Marginal quantization of an Euler diffusion process and its application

We propose a new approach to quantize the marginals of the discrete Euler process resulting from the discretization of a brownian diffusion process using the Euler scheme. The method is built recursively using the distribution of the marginals of the discrete Euler process. The quantization error associated to the marginals is shown to goes toward 0 at the optimal rate associated to the quantization of an R-valued random vector. In the one dimensional setting we illustrate how to perform the optimal grids using the Newton algorithm and show how to estimate the associated weights from a recursive formula. Numerical tests are carried out for the pricing of European options in a local volatility model and a comparison with the Monte Carlo simulations shows that the proposed method is more efficient than the Monte Carlo method.