Filtering the Wright-Fisher diffusion

Chaleyat-Maurel, Mireille; Genon-Catalot, Valentine

ESAIM: Probability and Statistics, Tome 13 (2009), pp. 197-217.

We consider a Wright-Fisher diffusion $\left(x\right(t\left)\right)$ whose current state cannot be observed directly. Instead, at times $t_1$ < $t_2$ < ..., the observations $y\left(t_i\right)$ are such that, given the process $\left(x\right(t\left)\right)$, the random variables $\left(y\left(t_i\right)\right)$ are independent and the conditional distribution of $y\left(t_i\right)$ only depends on $x\left(t_i\right)$. When this conditional distribution has a specific form, we prove that the model $((x\left(t_i\right),y\left(t_i\right))$, $i$$\ge$1) 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.

Classification : Primary 93E11, 60G35, secondary, 62C10
Mots clés : stochastic filtering, partial observations, diffusion processes, discrete time observations, hidden Markov models, prior and posterior distributions

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