# 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:**5236

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:**3336

## 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:**4017

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.

## Nichtstandard Analysis

**Author**: Dieter Landers,Lothar Rogge**Publisher:**Springer-Verlag**ISBN:**3642579159**Category:**Mathematics**Page:**488**View:**2352

Die Nichtstandard-Mathematik hat in den letzten Jahren einen gewaltigen Aufschwung erfahren und die Entwicklungen in den verschiedenartigsten Gebieten beeinflußt und befruchtet. Mit diesem Lehrbuch liegt nun die erste umfassende und leicht verständliche Einführung in dieses Thema in deutscher Sprache vor. An Vorkenntnissen braucht der Leser für ein gewinnbringendes Selbststudium nichts weiter als Grundkenntnisse in Linearer Algebra und Analysis, d.h. Kenntnisse des ersten Studienjahres. Ausführliche Beweise, viele Aufgaben mit Lösungen und eine gelungene didaktische Aufbereitung des Stoffes machen Methoden und Erkenntnisse durchsichtig und verständlich. Trotz der einfachen Lesbarkeit dieses Buches wird an mehreren Stellen bis zu neuesten Forschungsergebnissen vorgestoßen und viele Ergebnisse werden zum ersten Mal in Buchform vorgestellt. Mit diesem Lehrbuch wird der Leser in die Lage versetzt, schnell Nichtstandard-Methoden in den verschiedensten Bereichen selbständig anzuwenden. Es kann außerdem als Basis für ein- oder mehrsemestrige Vorlesungen verwendet werden. Aus dem Vorwort der Autoren: "Wir hoffen, daß unsere Leser beim Studium dieses Buches den Enthusiasmus der Autoren für die Schönheit, Eleganz und Wirksamkeit der Nichtstandard-Methoden teilen werden."

## 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:**5598

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:**7169

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.

## Mathematik, empirische Wissenschaft und Erkenntnistheorie

**Author**: Imre Lakatos**Publisher:**Springer-Verlag**ISBN:**3322910881**Category:**Science**Page:**282**View:**7990

## 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:**9251

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.

## Stochastic Integrals

*An Introduction*

**Author**: Heinrich von Weizsäcker**Publisher:**Springer-Verlag**ISBN:**3663139239**Category:**Mathematics**Page:**332**View:**3838

## Truth, Possibility and Probability

*New Logical Foundations of Probability and Statistical Inference*

**Author**: R. Chuaqui**Publisher:**Elsevier**ISBN:**9780080872773**Category:**Mathematics**Page:**483**View:**654

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.

## Stochastic Control of Hereditary Systems and Applications

**Author**: Mou-Hsiung Chang**Publisher:**Springer Science & Business Media**ISBN:**9780387758169**Category:**Mathematics**Page:**406**View:**4473

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.

## Grundbegriffe der Wahrscheinlichkeitsrechnung

**Author**: A. Kolomogoroff**Publisher:**Springer-Verlag**ISBN:**3642498884**Category:**Mathematics**Page:**62**View:**2466

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.

## Stability of Infinite Dimensional Stochastic Differential Equations with Applications

**Author**: Kai Liu**Publisher:**CRC Press**ISBN:**9781420034820**Category:**Mathematics**Page:**312**View:**9204

Stochastic differential equations in infinite dimensional spaces are motivated by the theory and analysis of stochastic processes and by applications such as stochastic control, population biology, and turbulence, where the analysis and control of such systems involves investigating their stability. While the theory of such equations is well established, the study of their stability properties has grown rapidly only in the past 20 years, and most results have remained scattered in journals and conference proceedings. This book offers a systematic presentation of the modern theory of the stability of stochastic differential equations in infinite dimensional spaces - particularly Hilbert spaces. The treatment includes a review of basic concepts and investigation of the stability theory of linear and nonlinear stochastic differential equations and stochastic functional differential equations in infinite dimensions. The final chapter explores topics and applications such as stochastic optimal control and feedback stabilization, stochastic reaction-diffusion, Navier-Stokes equations, and stochastic population dynamics. In recent years, this area of study has become the focus of increasing attention, and the relevant literature has expanded greatly. Stability of Infinite Dimensional Stochastic Differential Equations with Applications makes up-to-date material in this important field accessible even to newcomers and lays the foundation for future advances.

## Stochastic Modeling and Analysis of Telecom Networks

**Author**: Laurent Decreusefond,Pascal Moyal**Publisher:**John Wiley & Sons**ISBN:**1118563018**Category:**Technology & Engineering**Page:**387**View:**3367

This book addresses the stochastic modeling of telecommunicationnetworks, introducing the main mathematical tools for that purpose,such as Markov processes, real and spatial point processes andstochastic recursions, and presenting a wide list of results onstability, performances and comparison of systems. The authors propose a comprehensive mathematical construction ofthe foundations of stochastic network theory: Markov chains,continuous time Markov chains are extensively studied using anoriginal martingale-based approach. A complete presentation ofstochastic recursions from an ergodic theoretical perspective isalso provided, as well as spatial point processes. Using these basic tools, stability criteria, performance measuresand comparison principles are obtained for a wide class of models,from the canonical M/M/1 and G/G/1 queues to more sophisticatedsystems, including the current “hot topics” of spatialradio networking, OFDMA and real-time networks. Contents 1. Introduction. Part 1: Discrete-time Modeling 2. Stochastic Recursive Sequences. 3. Markov Chains. 4. Stationary Queues. 5. The M/GI/1 Queue. Part 2: Continuous-time Modeling 6. Poisson Process. 7. Markov Process. 8. Systems with Delay. 9. Loss Systems. Part 3: Spatial Modeling 10. Spatial Point Processes.

## 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:**3631

"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.

## The Mathematics of Computerized Tomography

**Author**: F. Natterer**Publisher:**Springer-Verlag**ISBN:**3663014096**Category:**Technology & Engineering**Page:**222**View:**2784

## 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:**3402

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.

## Stochastic Equations in Infinite Dimensions

**Author**: Guiseppe Da Prato,Jerzy Zabczyk**Publisher:**Cambridge University Press**ISBN:**9780521059800**Category:**Mathematics**Page:**454**View:**2984

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.

## 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:**8461

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.

## Zahlen und Kontinuum

*eine Einführung in die Infinitesimalmathematik*

**Author**: Detlef Laugwitz**Publisher:**N.A**ISBN:**N.A**Category:**Nonstandard mathematical analysis**Page:**269**View:**1101