An Introduction to the Modern Martingale Theory and Applications
An Analytic View
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Beschreibung
Martingale theory is a cornerstone of modern probability, offering a natural extension of the study of sums of independent random variables. Although its roots can be traced back to the work of Paul Lévy in 1937, it was Joseph L. Doob in the 1940s who formally developed the theory, culminating in his landmark book Stochastic Processes in 1953. Since then, martingale theory has evolved significantly, with deep contributions from mathematicians such as Donald L. Burkholder, Richard Gundy, and Burgess Davis, among others. This is what is now known as advanced martingale theory, which began with the publication of Burkholder’s seminal paper Martingale Transforms in 1966.
This book provides a comprehensive treatment of both classical and advanced martingale theory. It opens with a historical introduction, exploring foundational functions such as Rademacher, Haar, and Walsh functions, before delving into the core concepts of conditional probability. The classical theory, as developed by Doob, is meticulously presented, followed by an in-depth examination of modern advancements, including Burkholder’s inequalities, Burkholder-Davis-Gundy inequality, and their generalizations, as well as good-lambda inequalities. The final chapter showcases a wide range of applications, highlighting the theory’s profound impact on Banach space theory, harmonic analysis, and beyond.
Intended for graduate students and researchers in probability and analysis, this book serves as both an introduction and a reference, offering a clear and structured approach to a subject that continues to shape mathematical research and its applications.
Martingale theory is a cornerstone of modern probability, offering a natural extension of the study of sums of independent random variables. Although its roots can be traced back to the work of Paul Lévy in 1937, it was Joseph L. Doob in the 1940s who formally developed the theory, culminating in his landmark book Stochastic Processes in 1953. Since then, martingale theory has evolved significantly, with deep contributions from mathematicians such as Donald L. Burkholder, Richard Gundy, and Burgess Davis, among others. This is what is now known as advanced martingale theory, which began with the publication of Burkholder’s seminal paper Martingale Transforms in 1966.
This book provides a comprehensive treatment of both classical and advanced martingale theory. It opens with a historical introduction, exploring foundational functions such as Rademacher, Haar, and Walsh functions, before delving into the core concepts of conditional probability. The classical theory, as developed by Doob, is meticulously presented, followed by an in-depth examination of modern advancements, including Burkholder’s inequalities, Burkholder-Davis-Gundy inequality, and their generalizations, as well as good-lambda inequalities. The final chapter showcases a wide range of applications, highlighting the theory’s profound impact on Banach space theory, harmonic analysis, and beyond.
Intended for graduate students and researchers in probability and analysis, this book serves as both an introduction and a reference, offering a clear and structured approach to a subject that continues to shape mathematical research and its applications.
An accessible but deeply study of martingale theory with discrete parameter from an analytic point of view An unified treatment of the classical martingale theory and advanced topics using the notion of martingale transform A wide range of applications of discrete martingale theory into probability, analysis and functional analysis
Autor*in
Wilfredo Urbina-Romero
Themen in »An Introduction to the Modern Martingale Theory and Applications«
martingale with discrete parameter martingale transforms good lambda inequalities Burkholder-Davis-Gundy inequalities Haar and Walsh systems martingale limit theorems operators on martingales martingales and geometry in Banach spaces