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A new numerical scheme for the Zaka\"i equation

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.