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Malliavin calculus, geometric mixing, and expansion of diffusion functionals
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  • Published: April 2000

Malliavin calculus, geometric mixing, and expansion of diffusion functionals

  • Shigeo Kusuoka1 &
  • Nakahiro Yoshida1 

Probability Theory and Related Fields volume 116, pages 457–484 (2000)Cite this article

  • 219 Accesses

  • 41 Citations

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Abstract.

Under geometric mixing condition, we presented asymptotic expansion of the distribution of an additive functional of a Markov or an ε-Markov process with finite autoregression including Markov type semimartingales and time series models with discrete time parameter. The emphasis is put on the use of the Malliavin calculus in place of the conditional type Cramér condition, whose verification is in most case not easy for continuous time processes without such an infinite dimensional approach. In the second part, by means of the perturbation method and the operational calculus, we proved the geometric mixing property for non-symmetric diffusion processes, and presented a sufficient condition which is easily checked in practice. Accordingly, we obtained asymptotic expansion of diffusion functionals and proved the validity of it under mild conditions, e.g., without the strong contractivity condition.

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan., , , , , , JP

    Shigeo Kusuoka & Nakahiro Yoshida

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  1. Shigeo Kusuoka
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  2. Nakahiro Yoshida
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Received: 7 September 1997 / Revised version: 17 March 1999

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Kusuoka, S., Yoshida, N. Malliavin calculus, geometric mixing, and expansion of diffusion functionals. Probab Theory Relat Fields 116, 457–484 (2000). https://doi.org/10.1007/s004400070001

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  • Issue Date: April 2000

  • DOI: https://doi.org/10.1007/s004400070001

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Keywords

  • Asymptotic Expansion
  • Discrete Time
  • Mild Condition
  • Continuous Time
  • Perturbation Method
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