Stochastic Calculus with Infinitesimals

  • Frederik Herzberg

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

About this book

Introduction

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.

Keywords

03H05; 60G05; 91B25; 81Q30; 60G51; 60H05; 60H10 Feynman path integral Internal Set Theory asset pricing infinitesimals

Authors and affiliations

  • Frederik Herzberg
    • 1
  1. 1.Institute of Mathematical EconomicsBielefeld UniversityBielefeldGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-33149-7
  • Copyright Information Springer-Verlag Berlin Heidelberg 2013
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-33148-0
  • Online ISBN 978-3-642-33149-7
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • About this book