Issue ESAIM: PS Volume 13, 2009 Page(s) 197 - 217 DOI 10.1051/ps:2008006 Published online 11 June 2009

ESAIM: PS, June 2009, Vol. 13, p. 197-217
DOI: 10.1051/ps:2008006

Filtering the Wright-Fisher diffusion

Mireille Chaleyat-Maurel¹ and Valentine Genon-Catalot<sup>2

1</sup>  (Corresponding author) Laboratoire MAP5, Université Paris Descartes, UFR de Mathématique et Informatique, CNRS-UMR 8145 and Laboratoire de Probabilités et Modèles Aléatoires (CNRS-UMR 7599), 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; mcm@math-info.univ-paris5.fr
²  Laboratoire MAP5, Université Paris Descartes, UFR de Mathématique et Informatique, CNRS-UMR 8145, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; genon@math-info.univ-paris5.fr

Received April 27, 2007. Revised February 13, 2008. Published online 11 June 2009

Abstract
We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t₁ < t₂ < ..., the observations y(t_i) are such that, given the process (x(t)), the random variables (y(t_i)) are independent and the conditional distribution of y(t_i) only depends on x(t_i). When this conditional distribution has a specific form, we prove that the model ((x(t_i),y(t_i)), i1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations.

Mathematics Subject Classification. Primary 93E11, 60G35; secondary 62C10

Key words: Stochastic filtering, partial observations, diffusion processes, discrete time observations, hidden Markov models, prior and posterior distributions


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