Approximate Nonlinear Filtering for a Two-dimensional Di usion with One-dimensional Observations in a Low Noise Channel

The asymptotic behaviour of a nonlinear continuous time ltering problem is studied when the variance of the observation noise tends to 0. We suppose that the signal is a two-dimensional process from which only one of the components is noisy, and that a one-dimensional function of this signal, depending only of the unnoisy component, is observed in a low noise channel. An approximate lter is considered in order to solve this problem. Under some detectability assumptions we prove that the ltering error converges to 0 and an upper bound for the convergence rate is given. The eeciency of the approximate lter is compared with the eeciency of the optimal lter, and the order of magnitude of the error between the two lters, as the observation noise vanishes, is obtained. R esum e Nous etudions un probl eme de ltrage non lin eaire avec petit bruit d'observation o u le signal est un processus de dimension 2 dont seule une composante est bruit ee, et o u la fonction d'observation ne d epend que de la composante non bruit ee du signal. Nous etudions alors sous des hypoth eses de detectabilit e un ltre approch e. Nous montrons que l'erreur du ltre tend vers 0 et nous estimons la vitesse de convergence. L'eecacit e du ltre approch e est compar ee avec l'eecacit e du ltre optimal, et nous estimons l'ordre de grandeur de la dii erence entre ces deux ltres.