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Stochastic Calculus with Infinitesimals

  • Book
  • © 2013

Overview

Authors:
  1. Frederik Herzberg

    1. Institute of Mathematical Economics, Bielefeld University, Bielefeld, Germany

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  • A demonstrably consistent use of infinitesimals permits a radically simplified approach to stochastic calculus
  • Chapters on asset pricing, Lévy processes and the Feynman path integral introduce readers to applications
  • Appendixes explore the relationship with Internal Set Theory and Robinsonian nonstandard analysis
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2067)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xviii
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  2. Infinitesimal Calculus, Consistently and Accessibly

    • Frederik S. Herzberg
    Pages 1-5

  3. Radically Elementary Probability Theory

    • Frederik S. Herzberg
    Pages 7-17

  4. Radically Elementary Stochastic Integrals

    • Frederik S. Herzberg
    Pages 19-34

  5. The Radically Elementary Girsanov Theorem and the Diffusion Invariance Principle

    • Frederik S. Herzberg
    Pages 35-44

  6. Excursion to Financial Economics: A Radically Elementary Approach to the Fundamental Theorems of Asset Pricing

    • Frederik S. Herzberg
    Pages 45-53

  7. Excursion to Financial Engineering: Volatility Invariance in the Black–Scholes Model

    • Frederik S. Herzberg
    Pages 55-59

  8. A Radically Elementary Theory of Itô Diffusions and Associated Partial Differential Equations

    • Frederik S. Herzberg
    Pages 61-70

  9. Excursion to Mathematical Physics: A Radically Elementary Definition of Feynman Path Integrals

    • Frederik S. Herzberg
    Pages 71-75

  10. A Radically Elementary Theory of Lévy Processes

    • Frederik S. Herzberg
    Pages 77-92

  11. Final Remarks

    • Frederik S. Herzberg
    Pages 93-93

  12. Back Matter

    Pages 95-112
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Keywords

About this book

Stochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well. However, existing approaches to stochastic analysis either presuppose various concepts from measure theory and functional analysis or lack full mathematical rigour. This short book proposes to solve the dilemma: By adopting E. Nelson's "radically elementary" theory of continuous-time stochastic processes, it is based on a demonstrably consistent use of infinitesimals and thus permits a radically simplified, yet perfectly rigorous approach to stochastic calculus and its fascinating applications, some of which (notably the Black-Scholes theory of option pricing and the Feynman path integral) are also discussed in the book.

Authors and Affiliations

  • Institute of Mathematical Economics, Bielefeld University, Bielefeld, Germany

    Frederik Herzberg

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