Optimal quantization: a ride from Signal processing to Finance

Abstract

Originally devised in the Bell laboratories to optimize signal transmission in the 1950s, optimal vector quantization is also connected to data analysis as the ancestor of unsupervised classification method (k-means, big data,...). More generally, it provides optimal finite skeletons of a distribution of probability on R^d .Starting in the 1990’s quantization based numerical schemes of Markov chains have been introduced to solve non-linear problems of numerical probability especially arising in mathematical finance such as optimal stopping (American options), numerical schemes for nonlinear filtering (volatility estimation), stochastic control (portfolio optimization), solving Backward Stochastic Differential equations (dynamical hedging of derivatives). All these problems share that their numerical treatment need a spatial discretization of their dynamics in order to compute multi- (in practice medium-)dimensional conditional expectations. We will present, beyond the main results of optimal vector quantization theory, both marginally and recursively quantized numerical schemes and some recent numerical applications.