Beyond Brownian Motion
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| Author | : Sabir Umarov |
| Publisher | : World Scientific |
| Total Pages | : 192 |
| Release | : 2018-02-13 |
| Genre | : Mathematics |
| ISBN | : 9813230991 |
The book is devoted to the fundamental relationship between three objects: a stochastic process, stochastic differential equations driven by that process and their associated Fokker-Planck-Kolmogorov equations. This book discusses wide fractional generalizations of this fundamental triple relationship, where the driving process represents a time-changed stochastic process; the Fokker-Planck-Kolmogorov equation involves time-fractional order derivatives and spatial pseudo-differential operators; and the associated stochastic differential equation describes the stochastic behavior of the solution process. It contains recent results obtained in this direction.This book is important since the latest developments in the field, including the role of driving processes and their scaling limits, the forms of corresponding stochastic differential equations, and associated FPK equations, are systematically presented. Examples and important applications to various scientific, engineering, and economics problems make the book attractive for all interested researchers, educators, and graduate students.
| Author | : René L. Schilling |
| Publisher | : Walter de Gruyter GmbH & Co KG |
| Total Pages | : 424 |
| Release | : 2014-06-18 |
| Genre | : Mathematics |
| ISBN | : 3110307308 |
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.
| Author | : Peter Mörters |
| Publisher | : Cambridge University Press |
| Total Pages | : |
| Release | : 2010-03-25 |
| Genre | : Mathematics |
| ISBN | : 1139486578 |
This eagerly awaited textbook covers everything the graduate student in probability wants to know about Brownian motion, as well as the latest research in the area. Starting with the construction of Brownian motion, the book then proceeds to sample path properties like continuity and nowhere differentiability. Notions of fractal dimension are introduced early and are used throughout the book to describe fine properties of Brownian paths. The relation of Brownian motion and random walk is explored from several viewpoints, including a development of the theory of Brownian local times from random walk embeddings. Stochastic integration is introduced as a tool and an accessible treatment of the potential theory of Brownian motion clears the path for an extensive treatment of intersections of Brownian paths. An investigation of exceptional points on the Brownian path and an appendix on SLE processes, by Oded Schramm and Wendelin Werner, lead directly to recent research themes.
| Author | : David A. Baum |
| Publisher | : Princeton University Press |
| Total Pages | : 886 |
| Release | : 2017-03-21 |
| Genre | : Science |
| ISBN | : 069117587X |
The essential one-volume reference to evolution The Princeton Guide to Evolution is a comprehensive, concise, and authoritative reference to the major subjects and key concepts in evolutionary biology, from genes to mass extinctions. Edited by a distinguished team of evolutionary biologists, with contributions from leading researchers, the guide contains some 100 clear, accurate, and up-to-date articles on the most important topics in seven major areas: phylogenetics and the history of life; selection and adaptation; evolutionary processes; genes, genomes, and phenotypes; speciation and macroevolution; evolution of behavior, society, and humans; and evolution and modern society. Complete with more than 100 illustrations (including eight pages in color), glossaries of key terms, suggestions for further reading on each topic, and an index, this is an essential volume for undergraduate and graduate students, scientists in related fields, and anyone else with a serious interest in evolution. Explains key topics in some 100 concise and authoritative articles written by a team of leading evolutionary biologists Contains more than 100 illustrations, including eight pages in color Each article includes an outline, glossary, bibliography, and cross-references Covers phylogenetics and the history of life; selection and adaptation; evolutionary processes; genes, genomes, and phenotypes; speciation and macroevolution; evolution of behavior, society, and humans; and evolution and modern society
| Author | : Ubbo F. Wiersema |
| Publisher | : John Wiley & Sons |
| Total Pages | : 342 |
| Release | : 2008-12-08 |
| Genre | : Business & Economics |
| ISBN | : 0470021705 |
BROWNIAN MOTION CALCULUS Brownian Motion Calculus presents the basics of Stochastic Calculus with a focus on the valuation of financial derivatives. It is intended as an accessible introduction to the technical literature. The sequence of chapters starts with a description of Brownian motion, the random process which serves as the basic driver of the irregular behaviour of financial quantities. That exposition is based on the easily understood discrete random walk. Thereafter the gains from trading in a random environment are formulated in a discrete-time setting. The continuous-time equivalent requires a new concept, the Itō stochastic integral. Its construction is explained step by step, using the so-called norm of a random process (its magnitude), of which a motivated exposition is given in an Annex. The next topic is Itō’s formula for evaluating stochastic integrals; it is the random process counter part of the well known Taylor formula for functions in ordinary calculus. Many examples are given. These ingredients are then used to formulate some well established models for the evolution of stock prices and interest rates, so-called stochastic differential equations, together with their solution methods. Once all that is in place, two methodologies for option valuation are presented. One uses the concept of a change of probability and the Girsanov transformation, which is at the core of financial mathematics. As this technique is often perceived as a magic trick, particular care has been taken to make the explanation elementary and to show numerous applications. The final chapter discusses how computations can be made more convenient by a suitable choice of the so-called numeraire. A clear distinction has been made between the mathematics that is convenient for a first introduction, and the more rigorous underpinnings which are best studied from the selected technical references. The inclusion of fully worked out exercises makes the book attractive for self study. Standard probability theory and ordinary calculus are the prerequisites. Summary slides for revision and teaching can be found on the book website www.wiley.com/go/brownianmotioncalculus.
| Author | : Michael D. Graham |
| Publisher | : Cambridge University Press |
| Total Pages | : 282 |
| Release | : 2018-09-13 |
| Genre | : Science |
| ISBN | : 110861440X |
This is an introduction to the dynamics of fluids at small scales, the physical and mathematical underpinnings of Brownian motion, and the application of these subjects to the dynamics and flow of complex fluids such as colloidal suspensions and polymer solutions. It brings together continuum mechanics, statistical mechanics, polymer and colloid science, and various branches of applied mathematics, in a self-contained and integrated treatment that provides a foundation for understanding complex fluids, with a strong emphasis on fluid dynamics. Students and researchers will find that this book is extensively cross-referenced to illustrate connections between different aspects of the field. Its focus on fundamental principles and theoretical approaches provides the necessary groundwork for research in the dynamics of flowing complex fluids.
| Author | : Richard Durrett |
| Publisher | : Wadsworth Publishing Company |
| Total Pages | : 328 |
| Release | : 1984 |
| Genre | : Mathematics |
| ISBN | : 9780534030650 |
| Author | : Luke J. Harmon |
| Publisher | : |
| Total Pages | : 286 |
| Release | : 2018-05-23 |
| Genre | : |
| ISBN | : 9781719584463 |
An introduction to statistical analyses of phylogenetic trees using comparative methods.
| Author | : Alexander I. Saichev |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 176 |
| Release | : 2009-11-04 |
| Genre | : Business & Economics |
| ISBN | : 3642029469 |
Zipf’s law is one of the few quantitative reproducible regularities found in e- nomics. It states that, for most countries, the size distributions of cities and of rms (with additional examples found in many other scienti c elds) are power laws with a speci c exponent: the number of cities and rms with a size greater thanS is inversely proportional toS. Most explanations start with Gibrat’s law of proportional growth but need to incorporate additional constraints and ingredients introducing deviations from it. Here, we present a general theoretical derivation of Zipf’s law, providing a synthesis and extension of previous approaches. First, we show that combining Gibrat’s law at all rm levels with random processes of rm’s births and deaths yield Zipf’s law under a “balance” condition between a rm’s growth and death rate. We nd that Gibrat’s law of proportionate growth does not need to be strictly satis ed. As long as the volatility of rms’ sizes increase asy- totically proportionally to the size of the rm and that the instantaneous growth rate increases not faster than the volatility, the distribution of rm sizes follows Zipf’s law. This suggests that the occurrence of very large rms in the distri- tion of rm sizes described by Zipf’s law is more a consequence of random growth than systematic returns: in particular, for large rms, volatility must dominate over the instantaneous growth rate.
| Author | : Jean-François Le Gall |
| Publisher | : Springer |
| Total Pages | : 282 |
| Release | : 2016-04-28 |
| Genre | : Mathematics |
| ISBN | : 3319310895 |
This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments. Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.
