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We introduce a new approach to quantize the Euler scheme of an$\mathbb{R}^d$-valued diffusion process. This method is based on a Markovianand componentwise product quantization and allows us, from a numerical point ofview, to speak of {\em fast online quantization} in dimension greater than onesince the product quantization of the Euler scheme of the diffusion process andits companion weights and transition probabilities may be computed quiteinstantaneously. We show that the resulting quantization process is a Markovchain, then, we compute the associated companion weights and transitionprobabilities from (semi-) closed formulas. From the analytical point of view,we show that the induced quantization errors at the $k$-th discretization step$t_k$ is a cumulative of the marginal quantization error up to time $t_k$.Numerical experiments are performed for the pricing of a Basket call option,for the pricing of a European call option in a Heston model and for theapproximation of the solution of backward stochastic differential equations toshow the performances of the method.

Product Markovian quantization of an R^d -valued Euler scheme of a diffusion process with applications to finance