Information Theory, Inference, and Learning Algorithms

The fundamental step in a systematic development of analytical methods for Markov processes, particularly, for processes with stationary and independent increments (PSI), was taken by Kolmogorov (1931). Since 1990, many authors have referred to PSI as the theory of Lévy processes. After a period of intense development in 1930–1980, applications in mathematical finance have motivated more interest in Lévy processes (see Cont and Tankov 2004; Schoutens 2003). There is also a persistent interest by the physical community in the theory of Lévy processes, also called “Lévy flights” (see Shlesinger, Zaslavsky, and Frisch 1995). Both finanacial and physical motivations come from the fact that models with underlying Lévy processes can be (at least, potentially) more accurately fitted to real data than traditional models with underlying Brownian motion (but, still keeping the assumption on independent increments, which often simplifies calculations). This book presents an interplay between the classical theory of general Lévy processes described by Skorohod (1991), Bertoin (1996), Sato (2003), and modern stochastic analysis as presented by Liptser and Shiryayev (1989), Protter (2004), and others. Most applications in finance and physics do not require the full power of stochastic analysis, which allows us to consider discrete and continuous time models in one framework. I think the author made a wise decision to limit attention to the aspects of stochastic analysis related to stochastically continuous Lévy processes. This approach allows one to economically describe basic tools of stochastic analysis such as “Martingales, Stopping Times, and Random Measures” (Chap. 2); “Stochastic Integration” (Chap. 4); “Exponential Martingales, Change of Measure and Finance Applications” (Chap. 5); and “Stochastic Differential Equations” (Chap. 6). Chapter 1, “Lévy Processes,” contains a nice review of measure and probability including the detailed discussion of the key notions of infinite divisibility and Lévy processes. Chapter 3, “Markov Processes, Semigroups, and Generators,” is more analytical than the rest of the book and could be skipped by “readers with specific interest in finance,” as the author recommends in his preface. Actually, aspects of mathematical finance are outlined only briefly in Sections 5.4. There is no discussion of statistical aspects of fitting Lévy models, and it seems that this topic has not yet been treated in textbooks. The background of the readers should be, at least at the level of graduate students, a solid knowledge of probability, Fourier transforms, and, for the readers of Chapter 3, a basic knowledge of linear operators in Banach spaces (although Appendix 3.8 contains some key results on this topic). Most of the exposition is clearly presented and contains useful exercises (without solutions), notes, and suggestions for further reading. My only complaint is that some numberings of references in the range 100–312 are incorrect (although in most cases this can be easily recognized and corrected). The author has promised to post corrections on his website. I would recommend this book as a reference textbook for advanced courses like stochastic modeling or stochastic calculus in finance.