A versatile technique for the optimal approximation of random processes by Functional Quantization

This paper presents the mathematical foundation of a novel and versatile technique for the approximation of random processes using the Functional Quantization concept. In this approach, random processes are represented by optimal quantizers; that is, finite collections of deterministic functions and corresponding probability masses that are carefully constructed to attain certain optimal properties. Although the computational cost of obtaining such optimal quantizers is not negligible, their use in applications is simple and relatively inexpensive. The paper has two objectives. First, we present an accessible overview of Functional Quantization theory. Second, we introduce a novel methodology to compute optimal quantizers, which is based on the classical Lloyd’s Method and Monte Carlo Simulation. For validation purposes, the proposed methodology is tested using two fundamental Gaussian random processes (Brownian motion and fractional Brownian motion) for which optimal quantizers are known. The results obtained compare very well with previous results reported in the literature. Then, to demonstrate the unique versatility of the methodology, the application to a non-Gaussian process is presented.