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Stochastic calculus and differential equations for physics and finance. (English) Zbl 1321.60002

Cambridge: Cambridge University Press (ISBN 978-0-521-76340-0/hbk; 978-1-139-01946-0/ebook). xi, 206 p. (2013).
This book presents stochastic calculus used for the description of stochastic processes. These are applied, e.g., in physics or finance and often used by practitioners. Stochastic processes in this book are described by Itô calculus and Fokker-Planck equations in parallel, where the author focuses on nonstationary processes and general martingales.
The book contains 15 chapters:
1. Random variables and probability distributions
2. Martingales, Markov, and nonstationarity
3. Stochastic calculus
4. Itō processes and Fokker-Planck equations
5. Self-similar Itō processes
6. Fractional Brownian motion
7. Kolmogorov’s PDEs and Chapman-Kolmogorov equation
8. Non-Markov Itō processes
9. Black-Scholes, martingales, and Feynman-Kac
10. Stochastic calculus with martingales
11. Statistical physics and finance
12. Introduction to new financial economics
13. Statistical ensembles and time-series analysis
14. Econometrics
15. Semimartingales
Each chapter is divided into several sections and subsections and ends with some exercises. Alas, there are no solutions at the end of the book.
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.

MSC:

60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G18 Self-similar stochastic processes
60G22 Fractional processes, including fractional Brownian motion
60H30 Applications of stochastic analysis (to PDEs, etc.)
60G44 Martingales with continuous parameter
60G48 Generalizations of martingales
60J60 Diffusion processes
35Q84 Fokker-Planck equations
60K35 Interacting random processes; statistical mechanics type models; percolation theory
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P10 Applications of statistics to biology and medical sciences; meta analysis
91G80 Financial applications of other theories
91G70 Statistical methods; risk measures
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