  The aim of this paper is to propose a new method for numerical approximations
of the solution of the linear stochastic partial differential equation arising
in non-linear filtering problems: the Zaka\"i equation. The approximation
scheme is based on a representation of the solution of the Zaka\"i equation
involving a stochastic part arising from the observation process and a
deterministic partial differential equation in which are involved only the
parameters of the signal process. We may then employ a dynamic programming
principle in order to write down an approximation of this partial differential
equation. A quantization method based on the underlying diffusion process
(which is a not the signal itself) is used.
