Stochastic Calculus and Differential Equations for Physics and Finance

Stochastic calculus provides a powerful description of a specific class of stochastic processes in physics and finance. However, many econophysicists struggle to understand it. This book presents the subject simply and systematically, giving graduate students and practitioners a better understanding and enabling them to apply the methods in practice. The book develops Ito calculus and Fokker–Planck equations as parallel approaches to stochastic processes, using those methods in a unified way. The focus is on nonstationary processes, and statistical ensembles are emphasized in time series analysis. Stochastic calculus is developed using general martingales. Scaling and fat tails are presented via diffusive models. Fractional Brownian motion is thoroughly analyzed and contrasted with Ito processes. The Chapman–Kolmogorov and Fokker–Planck equations are shown in theory and by example to be more general than a Markov process. The book also presents new ideas in financial economics and a critical survey of econometrics.

Table of Contents:
1. Random variables and probability distributions; 2. Martingales, Markov, and nonstationarity; 3. Stochastic calculus; 4. Ito processes and Fokker–Planck equations; 5. Selfsimilar Ito processes; 6. Fractional Brownian motion; 7. Kolmogorov's PDEs and Chapman–Kolmogorov; 8. Non Markov Ito processes; 9. Black–Scholes, martingales, and Feynman–Katz; 10. Stochastic calculus with martingales; 11. Statistical physics and finance, a brief history of both; 12. Introduction to new financial economics; 13. Statistical ensembles and time series analysis; 14. Econometrics; 15. Semimartingales; References; Index.

About the Author :
Joseph L. McCauley is Professor of Physics at the University of Houston. During his career he has contributed to several fields, including statistical physics, superfluids, nonlinear dynamics, cosmology, econophysics, economics and finance theory.

Review :
'This new book by Joe McCauley is a most welcome and innovative contribution to the important field of mathematical finance theory. It presents a unified, rigorous and comprehensive framework of the dynamics of stochastic calculus that should underpin the mathematics of finance. The book's welcome focus on nonstationary processes and statistical ensembles in time series analysis, developing, inter alia, the Ito calculus and the Fokker-Planck equations as parallel approaches to stochastic processes, will make this the classic and indispensable textbook for any serious graduate courses in applied finance theory - not just for economists, but also for physicists interested in studying the world of finance.' Stefano Zambelli, Algorithmic Social Sciences Research Unit (ASSRU), University of Trento 'Joe McCauley's book fills a gap in the current literature by providing a clear and readable introduction to stochastic calculus and stochastic differential equations for physicists. His book is written in a style that will not deter physicists and other applied scientists from learning these important topics.' Enrico Scalas, University of East Piedmont 'Joe McCauley continues the tradition he has established for clarity of exposition, at the frontiers of research, in fields whose practitioners are in sore need of it. This book is an outstanding contribution to the mathematical needs of able financial theorists who are also interested in underpinning empirical work in sound mathematical theory. I do not think there is any other book that undertakes the difficult tasks McCauley has undertaken in this impeccably well crafted, yet deep and rigorous, book.' K. Vela Velupillai, The New School for Social Research 'This book represents a rare and successful effort to provide a unified treatment of continuous time stochastic processes derived from both finance and physics. It constitutes an effective guide for physicists trying to understand the models of modern finance and for students of mathematical finance looking for methods neglected by the traditional books on the subject. The intuitive presentation of models in terms of physical and financial phenomena and the constant attention to their practical applicability make this book extremely useful also for those already knowledgeable about the subject.' Giulio Bottazzi, Scuola Superiore Sant'Anna '… [this] book contains a wealth of useful information and most importantly helpful details. … [it] is further complemented by adding a discussion of historical developments of statistical physics and financial theory, taking into account their similarities and differences. … Stochastic Calculus and Differential Equations for Physics and Finance is a recommended title that both the physicist and the mathematician will find of interest.' Jesus Rogel-Salazar, Contemporary Physics 'The book gives a good introduction to stochastic calculus and is a helpful supplement to other well-known books on this topic. It may be recommended to graduate students in finance, stochastic analysis and physics, as well as practitioners of this field.' Oliver Janke, Zentralblatt MATH