具体描述
Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way. This fully revised edition now features a number of new topics. These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs.
Lévy Processes and Stochastic Calculus: A Deep Dive into the Foundations of Random Phenomena This comprehensive treatise, Lévy Processes and Stochastic Calculus, is an authoritative exploration into the mathematical frameworks underpinning the study of random phenomena. It delves into the intricate world of Lévy processes, a fundamental class of stochastic processes that exhibit independent and stationary increments, and their profound connections to the robust machinery of stochastic calculus. This work is meticulously crafted for advanced students, researchers, and anyone seeking a rigorous understanding of the probabilistic and analytical tools essential for modeling complex systems driven by randomness. The book begins by laying a solid groundwork in the theory of stochastic processes, introducing fundamental concepts such as martingales, Markov processes, and Brownian motion. It then transitions to the core of the matter: Lévy processes. The authors provide a detailed exposition of their defining properties, including continuity, càdlàg paths, and the crucial decomposition theorems that reveal their underlying structure. Key classes of Lévy processes are examined in depth, such as Poisson processes, compound Poisson processes, and stable processes, highlighting their distinct characteristics and applications. A significant portion of the book is dedicated to the development and application of stochastic calculus, the calculus of random functions. This section meticulously introduces the Itô integral, a cornerstone for integrating with respect to stochastic processes, and its fundamental properties. The celebrated Itô formula, a differential calculus for stochastic processes, is presented with rigorous proofs and numerous examples, demonstrating its power in transforming and analyzing stochastic differential equations. The authors also explore concepts such as stochastic differentials, quadratic variation, and the construction of stochastic integrals for general Lévy processes, extending the reach of stochastic calculus beyond Brownian motion. The interplay between Lévy processes and stochastic calculus is a central theme, with the book showcasing how these two areas mutually inform and enrich one another. Readers will discover how stochastic calculus provides the essential tools for analyzing the behavior of Lévy processes, solving stochastic differential equations driven by these processes, and deriving important properties of their distributions and trajectories. Conversely, the understanding of Lévy processes is crucial for developing and applying stochastic calculus in more general and realistic settings. Furthermore, Lévy Processes and Stochastic Calculus delves into advanced topics that showcase the breadth and depth of this field. This includes discussions on: The connection to partial differential equations: Exploring the Feynman-Kac formula and its role in linking parabolic partial differential equations with expectations of functionals of stochastic processes. Applications in finance and economics: Demonstrating how Lévy processes are used to model asset prices, option pricing, and risk management, capturing features such as jumps and heavy tails not present in standard Brownian motion models. Connections to Fourier analysis and spectral theory: Revealing the harmonic analysis aspects of Lévy processes and their characteristic functions. Infinitely divisible distributions: Understanding the broader class of distributions to which Lévy processes are intimately related. Specific classes of Lévy processes: Including detailed treatments of Gaussian processes, compound Poisson processes, and stable processes, with discussions on their paths, moments, and applications. Stochastic integration with respect to general Lévy processes: Expanding the domain of stochastic calculus to encompass a wider array of random phenomena. The book is rich with examples, exercises, and meticulously presented proofs, designed to foster a deep understanding of the underlying theory. The authors have taken great care to ensure that the material progresses logically, building from foundational concepts to more sophisticated ideas. Whether the reader is interested in theoretical advancements or practical applications, this volume offers an indispensable resource for navigating the sophisticated landscape of modern probability theory and its applications. It is an essential read for anyone seeking to master the probabilistic tools that drive innovation in fields ranging from finance and physics to biology and engineering.
