Bücher der Reihe Universitext

Gives an introduction to the basic theory of stochastic calculus and its applications. This book offers examples in order to motivate and illustrate the theory and show its importance for many applications in for example economics, biology and physics.

This text covers a first course in bilinear maps and tensor products intending to bring the reader from the beginning of functional analysis to the frontiers of exploration with tensor products. Tensor products, particularly in infinite-dimensional normed spaces, are heavily based on bilinear maps. The author brings these topics together by using bilinear maps as an auxiliary, yet fundamental, tool for accomplishing a consistent, useful, and straightforward theory of tensor products. The author¿s usual clear, friendly, and meticulously prepared exposition presents the material in ways that are designed to make grasping concepts easier and simpler. The approach to the subject is uniquely presented from an operator theoretic view. An introductory course in functional analysis is assumed. In order to keep the prerequisites as modest as possible, there are two introductory chapters, one on linear spaces (Chapter 1) and another on normed spaces (Chapter 5), summarizing the background material required for a thorough understanding. The reader who has worked through this text will be well prepared to approach more advanced texts and additional literature on the subject.The book brings the theory of tensor products on Banach spaces to the edges of Grothendieck's theory, and changes the target towards tensor products of bounded linear operators. Both Hilbert-space and Banach-space operator theory are considered and compared from the point of view of tensor products. This is done from the first principles of functional analysis up to current research topics, with complete and detailed proofs. The first four chapters deal with the algebraic theory of linear spaces, providing various representations of the algebraic tensor product defined in an axiomatic way. Chapters 5 and 6 give the necessary background concerning normed spaces and bounded bilinear mappings. Chapter 7 is devoted to the study of reasonable crossnorms on tensor product spaces, discussing in detail the important extreme realizations of injective and projective tensor products. In Chapter 8 uniform crossnorms are introduced in which the tensor products of operators are bounded; special attention is paid to the finitely generated situation. The concluding Chapter 9 is devoted to the study of the Hilbert space setting and the spectral properties of the tensor products of operators. Each chapter ends with a section containing ¿Additional Propositions" and suggested readings for further studies.

This textbook provides a self-contained introduction to numerical methods in probability with a focus on applications to finance.Topics covered include the Monte Carlo simulation (including simulation of random variables, variance reduction, quasi-Monte Carlo simulation, and more recent developments such as the multilevel paradigm), stochastic optimization and approximation, discretization schemes of stochastic differential equations, as well as optimal quantization methods. The author further presents detailed applications to numerical aspects of pricing and hedging of financial derivatives, risk measures (such as value-at-risk and conditional value-at-risk), implicitation of parameters, and calibration.Aimed at graduate students and advanced undergraduate students, this book contains useful examples and over 150 exercises, making it suitable for self-study.

Requiring no more than a basic knowledge of abstract algebra, this text presents the mathematics of number fields in a straightforward, pedestrian manner. It therefore avoids local methods and presents proofs in a way that highlights the important parts of the arguments.

This textbook presents the essential parts of the modern theory of nonlinear partial differential equations, including the calculus of variations.After a short review of results in real and functional analysis, the author introduces the main mathematical techniques for solving both semilinear and quasilinear elliptic PDEs, and the associated boundary value problems. Key topics include infinite dimensional fixed point methods, the Galerkin method, the maximum principle, elliptic regularity, and the calculus of variations. Aimed at graduate students and researchers, this textbook contains numerous examples and exercises and provides several comments and suggestions for further study.

This textbook provides a comprehensive introduction to the classical and modern calculus of variations, serving as a useful reference to advanced undergraduate and graduate students as well as researchers in the field.Starting from ten motivational examples, the book begins with the most important aspects of the classical theory, including the Direct Method, the Euler-Lagrange equation, Lagrange multipliers, Noether's Theorem and some regularity theory. Based on the efficient Young measure approach, the author then discusses the vectorial theory of integral functionals, including quasiconvexity, polyconvexity, and relaxation. In the second part, more recent material such as rigidity in differential inclusions, microstructure, convex integration, singularities in measures, functionals defined on functions of bounded variation (BV), and G-convergence for phase transitions and homogenization are explored.While predominantly designed as a textbook for lecture courses on the calculus of variations, this book can also serve as the basis for a reading seminar or as a companion for self-study. The reader is assumed to be familiar with basic vector analysis, functional analysis, Sobolev spaces, and measure theory, though most of the preliminaries are also recalled in the appendix.

This unique text for beginning graduate students gives a self-contained introduction to the mathematical properties of stochastics and presents their applications to Markov processes, coding theory, population dynamics, and search engine design.

Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory.

Using a `learning by calculating' approach, this comprehensive introductory text shows how stochastic computational methods are used across the field of finance. The revised and expanded fifth edition includes updates, as well as new material and exercises.

Introducing ergodic theory and dynamical systems, with an emphasis on chaotic dynamics, this book explores the fundamental ideas from basic notions like ergodicity, mixing and isomorphisms of dynamical systems, to chaotic transformations with hyperbolic dynamics, entropy, information theory, ergodic decomposition and measurable partitions.

Based on three series of lectures by the author at the University of Strasbourg, this book presents functional analysis in a non-traditional way by generalizing elementary theorems of plane geometry to spaces of arbitrary dimension, leading naturally to the basic notions and theorems. Most results are illustrated by the small lp spaces.

Designed for intermediate graduate studies, this text will broaden students' core knowledge of differential geometry providing foundational material to relevant topics in classical differential geometry.

Taking readers with a basic knowledge of probability and real analysis to the frontiers of a very active research discipline, this textbook provides all the necessary background from functional analysis and the theory of PDEs.

This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs).

This book offers an extensive description of the classical complex analysis, roughly meaning that sheaf theoretical and cohomological methods are omitted. Over 400 exercises are included, and the text has been heavily revised for this new edition.

Few books on Ordinary Differential Equations (ODEs) have the elegant geometric insight of this one, which puts emphasis on the qualitative and geometric properties of ODEs and their solutions, rather than on routine presentation of algorithms.

Graduate mathematics students will find this book an easy-to-follow, step-by-step guide to the subject. Rotman's book gives a treatment of homological algebra which approaches the subject in terms of its origins in algebraic topology.

This is a description of the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today.

The book presents the central facts of the local, projective and intrinsic theories of complex algebraic plane curves, with complete proofs and starting from low-level prerequisites.

Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject;

This book explains the mathematical background behind the Standard Model, translating ideas from physics into a mathematical language and vice versa. The first part of the book covers the mathematical theory of Lie groups and Lie algebras, fibre bundles, connections, curvature and spinors.

The core of the book covers stochastic calculus, including stochastic differential equations, the relationship to partial differential equations, numerical methods and simulation, as well as applications of stochastic processes to finance.

This established reference work continues to introduce its readers to some of the hottest topics in contemporary mathematical research. This sixth edition includes, among other new additions, a systematic treatment of eigenvalues of Riemannian manifolds.

This book is an introduction to stochastic analysis and quantitative finance; Stochastic Analysis for Finance with Simulations is designed for readers who want to have a deeper understanding of the delicate theory of quantitative finance by doing computer simulations in addition to theoretical study.

While the topic of principal bundles in differential geometry has become classic, even standard, material in the modern graduate mathematics curriculum, the unique approach taken in this text presents the material in a way that is intuitive for both students of mathematics and of physics.

Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity. The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature.

Set Theory