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Computable infinite-dimensional filters with applications to discretized diffusion processes

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  • Chaleyat-Maurel, Mireille
  • Genon-Catalot, Valentine

Abstract

Let us consider a pair signal-observation ((xn,yn),n>=0) where the unobserved signal (xn) is a Markov chain and the observed component is such that, given the whole sequence (xn), the random variables (yn) are independent and the conditional distribution of yn only depends on the corresponding state variable xn. The main problems raised by these observations are the prediction and filtering of (xn). We introduce sufficient conditions allowing us to obtain computable filters using mixtures of distributions. The filter system may be finite or infinite-dimensional. The method is applied to the case where the signal xn=Xn[Delta] is a discrete sampling of a one-dimensional diffusion process: Concrete models are proved to fit in our conditions. Moreover, for these models, exact likelihood inference based on the observation (y0,...,yn) is feasible.

Suggested Citation

  • Chaleyat-Maurel, Mireille & Genon-Catalot, Valentine, 2006. "Computable infinite-dimensional filters with applications to discretized diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1447-1467, October.
  • Handle: RePEc:eee:spapps:v:116:y:2006:i:10:p:1447-1467
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    Cited by:

    1. Lacour, Claire, 2008. "Nonparametric estimation of the stationary density and the transition density of a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 232-260, February.
    2. Denis Belomestny & Fabienne Comte & Valentine Genon-Catalot, 2019. "Sobolev-Hermite versus Sobolev nonparametric density estimation on $${\mathbb {R}}$$ R," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(1), pages 29-62, February.
    3. Kon Kam King, Guillaume & Pandolfi, Andrea & Piretto, Marco & Ruggiero, Matteo, 2024. "Approximate filtering via discrete dual processes," Stochastic Processes and their Applications, Elsevier, vol. 168(C).
    4. Lacour, Claire, 2008. "Adaptive estimation of the transition density of a particular hidden Markov chain," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 787-814, May.
    5. Fabienne Comte & Valentine Genon-Catalot & Mathieu Kessler, 2011. "Multiplicative Kalman filtering," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 389-411, August.

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