Best Books on Martingale Theory

“Martingales and Markov Chains: Solved Exercises and Elements of Theory” Book Review: This book focuses on applied probability and provides access to the study of more complex situations encountered in stochastic processes. It includes exercises that deepen the reader’s understanding of the theory, enabling them to tackle real-world problems. The book offers a variety of solved exercises with detailed solutions, ranging in difficulty from elementary to advanced, encouraging the reader’s initiative. Each section also presents a set of problems that serve as gateways to specific applications. Suitable for senior undergraduate and graduate level students, the book covers various approaches to problem-solving and addresses topics such as conditional expectations, martingales, stochastic processes, and Markov chains. Each section concludes with exercises and solutions.

“Measures, Integrals and Martingales” Book Review: This book provides a comprehensive overview of key theories and perspectives in a concise and accessible manner. Divided into two parts, the first part introduces basic theory including measures, integrals, convergence theorems, Lp-spaces, and multiple integrals. The second part delves into advanced topics such as martingales, Hardy-Littlewood maximal functions, Fourier series, Radon-Nikodym theorem, and Jacobi’s generalized transformation theorem. Suitable for readers with a background in undergraduate calculus and introductory rigorous analysis, the book is ideal for both lectures and self-study. It features numerous illustrations, exercises, and updated examples that illustrate key ideas across various topics, including counting, σ-algebras, measures, martingales, and conditional expectations. Additionally, it covers inner product spaces, Hilbert space, orthonormal systems, and their convergence properties.

“Martingale Limit Theory and Its Application” Book Review: This book explores the Martingale Limit Theory and its practical applications. It discusses the behavior and properties of martingales, which are mathematical concepts widely used in probability theory. The book examines the connections between martingales and processes related to Brownian motion. It covers topics such as inequalities, laws of large numbers, the central limit theorem, and estimation of parameters from stochastic processes. With its appendices and miscellaneous applications, this book is suitable for mathematicians, advanced students, and professors in the fields of higher mathematics or statistics who want to deepen their understanding of martingales and their applications.

“Probability Theory: Independence, Interchangeability, Martingales” Book Review: This book presents the fundamental theorems of the subject and builds upon the foundations of measure theory. It is unique because it combines both measure theory and probability theory. The book is intended for graduate students who are already familiar with measure theory. The text is clear and covers important topics such as different classes of sets, measures, and probability spaces. It also includes discussions on various concepts like independence, integration in a probability space, and sums of independent random variables. Advanced topics like martingales, distribution functions, and central limit theorems are also explored in this book.