Bücher der Reihe Probability Theory and Stochastic Modelling

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 space¿a 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.

Stochastic geometry is a relatively new branch of mathematics. Although its predecessors such as geometric probability date back to the 18th century, the formal concept of a random set was developed in the beginning of the 1970s. Theory of Random Sets presents a state of the art treatment of the modern theory, but it does not neglect to recall and build on the foundations laid by Matheron and others, including the vast advances in stochastic geometry, probability theory, set-valued analysis, and statistical inference of the 1990s. The book is entirely self-contained, systematic and exhaustive, with the full proofs that are necessary to gain insight. It shows the various interdisciplinary relationships of random set theory within other parts of mathematics, and at the same time, fixes terminology and notation that are often varying in the current literature to establish it as a natural part of modern probability theory, and to provide a platform for future development.

This book offers a systematic and rigorous treatment of continuous-time Markov decision processes, covering both theory and possible applications to queueing systems, epidemiology, finance, and other fields.

This textbook provides a broad overview of the present state of insurance mathematics and some related topics in risk management, financial mathematics and probability.

The first edition of this single volume on the theory of probability has become a highly-praised standard reference for many areas of probability theory. Chapters from the first edition have been revised and corrected, and this edition contains four new chapters.

This book is focused on the recent developments on problems of probability model uncertainty by using the notion of nonlinear expectations and, in particular, sublinear expectations.

This book presents broadly applicable methods for the large deviation and moderate deviation analysis of discrete and continuous time stochastic systems. By characterizing a large deviation principle in terms of Laplace asymptotics, one converts the proof of large deviation limits into the convergence of variational representations.

The construction of a Dynamic Markov Bridge, a useful extension of Markov bridge theory, addresses several important questions concerning how financial markets function, among them: how the presence of an insider trader impacts market efficiency;

This book is intended to make recent results on the derivation of higher order numerical schemes for random ordinary differential equations (RODEs) available to a broader readership, and to familiarize readers with RODEs themselves as well as the closely associated theory of random dynamical systems.

Offering the first comprehensive treatment of the theory of random measures, this book has a very broad scope, ranging from basic properties of Poisson and related processes to the modern theories of convergence, stationarity, Palm measures, conditioning, and compensation.

It shows the various interdisciplinary relationships of random set theory within other parts of mathematics, and at the same time fixes terminology and notation that often vary in the literature, establishing it as a natural part of modern probability theory and providing a platform for future development.

A simplified approach to Malliavin calculus adapted to Poisson random measures is developed and applied in this book. Thanks to the theory of Dirichlet forms, the authors develop a mathematical tool for a quite general class of random Poisson measures and significantly simplify computations of Malliavin matrices of Poisson functionals.

With a Contribution by M. Fuhrman and G. Tessitore

The authors present a concise but complete exposition of the mathematical theory of stable convergence and give various applications in different areas of probability theory and mathematical statistics to illustrate the usefulness of this concept.

The focus of the present volume is stochastic optimization of dynamical systems in discrete time where - by concentrating on the role of information regarding optimization problems - it discusses the related discretization issues.

Applications of the described theory include Gibbs fields, spin glasses, polymer models, image analysis and random shapes.Limit theorems form the backbone of probability theory and statistical theory alike.

Probability theory on compact Lie groups deals with the interaction between "chance" and "symmetry," a beautiful area of mathematics of great interest in its own sake but which is now also finding increasing applications in statistics and engineering (particularly with respect to signal processing).

This book offers a systematic introduction to the optimal stochastic control theory via the dynamic programming principle, which is a powerful tool to analyze control problems. First we consider completely observable control problems with finite horizons.