In this paper we are concerned with a very particular case of the following general filtering problem. The state process *X*is 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 π_{0} is a known distribution on ℝ^{d}, and α,β are known functions, and *W* is a *d*-dimensional Wiener process. We have noisy observations *Y*_{1},...,*Y*_{N} 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 ℝ^{d}-valued observation *Y*_{i} given by
$${Y_i} = h({X_i},{\varepsilon _i}),$$
where the ε_{i} 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 ℝ^{d} × ℝ^{q′} into ℝ*q*. We denote by π_{Y,N} the filter for *X*_{N}, that is a regular version of the conditional distribution of the random variable *X*_{N} knowing *Y*_{1},…,*Y*_{N}.