Optimal quantization methods with applications to finance.

Summary This thesis is devoted to quantification with applications to finance. Chapter 1 recalls the basics of quantization and the methods for finding optimal quantizers. In chap. 2 we study the asymptotic behavior, in s, of the quantization error associated with a linear transformation of an optimal quantifier sequence in r. We show that such a transformation makes the transformed sequence l's rate optimal for all s, for a large family of probabilities. Chap. 3 studies the asymptotic behavior of the maximal radius sequence associated to an optimal l'r quantifier sequence. We show that as soon as supp(p) is unbounded this sequence tends to infinity. We give, for a large family of probabilities, the speed of convergence to infinity. Chapter 4 is devoted to the pricing of lookback and barrier options. We write these prices in a form that allows us to estimate them by monte carlo, by a hybrid monte carlo-quantification method and by pure quantization.