A modern theory of random variation. With applications in stochastic calculus, financial mathematics, and Feynman integration
type: Item
instance of: scholarly article
zbMATH Open document ID: 1268.60002
DOI: 10.1002/9781118345955
author: Patrick Muldowney
publication date: 2012-08-21T00:00:00Z
Mathematics Subject Classification ID: 60-01
Mathematics Subject Classification ID: 60A05
Mathematics Subject Classification ID: 60G07
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 81Q30
Mathematics Subject Classification ID: 91G70
zbMATH DE Number: 6069715
zbMATH Keywords: integration theory
zbMATH Keywords: random variability
zbMATH Keywords: Brownian motion
zbMATH Keywords: Feynman path integrals
zbMATH Keywords: stochatic calculus
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1002/9781118345955
OpenAlex ID: W4205974323
title: A modern theory of random variation. With applications in stochastic calculus, financial mathematics, and Feynman integration
review text: The book presents the Riemann sum approach to the theory of random variation in a self-contained and illustrative way. To this end, Riemann-complete integration theory is studied in detail which is based on the Henstock-Kurzweil integral. With this theory at hand, experiments, or random variables, are analyzed without the classical measure theoretic probability but based on integration of finitely additive distribution functions. As two main applications, the Feynman path integrals and the stochastic calculus of Brownian motion are treated with this new approach. In the first two chapters, the basic concepts and ideas of the Riemann-complete integral and of random variation are illustrated. The relation to the Riemann, Stieltjes and Burkill integrals is discussed as well as the comparison to the axiomatic theory of probability. In the following chapters Henstock's general integration theory and the reformulation of the classical theory of random variation are rigorously introduced and extensively studied, including complete proofs and various examples. The main part of the book, consisting of Chapters 6 to 8, considers Brownian motion and stochastic integrals in this new framework. The application ranges from Feynman's integral, which is important in quantum mechanics, to the Black-Scholes equation for option pricing. Both topics are discussed in detail. A final chapter contains numerical calculations. Altogether, the book gives a complete and interesting picture of a new approach to random variation that applies in a broad field of stochastic research areas.
reviewed by: Mathias Trabs
description: scientific article; zbMATH DE number 6069715