Stochastic Calculus with Infinitesimals
By: by Frederik Herzberg
Contributors: SpringerLink (Online service)
Series: Lecture Notes in Mathematics -- 2067
Publisher: Springer Berlin Heidelberg; Imprint: Springer
Published: 2013
Place of publication: Berlin, Heidelberg
Language: eng
Format: [electronic resource] /
Number of pages: 112
ISBN-13: 9783642331497
Subjects: Probability Theory and Stochastic Processes; Mathematical physics; Symbolic and mathematical Logic; Game Theory, Economics, Social and Behav. Sciences; Mathematics; Distribution (Probability theory); Mathematical Logic and Foundations
Description: 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.
Table of contents: title: 1 Infinitesimal calculus, consistently and accessibly; title: 2 Radically elementary probability theory; title: 3 Radically elementary stochastic integrals; title: 4 The radically elementary Girsanov theorem and the diffusion invariance principle; title: 5 Excursion to nancial economics: A radically elementary approach to the fundamental theorems of asset pricing; title: 6 Excursion to financial engineering: Volatility invariance in the Black-Scholes model; title: 7 A radically elementary theory of Itô diffusions and associated partial differential equations; title: 8 Excursion to mathematical physics: A radically elementary definition of Feynman path integrals; title: 9 A radically elementary theory of Lévy processes; title: 10 Final remarks.