# Search Results for "foundations-of-infinitesimal-stochastic-analysis"

## Foundations of Infinitesimal Stochastic Analysis

**Author**: K.D. Stroyan,J.M. Bayod**Publisher:**Elsevier**ISBN:**0080960421**Category:**Computers**Page:**491**View:**6884

This book gives a complete and elementary account of fundamental results on hyperfinite measures and their application to stochastic processes, including the *-finite Stieltjes sum approximation of martingale integrals. Many detailed examples, not found in the literature, are included. It begins with a brief chapter on tools from logic and infinitesimal (or non-standard) analysis so that the material is accessible to beginning graduate students.

## An Infinitesimal Approach to Stochastic Analysis

**Author**: H. Jerome Keisler**Publisher:**American Mathematical Soc.**ISBN:**0821822977**Category:**Mathematics**Page:**184**View:**7740

## Nonstandard Methods in Stochastic Analysis and Mathematical Physics

**Author**: Sergio Albeverio,Jens Erik Fenstad,Raphael Høegh-Krohn,Tom Lindstrøm**Publisher:**Courier Dover Publications**ISBN:**0486468992**Category:**Mathematics**Page:**526**View:**8930

Two-part treatment begins with a self-contained introduction to the subject, followed by applications to stochastic analysis and mathematical physics. "A welcome addition." — Bulletin of the American Mathematical Society. 1986 edition.

## Model Theory of Stochastic Processes

**Author**: Sergio Fajardo,H. Jerome Keisler**Publisher:**Cambridge University Press**ISBN:**1108619266**Category:**Mathematics**Page:**136**View:**1836

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the fourteenth publication in the Lecture Notes in Logic series, Fajardo and Keisler present new research combining probability theory and mathematical logic. It is a general study of stochastic processes using ideas from model theory, a key central theme being the question, 'When are two stochastic processes alike?' The authors assume some background in nonstandard analysis, but prior knowledge of model theory and advanced logic is not necessary. This volume will appeal to mathematicians willing to explore new developments with an open mind.

## An Introduction to Mathematical Analysis for Economic Theory and Econometrics

**Author**: Dean Corbae,Maxwell B. Stinchcombe,Juraj Zeman**Publisher:**Princeton University Press**ISBN:**1400833086**Category:**Business & Economics**Page:**688**View:**2660

Providing an introduction to mathematical analysis as it applies to economic theory and econometrics, this book bridges the gap that has separated the teaching of basic mathematics for economics and the increasingly advanced mathematics demanded in economics research today. Dean Corbae, Maxwell B. Stinchcombe, and Juraj Zeman equip students with the knowledge of real and functional analysis and measure theory they need to read and do research in economic and econometric theory. Unlike other mathematics textbooks for economics, An Introduction to Mathematical Analysis for Economic Theory and Econometrics takes a unified approach to understanding basic and advanced spaces through the application of the Metric Completion Theorem. This is the concept by which, for example, the real numbers complete the rational numbers and measure spaces complete fields of measurable sets. Another of the book's unique features is its concentration on the mathematical foundations of econometrics. To illustrate difficult concepts, the authors use simple examples drawn from economic theory and econometrics. Accessible and rigorous, the book is self-contained, providing proofs of theorems and assuming only an undergraduate background in calculus and linear algebra. Begins with mathematical analysis and economic examples accessible to advanced undergraduates in order to build intuition for more complex analysis used by graduate students and researchers Takes a unified approach to understanding basic and advanced spaces of numbers through application of the Metric Completion Theorem Focuses on examples from econometrics to explain topics in measure theory

## Principles of Infinitesimal Stochastic and Financial Analysis

**Author**: Imme van den Berg**Publisher:**World Scientific**ISBN:**9789810243586**Category:**Mathematics**Page:**136**View:**8082

There has been a tremendous growth in the volume of financial transactions based on mathematics, reflecting the confidence in the Nobel-Prize-winning Black-Scholes option theory. Risks emanating from obligatory future payments are covered by a strategy of trading with amounts not determined by guessing, but by solving equations, and with prices not resulting from offer and demand, but from computation. However, the mathematical theory behind that suffers from inaccessibility. This is due to the complexity of the mathematical foundation of the Black-Scholes model, which is the theory of continuous-time stochastic processes: a thorough study of mathematical finance is considered to be possible only at postgraduate level. The setting of this book is the discrete-time version of the Black-Scholes model, namely the Cox-Ross-Rubinstein model. The book gives a complete description of its background, which is now only the theory of finite stochastic processes. The novelty lies in the fact that orders of magnitude -- in the sense of nonstandard analysis -- are imposed on the parameters of the model. This not only makes the model more economically sound (such as rapid fluctuations of the market being represented by infinitesimal trading periods), but also leads to a significant simplification: the fundamental results of Black-Scholes theory are derived in full generality and with mathematical rigour, now at graduate level. The material has been repeatedly taught in a third-year course to econometricians.

## Stochastic Control of Hereditary Systems and Applications

**Author**: Mou-Hsiung Chang**Publisher:**Springer Science & Business Media**ISBN:**9780387758169**Category:**Mathematics**Page:**406**View:**2357

This monograph develops the Hamilton-Jacobi-Bellman theory via dynamic programming principle for a class of optimal control problems for stochastic hereditary differential equations (SHDEs) driven by a standard Brownian motion and with a bounded or an infinite but fading memory. These equations represent a class of stochastic infinite-dimensional systems that become increasingly important and have wide range of applications in physics, chemistry, biology, engineering and economics/finance. This monograph can be used as a reference for those who have special interest in optimal control theory and applications of stochastic hereditary systems.

## Truth, Possibility and Probability

*New Logical Foundations of Probability and Statistical Inference*

**Author**: R. Chuaqui**Publisher:**Elsevier**ISBN:**9780080872773**Category:**Mathematics**Page:**483**View:**1089

Anyone involved in the philosophy of science is naturally drawn into the study of the foundations of probability. Different interpretations of probability, based on competing philosophical ideas, lead to different statistical techniques, and frequently to mutually contradictory consequences. This unique book presents a new interpretation of probability, rooted in the traditional interpretation that was current in the 17th and 18th centuries. Mathematical models are constructed based on this interpretation, and statistical inference and decision theory are applied, including some examples in artificial intelligence, solving the main foundational problems. Nonstandard analysis is extensively developed for the construction of the models and in some of the proofs. Many nonstandard theorems are proved, some of them new, in particular, a representation theorem that asserts that any stochastic process can be approximated by a process defined over a space with equiprobable outcomes.

## Five Lectures in Complex Analysis

*Second Winter School on Complex Analysis and Operator Theory, February 5-9, 2008, University of Sevilla, Sevilla, Spain*

**Author**: Contreras Márquez Contreras,Santiago Díaz-Madrigal**Publisher:**American Mathematical Soc.**ISBN:**0821848097**Category:**Mathematics**Page:**161**View:**3284

"This volume contains state-of-art survey papers in complex analysis based on lectures given at the second Winter School on Complex Analysis and Operator Theory held in February 2008 at the University of Sevilla, Sevilla, Spain." "Complex analysis is oneof the most classical branches of mathematical analysis and is closely related to many other areas of mathematics, including operator theory, harmonic analysis, probability theory, functional analysis and dynamical systems. Undoubtedly, the interplay among all these branches gives rise to very beautiful and deep results in complex analysis and its neighboring fields. This interdisciplinary aspect of complex analysis is the central topic of this volume." "This book collects the latest advances in five significant areas of rapid development in complex analysis. The papers are: Local holomorphic dynamics of diffeomorphisms in dimension one, by F. Bracci, Nonpostive curvature and complex analysis, by S. M. Buckley, Virasoro algebra and dynamics in the space of univalent functions, by I. Markina and A. Vasil'ev, Composition operators Toeplitz operators, by J. H. Shapir, and Two applications of the Bergman spaces techniques, by S. Shimorin." "The papers are aimed, in particular, at graduate students with some experince in basic complex analysis. They might also serve as introductions for general researchers in mathematical analysis who may be interested in the specific areas addressed by the authors. Indeed, the contributions can be considered as up-to-the minute reports on the current state of the fields, each of them including many recent results which may be difficult to find in the literature."--BOOK JACKET.

## Interacting Stochastic Systems

**Author**: Jean-Dominique Deuschel,DFG-Schwerpunkt: Interacting Stochastic Systems of High Complexity,Andreas Greven,Research Network on "Interacting Stochastic Systems of High Complexity.",Research Network on "Interacting Stochastic Systems of High Complexity".**Publisher:**Springer Science & Business Media**ISBN:**9783540230335**Category:**Mathematics**Page:**450**View:**513

The Research Network on "Interacting stochastic systems of high complexity" set up by the German Research Foundation aimed at exploring and developing connections between research in infinite-dimensional stochastic analysis, statistical physics, spatial population models from mathematical biology, complex models of financial markets or of stochastic models interacting with other sciences. This book presents a structured collection of papers on the core topics, written at the close of the 6-year programme by the research groups who took part in it. The structure chosen highlights the interweaving of certain themes and certain interconnections discovered through the joint work. This yields a reference work on results and methods that will be useful to all who work between applied probability and the physical, economic, and life sciences.

## Diffusions, Markov Processes, and Martingales: Volume 1, Foundations

**Author**: L. C. G. Rogers,David Williams**Publisher:**Cambridge University Press**ISBN:**1107717493**Category:**Mathematics**Page:**406**View:**1478

Now available in paperback, this celebrated book has been prepared with readers' needs in mind, remaining a systematic guide to a large part of the modern theory of Probability, whilst retaining its vitality. The authors' aim is to present the subject of Brownian motion not as a dry part of mathematical analysis, but to convey its real meaning and fascination. The opening, heuristic chapter does just this, and it is followed by a comprehensive and self-contained account of the foundations of theory of stochastic processes. Chapter 3 is a lively and readable account of the theory of Markov processes. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

## The Strength of Nonstandard Analysis

**Author**: Imme van den Berg,Vitor Neves**Publisher:**Springer Science & Business Media**ISBN:**3211499059**Category:**Mathematics**Page:**401**View:**4448

This book reflects the progress made in the forty years since the appearance of Abraham Robinson’s revolutionary book Nonstandard Analysis in the foundations of mathematics and logic, number theory, statistics and probability, in ordinary, partial and stochastic differential equations and in education. The contributions are clear and essentially self-contained.

## Loeb Measures in Practice: Recent Advances

*EMS Lectures 1997*

**Author**: Nigel J. Cutland**Publisher:**Springer Science & Business Media**ISBN:**9783540413844**Category:**Business & Economics**Page:**111**View:**5174

This expanded version of the 1997 European Mathematical Society Lectures given by the author in Helsinki, begins with a self-contained introduction to nonstandard analysis (NSA) and the construction of Loeb Measures, which are rich measures discovered in 1975 by Peter Loeb, using techniques from NSA. Subsequent chapters sketch a range of recent applications of Loeb measures due to the author and his collaborators, in such diverse fields as (stochastic) fluid mechanics, stochastic calculus of variations ("Malliavin" calculus) and the mathematical finance theory. The exposition is designed for a general audience, and no previous knowledge of either NSA or the various fields of applications is assumed.

## Kalman-Bucy-Filter

*determinist. Beobachtung u. stochast. Filterung*

**Author**: Karl Brammer,Gerhard Siffling**Publisher:**N.A**ISBN:**N.A**Category:**Control theory**Page:**232**View:**4928

Das Buch will die mannigfaltigen Aufgaben der heutigen Regelungs- und Steuerungstechnik und ihre Lösung nahebringen. Das soll mit möglichst geringem Zeit- und Arbeitsaufwand für den Leser verbunden sein. Leichte Verständlichkeit, Anschaulichkeit und Anwendungsnähe sind deshalb Hauptgesichtspunkt der Darstellung. Vollständigkeit ist nicht angestrebt, vielmehr Darstellung des Wesentlichen. Mathematische Methoden werden auf das Notwendige beschränkt.

## Stochastic Integrals

*An Introduction*

**Author**: Heinrich von Weizsäcker**Publisher:**Springer-Verlag**ISBN:**3663139239**Category:**Mathematics**Page:**332**View:**5805

## Mathematik, empirische Wissenschaft und Erkenntnistheorie

**Author**: Imre Lakatos**Publisher:**Springer-Verlag**ISBN:**3322910881**Category:**Science**Page:**282**View:**9244

## Nonstandard methods and applications in mathematics

**Author**: Nigel Cutland,Mauro Di Nasso,David A. Ross**Publisher:**A K Peters Ltd**ISBN:**9781568812915**Category:**Mathematics**Page:**248**View:**6210

This book is a collection of peer-reviewed papers from a conference on Nonstandard Methods and Applications in Mathematics (NS2002) that was held in Pisa, Italy. The papers address nonstandard analysis, which is one of the great achievements of modern applied mathematical logic. They focus on its important philosophical achievement of providing a sound mathematical basis for using infinitesimals in analysis, and they show how this methodology is now well established as a tool for both research and teaching.

## Markov Processes

**Author**: Evgenij Borisovic Dynkin**Publisher:**Springer Science & Business Media**ISBN:**3662000318**Category:**Mathematics**Page:**366**View:**1524

The modem theory of Markov processes has its origins in the studies of A. A. MARKOV (1906-1907) on sequences of experiments "connected in a chain" and in the attempts to describe mathematically the physical phenomenon known as Brownian motion (L. BACHELlER 1900, A. EIN STEIN 1905). The first correct mathematical construction of a Markov process with continuous trajectories was given by N. WIENER in 1923. (This process is often called the Wiener process.) The general theory of Markov processes was developed in the 1930's and 1940's by A. N. KOL MOGOROV, W. FELLER, W. DOEBLlN, P. LEVY, J. L. DOOB, and others. During the past ten years the theory of Markov processes has entered a new period of intensive development. The methods of the theory of semigroups of linear operators made possible further progress in the classification of Markov processes by their infinitesimal characteristics. The broad classes of Markov processes with continuous trajectories be came the main object of study. The connections between Markov pro cesses and classical analysis were further developed. It has become possible not only to apply the results and methods of analysis to the problems of probability theory, but also to investigate analytic problems using probabilistic methods. Remarkable new connections between Markov processes and potential theory were revealed. The foundations of the theory were reviewed critically: the new concept of strong Markov process acquired for the whole theory of Markov processes great importance.

## Grundbegriffe der Wahrscheinlichkeitsrechnung

**Author**: A. Kolomogoroff**Publisher:**Springer-Verlag**ISBN:**3642498884**Category:**Mathematics**Page:**62**View:**2019

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

## Stochastic Equations in Infinite Dimensions

**Author**: Guiseppe Da Prato,Jerzy Zabczyk**Publisher:**Cambridge University Press**ISBN:**9780521059800**Category:**Mathematics**Page:**454**View:**8379

The aim of this book is to give a systematic and self-contained presentation of the basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. These are a generalization of stochastic differential equations as introduced by Itô and Gikhman that occur, for instance, when describing random phenomena that crop up in science and engineering, as well as in the study of differential equations. The book is divided into three parts. In the first the authors give a self-contained exposition of the basic properties of probability measures on separable Banach and Hilbert spaces, as required later; they assume a reasonable background in probability theory and finite dimensional stochastic processes. The second part is devoted to the existence and uniqueness of solutions of a general stochastic evolution equation, and the third concerns the qualitative properties of those solutions. Appendices gather together background results from analysis that are otherwise hard to find under one roof.