In this paper we are concerned with a very particular case of the following general filtering problem. The state process Xis the solution of a stochastic differential equation of the form
$$\begin{array}{*{20}{c}}
{d{X_t} = \alpha ({X_t})dt + \beta ({X_t})d{W_t},}&{\mathcal{L}({X_0}) = {\pi _0},}
\end{array}$$
where π0 is a known distribution on ℝd, and α,β are known functions, and W is a d-dimensional Wiener process. We have noisy observations Y1,...,YN at N regularly spaced times, and without loss of generality we will assume that these times are. That is, at each time i ∈ ℕ*
we have an ℝd-valued observation Yi given by
$${Y_i} = h({X_i},{\varepsilon _i}),$$
where the εi are i.i.d. q′-dimensional variables, independent of X and with a law having a known density g, and h is a known function from ℝd × ℝq′ into ℝq. We denote by πY,N the filter for XN, that is a regular version of the conditional distribution of the random variable XN knowing Y1,…,YN.