Stochastic calculus with infinitesimals

Abstract

This short monograph develops basic stochastic analysis including Itô's formula, Girsanov's theorem, the Feynman Kac formula, and results about Lévy processes with finite-variation jump part and select applications in the framework of Edward Nelson's Radically elementary probability theory [Annals of Mathematics Studies, 117, Princeton, NJ: Princeton University Press, 1987]. This approach requires neither measure-theoretic probability theory nor functional analysis, but is based on a rigorous, yet elementary use of unlimited natural numbers and infinitesimals. The underlying axiomatic framework, a modest subsystem of Nelson's Internal Set Theory (IST) [Bulletin of the American Mathematical Society, 83(6):1165 1198, 1977] and hence called Minimal Internal Set Theory, is truly elementary and can be easily motivated through the incompleteness of the Peano axioms or an ultrapower construction. (As a subsystem of IST, it is also conservative over and hence consistent relative to conventional mathematics, i.e. ZFC; moreover, a substantial fragment of it also admits an accessible relative consistency proof.) In an excursion, the "radically elementary" approach to stochastic analysis will be employed to provide a "radically elementary" proof of the fundamental theorems of asset pricing. As an example for applications of Minimal Internal Set Theory in mathematical physics, a fully rigorous "radically elementary" definition of the Feynman path integral is proposed. All these features recommend Minimal Internal Set Theory as a suitable framework for teaching stochastic analysis to finance or physics students without previous training in pure mathematics. The book is self-contained and written in expository style; in particular, it assumes no prior knowledge of nonstandard analysis. © Springer-Verlag Berlin Heidelberg 2013.