Book Reviews of Chicago), a student of Yuri Manin in the 1970s, and ally Manin accepted. Another co-director of the Max Pierre Deligne (Princeton), who visited Manin in Mos- Planck Institute of Mathematics, Don Zagier, talks in the cow in 1972. “Manin was not in danger,” says Deligne in film very warmly about Manin’s style, about his greatest the film, “but he suffered because his students suffered talent: how Manin was often able to see unexpected and … their possibilities were very limited although they surprising connections between two different branches were very good.” One of them was Michael Tsfasman, of mathematics where no connections had appeared be- who had started his studies at Moscow State Universi- fore – and establishing them successfully with creative ty in 1971. He reports in the film how two thirds of the ideas and persistency. students from Highschool No. 2, a top level college in The last part of the movie is about Manin’s childhood; Moscow, were not given the possibility to study because it shows where and how he grew up. When Manin was they were Jews, not communists or simply independent five years old, he lost his father in World War II. His edu- thinkers. Michael Tsfasman says he only got a chance cation started at School No. 7 in Simferopol, the capital to study at Moscow University because he was selected of the Crimea; his first contact with another language was for the Russian team at the International Mathemati- when he read Gulliver’s Travels in English. Manin takes cal Olympiad (IMO); IMO participants were allowed to the viewers of the film back to his hometown Simferopol. enter Russian top universities without exams and other We see Manin visiting the two houses of his childhood selection processes. after decades of absence. Manin’s life changed a lot after the fall of the Iron Then the title “Late Style” is explained at last. “Late Curtain when he was allowed to travel again; for many style I think quite well explains this emotional atmos- years he had received invitations to congresses and meet- phere of returning to one’s remote past,” says Manin. “It’s ings without being able to go to them. Friedrich Hirze- kind of you want to connect the beginning and the end of bruch, professor of mathematics at Bonn University and your life … you want to see some kind of entirety …” the founder of the Max Planck Institute for Mathematics in Bonn, had invited Russian Mathematicians to Bonn every year since 1957, among them Yuri Manin. And Thomas Vogt [[email protected]] stud- Manin went to Bonn with four other mathematicians, ied geology, German literature and science on his way back to Russia from Paris in 1967. He gave journalism in Berlin. He has written for a a lecture about a problem in algebraic geometry at one Berlin daily newspaper and has worked as of Hirzebruch’s Arbeitstagungen. Hirzebruch reports in a press officer for the Helmholtz Associa- the film that he invited Manin again and again during tion and the Leibniz Association. In 2008 the Cold War. After the Iron Curtain had finally fallen he provided content (print and online) for Photo taken by Manin accepted some invitations from Western univer- Michael Ebner Germany’s “Year of Mathematics” – a na- sities and – enjoying the work abroad – resigned from tional campaign to present mathematics to his professorship at Moscow State University. One day society at large. Since then he has been press officer of the Hirzebruch – close to his own retirement – visited Manin German Mathematical Society (DMV) and has provided at Harvard and asked him to become his successor at the news on mathematics for the media and the general public Max Planck Institute for Mathematics in Bonn; eventu- via DMV’s mathematics media office.

tion between pure mathematics and a domain of human knowledge in need of a rational and quantitative formu- lation of natural questions associated to it. Since Galil- ean times and up to the last century, this symbiosis link- Ole E. Barndorff-Nielsen ing mathematics to other disciplines took place mainly Albert Shiryaev in the arenas of natural sciences and engineering. The important development during the last century of the Change of Time and mathematics of randomness (probability theory, theory Change of Measure of stochastic processes, statistical modelling, to give a few names) has created an array of powerful tools capable World Scientific, 2010 of handling the complex phenomena and uncertain out- 305 pages comes usually studied within the realm of social sciences. ISBN-13 978-981-4324-47-2 Indeed, experience has shown that a probabilistic treat- ment of many problems arising in, for example, demogra- Reviewer: Juan-Pablo Ortega phy, economics, epidemiology and finance is much more pertinent than the one coming from classical determinis- tic methods. Even though this conceptual leap took some Mathematical finance is one of the most recent examples time to be assimilated, its result seems to be by now an in the long list of successful instances of cross-fertilisa- established certitude.

EMS Newsletter December 2012 57 Book Reviews

This book takes two major ideas at the core of math- a simpler one by using in this case an equivalent prob- ematical finance, namely the use of equivalent (martin- ability measure; more specifically, the procedure aims at gale) measures and the equivalent (change of time and constructing a new measure that is absolutely continuous stochastic volatility) representations of a stochastic proc- with respect to the original one, such that the law of the ess, and uses them as a motivation to describe a variety of stochastic process under study “seen with the eyes” of topics in stochastic analysis. The result is a beautiful and this new measure coincides with the one coming from a well-articulated monograph, full of information, where simpler process like Brownian motion. There are several the interplay between deep mathematical ideas and ex- constructions available in the literature that serve this tremely explicit and applied financial problems and their purpose. The book examines two of them: the Girsanov solutions is generously exemplified. Theorem, of much use in mathematical finance, and the The change of time and the stochastic integral rep- Esscher Transform, introduced initially in the context of resentation problems aim at rewriting a given stochastic actuarial sciences. Both procedures are constructive and process in terms of a simpler one (it can be a Brownian provide new measures under which the stochastic proc- motion or, in general, a semimartingale) via a stochas- ess we are interested in becomes a martingale. tic change of time and stochastic integration, respec- Again, the impact of these captivating mathematical tively. The first approach is analysed in Chapter 1 of the constructions goes beyond a pure mathematical inter- book, in which all the necessary concepts having to do est and has far-reaching implications in mathematical fi- with stopping times and random changes of time are de- nance that are spelled out in Chapters 10 and 11. Indeed, scribed, as well as the Dambis-Dubins-Schwarz Theorem the importance of the existence of an equivalent martin- that explains how any continuous martingale can be ob- gale measure for a (discounted) price process is captured tained out of a Brownian motion via a random change of in the so-called First Fundamental Theorem of Arbitrage time. The integral representation problem is tackled in Theory that establishes the equivalence between the Chapters 2 and 3 and gives the authors the opportunity availability of these measures and the absence of arbi- to construct in a reduced number of pages a delightful trage opportunities in the market under study. This link presentation of the most relevant results in relation with is of paramount importance because many constructions stochastic integration and stochastic differential equa- in finance have the nonexistence of arbitrage opportuni- tions. ties, i.e. the impossibility of guaranteed profits without The existence and uniqueness results that are pre- risk taking, as a fundamental hypothesis; this assumption sented in those pages and that solve the described repre- is very plausible in highly liquid and efficient markets sentation problems are not just deep and beautiful math- where results coming from mathematical finance are ematical theorems but they also constitute key results in regularly applied. the understanding of important questions in mathemati- Another fascinating link between two of the core ide- cal finance. This interplay is presented mainly in Chap- as in the book, namely the existence of martingale meas- ters 10 and 11. Indeed, the change of time representation ures and of integral representations and their application provides mathematical legitimacy to a common inter- in mathematical finance, is provided by the Second Fun- pretation of volatility among finance practitioners that damental Theorem of Arbitrage Theory, also examined consists of visualising moments of high fluctuation of the in Chapter 10. When we are in the presence of an un- prices with an acceleration of the so-called operational derlying asset whose dynamics is such that there exists or business time. Moreover, the existence of an integral an integral representation for any contingent product, representation for a process with a predictable integrand we say that the corresponding market is complete; ac- amounts in financial language to the availability of a self- cording to what we said above, in complete markets any financing trading strategy that replicates (or hedges, in derivative product can be perfectly hedged/replicated. financial jargon) a derivative product whose underlying The Second Fundamental Theorem of Arbitrage Theory asset has a dynamical behaviour described by the process establishes the equivalence between the completeness of in question. This makes of this mathematical construc- a market and the uniqueness of an equivalent martingale tion, together with the notion of martingale or risk-neu- measure. tral measure that we will review later on, a main building The book does not restrict itself to stating the basic block of the no-arbitrage pricing and hedging theory of concepts underlying mathematical finance and their con- derivative securities. Indeed, when the underlying asset nection with stochastic calculus that are available in so is described by a lognormal process (also called geomet- many other monographs. Indeed, it manages to present in ric Brownian motion), it gives rise to the Black-Scholes- a reduced number of pages most models that are used by Merton formulas for the price and the hedges of Euro- practitioners when trying to solve the deficiencies of the pean style options, for which Robert Merton and Myron Black-Scholes-Merton model that have been profusely Scholes received in 1997 the Sveriges Riksbank Prize in documented over the years. Chapter 9 opens this discus- Economic Sciences in Memory of Alfred Nobel (usually sion in the discrete time setup by introducing the ARMA/ referred to as the Nobel Prize in Economics) and that GARCH parametric family of time series models driven made viable the massive and industrial scale trading of by a variety of different innovations (Inverse Gaussian, these securities. Generalised Inverse Gaussian, Generalised Hyperbolic, Changes of measure are treated in Chapters 6 and 7. etc.) whose use aims at appropriately modelling stylised The goal is again reducing a given stochastic process to features of time series of financial returns that are em-

58 EMS Newsletter December 2012 Book Reviews pirically observed, like, for example, leptokurtosis (the the volume. The style of presentation and the contents likelihood of extreme events is higher than the one asso- makes this enjoyable book ideal for a reader interested ciated to the Gaussian distribution), volatility clustering in going deeper in understanding the area or in having (moments of high volatility tend to accumulate in time) a panoramic view of it. For that category of readers, this or asymmetry (bad market days create more volatility book will certainly provide an opportunity to learn ad- than good ones). The analogue of this discussion in con- ditional results and to establish beautiful connections be- tinuous time is carried out in Chapter 12, which contains tween them, all of it elegantly illustrated with one of the a pleasantly readable presentation of stochastic volatility most relevant and impressive applications of this math- and Lévy processes introduced in the light of the integral ematical field. and change of time representations, respectively, at the heart of the book. Even though the book includes self-contained ac- Juan-Pablo Ortega [juan-pablo.ortega@ counts of a variety of topics in stochastic analysis, like univ-fcomte.fr] is a Chargé de Recherche stochastic integration, stochastic differential equations, at the French Centre National de la Recher- equivalent martingale measures, Lévy processes and che Scientifique. He has a first degree in stochastic volatility models, to name a few, it is not writ- physics from the Universidad de Zaragoza ten as a textbook and is not well suited for a reader who (Spain) and a MA and PhD in mathemat- would like to use it as an entry door to the field; the ics from the University of California, Santa proofs for most results are not included, the book is not Cruz (USA). He was a recipient of the 2000 edition of the linear and, despite the presence of numerous examples, Ferran Sunyer i Balaguer Prize (with Tudor S. Ratiu) and the book contains no exercises. Like most of the titles in was an invited speaker at the ECM, Barcelona, 2000. His World Scientific’s Advanced Series on Statistical Science research focuses on geometric mechanics and, more re- and Applied Probability, the reader is expected to have cently, statistical modelling, financial econometrics and some acquaintance with the various subjects covered by mathematical finance.

Andrzej Skowro´nski (Toru´n, Poland) Kunio Yamagata (Tokyo, Japan) Frobenius Algebras I Basic Representation Theory (EMS Textbooks in Mathematics)

ISBN 978-3-03719-102-6 2011. 661 pages Hardcover. 16.5 x 23.5 cm 58.00 Euro This is the first of two volumes which will provide a comprehensive intro- duction to the modern representation theory of Frobenius algebras. The first part of the book serves as a general introduction to basic results and techniques of the modern representation theory of finite dimensional associative algebras over fields. The second part is devoted to fundamen- tal classical and recent results concerning the Frobenius algebras and their module categories. Moreover, the prominent classes of Frobenius algebras, the Hecke algebras of Coxeter groups and the finite dimensional Hopf algebras over fields are exhibited. This volume is self-contained and the only prerequisite is a basic know- ledge of linear algebra. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well mathematicians working in other fields.

European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich, Switzerland [email protected] | www.ems-ph.org

EMS Newsletter December 2012 59