Product Markovian Quantization of a Diffusion Process with Applications to Finance

We introduce a new methodology for the quantization of the Euler scheme for a d-dimensional diffusion process. This method is based on a Markovian and componentwise product quantization and allows us, from a numerical point of view, to speak of fast online quantization in a dimension greater than one since the product quantization of the Euler scheme of the diffusion process and its companion weights and transition probabilities may be computed quite instantaneously. We show that the resulting quantization process is a Markov chain, then we compute the associated weights and transition probabilities from (semi-) closed formulas. From the analytical point of view, we show that the induced quantization errors at the k-th discretization step is a cumulative of the marginal quantization error up to that time. Numerical experiments are performed for the pricing of a Basket call option in a correlated Black Scholes framework, for the pricing of a European call option in a Heston model and for the approximation of the solution of backward stochastic differential equations in order to show the performances of the method.