRJE-F-883, CIRJE, Faculty of Economics, University of Tokyo.
    2. Masaaki Fujii, 2014. "Momentum-space approach to asymptotic expansion for stochastic filtering," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(1), pages 93-120, February.

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