A modern theory of random variation. With applications in stochastic calculus, financial mathematics, and Feynman integration
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.
scientific article; zbMATH DE number 6069715