Ralf Korn and Elke Korn, University of Kaiserslautern, Germany
Options Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics
Description
Understanding and working with the current models of financial markets requires a sound knowledge of the mathematical tools and ideas from which they are built. Banks and financial houses all over the world recognize this and are avidly recruiting mathematicians, physicists, and other scientists with these skills.
The mathematics involved in modern finance springs from the heart of probability and analysis: the It・calculus, stochastic control, differential equations, martingales, and so on. The authors give rigorous treatments of these topics, while always keeping the applicatio s in mind. Thus, the way in which the mathematics is developed is governed by the way it will be used, rather than by the goal of optimal generality. Indeed, most of purely mathematical topics are treated in extended "excursions" from the applications into the theory. Thus, with the main topic of financial modelling and optimization in view, the reader also obtains a self-contained and complete introduction to the underlying mathematics.
This book is specifically designed as a graduate textbook. It could be used for the second part of a course in probability theory, as it includes as applied introduction to the basics of stochastic processes (martingales and Brownian motion) and stochastic calculus. It would also be suitable for a course in continuous-time finance that assumes familiarity with stochastic processes.
The prerequisites are basic probability theory and calculus. Some background in stochastic processes would be useful, but not essential.
Contents
The mean-variance approach in a one-period model
The continuous-time market model
Option pricing
Pricing of exotic options and numerical algorithms
Optimal portfolios
Bibliography
Index
Details:
Series: Graduate Studies in Mathematics, Publication Year: 2001
ISBN: 0-8218-2123-7
Paging: approximately 272 pp.
Binding: Hardcover
Minoru Wakimoto, Kyushu University, Fukuoka, Japan
Infinite-Dimensional Lie Algebras
Iwanami Series in Modern Mathematics
Description
This volume begins with an introduction to the structure of finite-dimensional simple Lie algebras, including the representation of ${\widehat {\mathfrak \sl}}(2, {\mathbb C})$, root systems, the Cartan matrix, and a Dynkin diagram of a finite-dimensional simple Lie algebra. Continuing on, the main subjects of the book are the structure (real and imaginary root systems) of and the character formula for Kac-Moody superalge ras, which is explained in a very general setting. Only elementary linear algebra and group theory are assumed. Also covered is modular property and asymptotic behavior of integrable characters of affine Lie algebras.
The exposition is self-contained and includes examples. The book can be used in a graduate-level course on the topic.
Contents
Opening
Structures and representations of BKM (super-)algebras
Affine Lie algebras
Modular transformations of characters of affine Lie algebras
Fusion algebras
In lieu of postscript-Virasoro algebra
Further developments
Bibliography
Index
Details:
Series: Translations of Mathematical Monographs,
Subseries: Iwanami Series in Modern Mathematics
Publication Year: 2001
ISBN: 0-8218-2654-9
Paging: approximately 320 pp.
Binding: Softcover
Bojko Bakalov, University of California, Berkeley, CA,
and Alexander Kirillov, Jr., SUNY at Stony Brook, NY
Lectures on Tensor Categories and Modular Functors
Description
This book gives an exposition of the relations among the following three topics: monoidal tensor categories (such as a category of representations of a quantum group), 3-dimensional topological quantum field theory, and 2-dimensional modular functors (which naturally arise in 2-dimensional conformal field theory). The
following examples are discussed in detail: the category of representations of a quantum group at a root of unity and the Wess-Zumino-Witten modular functor.
The idea that these topics are related first appeared in the physics literature in the study of quantum field theory. Pioneering works of Witten and Moore-Seiberg triggered an avalanche of papers, both physical and mathematical, exploring various aspects of these relations. Upon preparing to lecture on the topic at MIT,
however, the authors discovered that the existing literature was difficult and that there were gaps to fill.
The text is wholly expository and finely succinct. It gathers results, fills existing gaps, and simplifies some proofs. The book makes an important addition to the existing literature on the topic. It would be suitable as a course text at the advanced-graduate level.
Contents
Braided tensor categories
Ribbon categories
Modular tensor categories
3-dimensional topological quantum field theory
Modular functors
Moduli spaces and complex modular functors
Wess-Zumino-Witten model
Bibliography
Index
Index of notation
Details:
Series: University Lecture Series, Volume: 21
Publication Year: 2001
ISBN: 0-8218-2686-7
Paging: approximately 222 pp.
Binding: Softcover
Valery B. Nevzorov, Saint Petersburg State University, St. Petersburg, Russia
Records: Mathematical Theory
Description
This volume is based on a course of lectures delivered at the St. Petersburg State University (Russia) and at Ohio State University (Columbus). It is intended as a textbook for graduate students and postdocs.
The book presents the theory of records and some information on order statistics. Also included are exercises illustrating the examples and developing the ideas.
The past 20 years has seen tremendous progress in the topic, giving forth a large number of new models that reflect the dynamics of records in a wide range of areas. This volume presents systematic main results with a special emphasis on non-classical record schemes. The material is presented in a comprehensive style succinctly outlining the current state of the theory. The work is geared toward statisticians, actuarians, engineers, hydrologists, meteorologists, and sports and market analysts.
Contents
Introduction
Order statistics
Record times and record values
Theory of records: Historical review
Hints, solutions, and answers
Bibliography
Details:
Series: Translations of Mathematical Monographs, Volume: 194
Publication Year: 2001
ISBN: 0-8218-1945-3
Paging: 164 pp.
Binding: Hardcover
Kirsten A. Morris
Title: Introduction to Feedback Control
ISBN: 0125076606
Cover: CaseBound
What is often referred to as industrial mathematics is becoming a more important focus of applied mathematics. An increased interest in undergraduate control theory courses for mathematics students is part of this trend. This is due to the fact that control theory is both quite mathematical and very important in applications.
Introduction to Feedback Control is the first survey of input/output controller design aimed at a mathematical audience. The text provides a rigorous introduction to input/output controller design for linear systems. The author has written a truly unique reference for professionals, researchers, the interested reader, and librarians.
Pierre Berthelot, IRMAR, Universitat de Rennes I, France
D-Modules Arithmetiques II Descente Par Frobenius
A publication of the Societe Mathematique de France.
Description
In algebraic geometry, regardless of the characteristic, the theory of modules over suitable rings of differential operators, generically called "$\mathcal D$-modules", is an essential tool in the study of de Rham cohomology and other theories derived from it (crystalline and rigid cohomologies). In this memoir, the author studies the
particular properties of the action of a lifting of the Frobenius morphism on the category of $\mathcal D$-modules when the base is a scheme annihilated by a power of a fixed prime $p$ or a $p$-adic formal scheme. The main result is a descent theorem for the Frobenius morphism, allowing to reduce the study of modules endowed with an action of usual differential operators of order $\leq p^{m}$ to that of modules endowed with an action of derivations. The author proves the compatibility of this descent with all cohomological operations from $\mathcal D$-module theory, which shows that any $\mathcal D$-module of geometric origin can be endowed, in a suitable sense, with a natural Frobenius action. Some applications are included. Text is in French.
Contents
Introduction
${\mathcal D}$-modules ・droite
Theories de descente
Theories de commutation ・l'action de Frobenius
$F-{\mathcal D}^\dag$-modules
Appendice: Filtration m-PD-adique
Bibliographie
Details:
Series: Memoires de la Societe・Mathematique de France, Number: 81
Publication Year: 2000
ISBN: 2-85629-086-8
Paging: 136 pp.
Binding: Softcover