A Constructive Sharp Approach to Functional Quantization of Stochastic Processes

Abstract

We present a constructive approach to the functional quantization problem of stochastic processes, with an emphasis on Gaussian processes. The approach is constructive, since we reduce the infinite-dimensional functional quantization problem to a finite-dimensional quantization problem that can be solved numerically. Our approach achieves the sharp rate of the minimal quantization error and can be used to quantize the path space for Gaussian processes and also, for example, Lévy processes.

Harmonic analysis meets stationarity: A general framework for series expansions of special Gaussian processes

“…When the basis (e k ) k∈N chosen in the series expansion is not orthonormal, an alternative rate-optimal quantization method is presented in [Junglen and Luschgy(2010)]. The idea consists in truncating the series up to the optimal order m log N and to consider the finite-dimensional covariance operator K m of the truncation given by…”

“…Following the same argument as in [Junglen and Luschgy(2010)], if we set m log N and replace the process by a rate-optimal quantizer of m i=0…”