Computable infinite-dimensional filters with applications to discretized diffusion processes |
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Authors: | Mireille Chaleyat-Maurel Valentine Genon-Catalot |
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Affiliation: | 1. Université René Descartes Paris 5, U.F.R. de Mathématiques et Informatique, Laboratoire MAP5 (CNRS-UMR 8145) et Laboratoire de Probabilités et Modèles Aléatoires (CNRS-UMR 7599), 45, rue des Saints-Pères, 75270 Paris Cedex 06, France;2. Université René Descartes Paris 5, U.F.R. de Mathématiques et Informatique, Laboratoire MAP5 (CNRS-UMR 8145), 45, rue des Saints-Pères, 75270 Paris Cedex 06, France |
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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=XnΔ 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. |
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Keywords: | primary 93E11 60G35 secondary 62C10 |
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