Marginal and Functional Quantization of Stochastic Processes

Vector Quantization, a pioneering discretization method based on nearest neighbor search, emerged in the 1950s primarily in signal processing, electrical engineering, and information theory. Later in the 1960s, it evolved into an automatic classification technique for generating prototypes of extensive datasets. In modern terms, it can be recognized as a seminal contribution to unsupervised learning through the k-means clustering algorithm in data science.

In contrast, Functional Quantization, a more recent area of study dating back to the early 2000s, focuses on the quantization of continuous-time stochastic processes viewed as random vectors in Banach function spaces. This book distinguishes itself by delving into the quantization of random vectors with values in a Banach spacea unique feature of its content.

Its main objectives are twofold: first, to offer a comprehensive and cohesive overview of the latest developments as well as several new results in optimal quantization theory, spanning both finite and infinite dimensions, building upon the advancements detailed in Graf and Luschgy's Lecture Notes volume. Secondly, it serves to demonstrate how optimal quantization can be employed as a space discretization method within probability theory and numerical probability, particularly in fields like quantitative finance. The main applications to numerical probability are the controlled approximation of regular and conditional expectations by quantization-based cubature formulas, with applications to time-space discretization of Markov processes, typically Brownian diffusions, by quantization trees.

While primarily catering to mathematicians specializing in probability theory and numerical probability, this monograph also holds relevance for data scientists, electrical engineers involved in data transmission, and professionals in economics and logistics who are intrigued by optimal allocation problems.

State of the art monograph on the quantization of stochastic processes Contains applications to numerical probability and mathematical finance Presents deep connections between optimal quantization and information theory

Auteur
Harald Luschgy studied mathematics, physics and mathematical logic at the universities of Bonn and Münster. He received his doctorate in mathematics in 1976 from the University of Münster. He held visiting positions at the Universities of Hamburg, Bayreuth, Dortmund, Oldenburg, Passau and Wien and was a recipient of a Heisenberg grant from the DFG. Since 1995 he is Professor of Mathematics at the University of Trier where he teaches probability and mathematical statistics. He is the author of 3 books on probability theory.

Gilles Pagès studied at Sorbonne Université, where he is Professor since 2001, specializing in probability theory, numerical probability and mathematical finance. He was the director of the Laboratoire de Probabilités, Statistique & Modélisation from 2009 to 2014, and has been the head of the Master 2 Probabilités & Finance (also known as the "Master El Karoui") since 2001. He has published over 120 research articles and is also theauthor of several graduate and undergraduate textbooks in statistics, applied and numerical probability and mathematical finance.

Contenu

Preface.- Notation Index.- Part I. Basics and Marginal Quantization.- 1. Optimal and Stationary Quantizers.- 2. The Finite-Dimensional Setting I.- 3. The Finite-Dimensional Setting II.- Part II. Functional Quantization.- 4. Functional Quantization, Small Ball Probabilities, Metric Entropy and Series Expansions for Gaussian Processes.- 5. Spectral Methods for Gaussian Processes.- 6. Geometry of Optimal and Rate-Optimal Quantizers for Gaussian Processes.- 7. Mean Regular Processes.- Part III. Algorithmic Aspects and Applications:- 8. Optimal Quantization from the Numerical Side (Static).- 9. Applications: Quantization-Based Cubature Formulas.- 10. Quantization-Based Numerical Schemes.- Appendices.- A. Radon Random Vectors, Stochastic Processes and Inequalities.- B. Miscellany.- References.- Index.

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