Imme van den Berg's Books

Neutrices and External Numbers

A Flexible Number System

By: Bruno Dinis,Imme van den Berg

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Neutrices and External Numbers

A Flexible Number System

By: Bruno Dinis,Imme van den Berg

Neutrices and External Numbers: A Flexible Number System introduces a new model of orders of magnitude and of error analysis, with particular emphasis on behaviour under algebraic operations. The model is formulated in terms of scalar neutrices and external numbers, in the form of an extension of the nonstandard set of real numbers. Many illustrative examples are given. The book starts with detailed presentation of the algebraic structure of external numbers, then deals with the generalized Dedekind completeness property, applications in analysis, domains of validity of approximations of solutions of differential equations, particularly singular perturbations. Finally, it describes the family of algebraic laws characterizing the practice of calculations with external numbers. Features Presents scalar neutrices and external numbers, a mathematical model of order of magnitude within the real number system. Outlines complete algebraic rules for the neutrices and external numbers Conducts operational analysis of convergence and integration of functions known up to orders of magnitude Formalises a calculus of error propagation, covariant with algebraic operations Presents mathematical models of phenomena incorporating their necessary imprecisions, in particular related to the Sorites paradox

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Page Count

361

Publisher

CRC Press

Published Date

ISBN 10

1000005380

ISBN 13

9781000005387

Principles Of Infinitesinal Stochastic & Financial Analysis

By: Imme Van Den Berg

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Principles Of Infinitesinal Stochastic & Financial Analysis

By: Imme Van Den Berg

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.

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Page Count

150

Publisher

World Scientific

Published Date

ISBN 10

9814492779

ISBN 13

9789814492775

Principles of Infinitesimal Stochastic and Financial Analysis

By: Imme van den Berg

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Principles of Infinitesimal Stochastic and Financial Analysis

By: Imme van den Berg

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.

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Page Count

156

Publisher

World Scientific

Published Date

ISBN 10

9810243588

ISBN 13

9789810243586

Nonstandard Asymptotic Analysis

By: Imme van den Berg

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Nonstandard Asymptotic Analysis

By: Imme van den Berg

This research monograph considers the subject of asymptotics from a nonstandard view point. It is intended both for classical asymptoticists - they will discover a new approach to problems very familiar to them - and for nonstandard analysts but includes topics of general interest, like the remarkable behaviour of Taylor polynomials of elementary functions. Noting that within nonstandard analysis, "small", "large", and "domain of validity of asymptotic behaviour" have a precise meaning, a nonstandard alternative to classical asymptotics is developed. Special emphasis is given to applications in numerical approximation by convergent and divergent expansions: in the latter case a clear asymptotic answer is given to the problem of optimal approximation, which is valid for a large class of functions including many special functions. The author's approach is didactical. The book opens with a large introductory chapter which can be read without much knowledge of nonstandard analysis. Here the main features of the theory are presented via concrete examples, with many numerical and graphic illustrations. N

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Page Count

192

Publisher

Springer

Published Date

ISBN 10

3540478108

ISBN 13

9783540478102

Neutrices and External Numbers

A Flexible Number System

By: Bruno Dinis,Imme van den Berg

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Neutrices and External Numbers

A Flexible Number System

By: Bruno Dinis,Imme van den Berg

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Page Count

361

Publisher

CRC Press

Published Date

ISBN 10

1000005380

ISBN 13

9781000005387

The Strength of Nonstandard Analysis

By: Imme van den Berg,Vitor Neves

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The Strength of Nonstandard Analysis

By: Imme van den Berg,Vitor Neves

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.

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Page Count

401

Publisher

Springer

Published Date

ISBN 10

3211499040

ISBN 13

9783211499047

Principles of Infinitesimal Stochastic and Financial Analysis

By: Imme van den Berg

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Principles of Infinitesimal Stochastic and Financial Analysis

By: Imme van den Berg

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.

Average ratings

N/A

Write a review

Solve jigsaw puzzle

Age

Page Count

156

Publisher

World Scientific

Published Date

ISBN 10

9810243588

ISBN 13

9789810243586

Principles Of Infinitesinal Stochastic & Financial Analysis

By: Imme Van Den Berg

Write a review Read reviews Take a Quiz Solve Book Puzzle

Principles Of Infinitesinal Stochastic & Financial Analysis

By: Imme Van Den Berg

Average ratings

N/A

Write a review

Solve jigsaw puzzle

Age

Page Count

150

Publisher

World Scientific

Published Date

ISBN 10

9814492779

ISBN 13

9789814492775

Book's by Imme van den Berg

Neutrices and External Numbers

Neutrices and External Numbers

Principles Of Infinitesinal Stochastic & Financial Analysis

Principles Of Infinitesinal Stochastic & Financial Analysis

Principles of Infinitesimal Stochastic and Financial Analysis

Principles of Infinitesimal Stochastic and Financial Analysis

Nonstandard Asymptotic Analysis

Nonstandard Asymptotic Analysis

Neutrices and External Numbers

Neutrices and External Numbers

The Strength of Nonstandard Analysis

The Strength of Nonstandard Analysis

Principles of Infinitesimal Stochastic and Financial Analysis

Principles of Infinitesimal Stochastic and Financial Analysis

Principles Of Infinitesinal Stochastic & Financial Analysis

Principles Of Infinitesinal Stochastic & Financial Analysis

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