This is a concise and elementary introduction to contemporary measure and integration theory as it is needed in many parts of analysis and probability theory. Undergraduate calculus and an introductory course on rigorous analysis in R are the only essential prerequisites, making the text suitable for both lecture courses and for self-study. Numerous illustrations and exercises are included to consolidate what has already been learned and to discover variants and extensions to the main material. Hints and solutions can be found on the authors website, which can be reached at http://www.motapa.de/measures_integrals_and_martingales/index.html

Measures, Integrals and Martingales

Science & Math

Schilling, René L.

Mathematics

Cambridge University Press

Counterexamples in Measure and Integration

Often it is more instructive to know 'what can go wrong' and to understand 'why a result fails' than to plod through yet another piece of theory. In this text, the authors gather more than 300 counterexamples - some of them both surprising and amusing - showing the limitations, hidden traps and pitfalls of measure and integration. Many examples are put into context, explaining relevant parts of the theory, and pointing out further reading. The text starts with a self-contained, non-technical overview on the fundamentals of measure and integration. A companion to the successful undergraduate textbook Measures, Integrals and Martingales, it is accessible to advanced undergraduate students, requiring only modest prerequisites. More specialized concepts are summarized at the beginning of each chapter, allowing for self-study as well as supplementary reading for any course covering measures and integrals. For researchers, it provides ample examples and warnings as to the limitations of general measure theory. This book forms a sister volume to René Schilling's other book Measures, Integrals and Martingales (www.cambridge.org/9781316620243).

Kühn, Franziska

A concise yet elementary introduction to measure and integration theory, which are vital in many areas of mathematics, including analysis, probability, mathematical physics and finance. In this highly successful textbook, core ideas of measure and integration are explored, and martingales are used to develop the theory further. Other topics are also covered such as Jacobi's transformation theorem, the Radon-Nikodym theorem, differentiation of measures and Hardy-Littlewood maximal functions. In this second edition, readers will find newly added chapters on Hausdorff measures, Fourier analysis, vague convergence and classical proofs of Radon-Nikodym and Riesz representation theorems. All proofs are carefully worked out to ensure full understanding of the material and its background. Requiring few prerequisites, this book is suitable for undergraduate lecture courses or self-study. Numerous illustrations and over 400 exercises help to consolidate and broaden knowledge. Full solutions to all exercises are available on the author's webpage at www.motapa.de.

This text is a self-contained and unified approach to Bernstein functions and their subclasses, bringing together old and establishing new connections. Applications of Bernstein functions in different fields of mathematics are given, with special attention to interpretations in probability theory. An extensive list of complete Bernstein functions with their representations is provided.

Bernstein Functions: Theory and Applications (De Gruyter Studies in Mathematics, 37)

Walter de Gruyter

Bernstein functions and the important subclass of complete Bernstein functions appear in various fields of mathematics - often with different definitions and under different names. Probabilists, for example, know Bernstein functions as Laplace exponents, and in harmonic analysis they are called negative definite functions. Complete Bernstein functions are used in complex analysis under the name Pick or Nevanlinna functions, while in matrix analysis and operator theory, the name operator monotone function is more common. When studying the positivity of solutions of Volterra integral equations, various types of kernels appear which are related to Bernstein functions. There exists a considerable amount of literature on each of these classes, but only a handful of texts observe the connections between them or use methods from several mathematical disciplines. This monograph - now in its second revised and extended edition - is about these connections. Compared to the first edition the authors added a substantial amount of new material. Their aim was to retain the overall structure of the text with theoretical material in Chapters 1 to 11 and more specific applications in Chapters 12 to 15.

Song, Renming

Vondracek, Zoran

de Gruyter

Brownian motion is one of the most important stochastic processes in continuous time and with continuous state space. Within the realm of stochastic processes, Brownian motion is at the intersection of Gaussian processes, martingales, Markov processes, diffusions and random fractals, and it has influenced the study of these topics. Its central position within mathematics is matched by numerous applications in science, engineering and mathematical finance. Often textbooks on probability theory cover, if at all, Brownian motion only briefly. On the other hand, there is a considerable gap to more specialized texts on Brownian motion which is not so easy to overcome for the novice. The authors' aim was to write a book which can be used as an introduction to Brownian motion and stochastic calculus, and as a first course in continuous-time and continuous-state Markov processes. They also wanted to have a text which would be both a readily accessible mathematical back-up for contemporary applications (such as mathematical finance) and a foundation to get easy access to advanced monographs. This textbook, tailored to the needs of graduate and advanced undergraduate students, covers Brownian motion, starting from its elementary properties, certain distributional aspects, path properties, and leading to stochastic calculus based on Brownian motion. It also includes numerical recipes for the simulation of Brownian motion.

Brownian Motion: An Introduction To Stochastic Processes

Many areas of mathematics and their application in physics, economics, and computer science require a firm grasp of measure theory. This textbook provides a concise, reliable, and accurate introduction to the most important elements of measure theory.

Maß und Integral (De Gruyter Studium) (German Edition)

Brownian motion is among the most important stochastic processes, occurring in continuous time and with continuous phase space. This textbook offers a rapid, reliable, and detailed introduction to the most important elements in the theory of Brownian motion.

Martingale und Prozesse (De Gruyter Studium) (German Edition)

See the best price to sell, buy, or rent books by Schilling, René L.. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by Schilling, René L.

See the best price to sell, buy, or rent books by Schilling, René L.. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Measures, Integrals and Martingales

Measures, Integrals and Martingales

Counterexamples in Measure and Integration

Counterexamples in Measure and Integration

Measures, Integrals and Martingales

Bernstein Functions: Theory and Applications (De Gruyter Studies in Mathematics, 37)

Bernstein Functions: Theory and Applications (De Gruyter Studies in Mathematics, 37)

Brownian Motion: An Introduction To Stochastic Processes

Maß und Integral (De Gruyter Studium) (German Edition)

Martingale und Prozesse (De Gruyter Studium) (German Edition)