Introduction to Malliavin Calculus

This textbook provides a comprehensive introduction to Malliavin calculus, covering applications in density formulas, probability laws, Brownian motion, and jump processes. It is accessible to non-experts and suitable for graduate students and researchers.

Format: Paperback / softback
Length: 246 pages
Publication date: 27 September 2018
Publisher: Cambridge University Press

This comprehensive textbook provides a concise yet thorough introduction to Malliavin calculus, a vibrant and influential field of research. It delves into diverse applications, encompassing density formulas, the regularity of probability laws, central and non-central limit theorems for Gaussian functionals, convergence of densities, and non-central limit theorems for the local time of Brownian motion. Additionally, the book offers a self-contained presentation of Brownian motion and stochastic calculus, along with Lévy processes and stochastic calculus for jump processes. Designed with accessibility in mind, it serves as an invaluable resource for both non-experts and graduate students seeking to enhance their proficiency in the fundamental techniques required for advanced study in this domain.

Introduction:
Malliavin calculus, a branch of mathematical analysis, has gained significant attention in recent years due to its wide range of applications in fields such as probability theory, statistics, finance, and physics. This textbook aims to provide a comprehensive and accessible introduction to Malliavin calculus, catering to both non-experts and graduate students.
Scope and Coverage:
The textbook covers a broad range of topics, including recent applications of Malliavin calculus. It begins by discussing density formulas, which are fundamental tools in the study of probability distributions. It then explores the regularity of probability laws, which play a crucial role in establishing the convergence of probability sequences. The book further discusses central and non-central limit theorems for Gaussian functionals, which provide insights into the behavior of random variables and processes. Additionally, it covers convergence of densities and non-central limit theorems for the local time of Brownian motion, a fundamental process in stochastic analysis.
Self-Contained Presentation:
In addition to the theoretical discussions, the textbook includes a self-contained presentation of Brownian motion and stochastic calculus. It provides a clear and concise explanation of these concepts, making them accessible to students with a basic background in mathematics. The book also introduces Lévy processes and stochastic calculus for jump processes, which are important tools in the study of complex systems and financial markets.
Accessible to Non-Experts:
The textbook is designed to be accessible to non-experts, with a clear and concise writing style that avoids technical jargon. It includes numerous examples and exercises to help students understand the concepts and apply them to real-world situations. The book also provides a comprehensive bibliography for further reading, allowing students to explore the topic in greater depth.
Applications:
Malliavin calculus has numerous applications in various fields. It is used in probability theory to study the behavior of random variables and processes, such as Brownian motion, stochastic differential equations, and martingales. In statistics, it is used to analyze the distribution of random variables and to develop statistical models. In finance, it is used to model risk and to develop financial instruments such as options and futures. In physics, it is used to study the behavior of particles in chaotic systems and to develop models for complex systems.
Conclusion:
This textbook offers a comprehensive and accessible introduction to Malliavin calculus, a field with significant potential for future research and applications. It provides a solid foundation in the core techniques necessary for further study in this area, making it an invaluable resource for both non-experts and graduate students. By exploring the recent applications of Malliavin calculus, the book demonstrates the power of this powerful tool in solving real-world problems and advancing our understanding of complex systems.

Weight: 344g
Dimension: 153 x 228 x 13 (mm)
ISBN-13: 9781107611986