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Mathematical methods for financial markets. (English) Zbl 1205.91003

Springer Finance. London: Springer (ISBN 978-1-85233-376-8/hbk; 978-1-4471-2524-2/pbk; 978-1-84628-737-4/ebook). xxv, 732 p. (2009).
This is another great book written by leading scientists in both stochastic analysis and stochastic financial modelling. The goal of the authors is to present the financial methodology and the relevant tools from mathematical stochastics. Fundamental financial concepts such as arbitrage opportunities, contingent claims, option pricing, strategies/portfolio policies, default risk, etc., are introduced and well mixed with fundamental objects such as Brownian motion, diffusion processes, and Lévy processes. The reader obviously needs more than a basic knowledge in probability and stochastic processes in order to go successfully through these topics at advanced level, but also to understand correctly the main ideas and concepts used to build and analyse stochastic financial models.
In a natural way, the material is divided into two parts. The first part deals with continuous path stochastic processes and a variety of continuous stochastic financial models. The second part is based on discontinuous stochastic processes and treats models with discontinuities.
I am not going to give a detailed description of what is in the book. I just suggest, take the book and start reading. Then \(\dots\) you do not stop for hours \(\dots\)
This book is a panorama of contemporary stochastics and finance. We find comprehensive theory applied convincingly to the analysis of sophisticated stochastic financial models. It is worth mentioning that a large number of new and nontrivial problems have arisen from the area of finance. The solutions of these problems needed new ideas and techniques. A huge progress was made over the last about 35 years. On one hand, we have today the wonderful area of mathematics called stochastic analysis. On the other hand, mathematical finance itself reached a higher level thus becoming one of the not easy, but still attractive, areas of applications of stochastics.
The book is well structured and carefully written. The text is smooth and clear. All notions, concepts, results, corollaries, illustrations, etc. are well presented. For many results complete proofs are given, and if not, appropriate references indicated. There is a user’s guide, authors, symbols and subject indices. Extremely useful is the extensive list of references containing 878 items, papers and books. A little strange, but missing from the list are the classical works by A. N. Kolmogorov, A. Ya. Khintchine and J. L. Doob. Perhaps the authors relied on the fact that everybody knows this. Still, it would be better to include these works in the list and make appropriate references in the text, as is done for many other contributors.
This voluminous book should be read, used and referred to on any occasion. For the reviewer, the book is a source of real intellectual pleasure and inspiration for further work. The book will be useful for a wide audience, from graduate and postgraduate students to researchers in stochastics and finance, as well as to applied scientists in other areas.

MSC:

91-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
91Gxx Actuarial science and mathematical finance
60H30 Applications of stochastic analysis (to PDEs, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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