Titlebook: Stochastic Calculus with Infinitesimals; Frederik Herzberg Book 2013 Springer-Verlag Berlin Heidelberg 2013 03H05; 60G05; 91B25; 81Q30; 60

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978-3-642-33148-0Springer-Verlag Berlin Heidelberg 2013

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Stochastic Calculus with Infinitesimals978-3-642-33149-7Series ISSN 0075-8434 Series E-ISSN 1617-9692

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Radically Elementary Probability Theory,The expressive power of . comes from the fact that it allows for the notions of finite sets with unlimited cardinality, and finite subsets of the reals whose distance is at most an infinitesimal from every point in some non-empty open interval.

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Radically Elementary Stochastic Integrals,For any two processes ., the . of . with respect to ξ is the process . defined by . for all ..

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Frederik HerzbergA demonstrably consistent use of infinitesimals permits a radically simplified approach to stochastic calculus.Chapters on asset pricing, Lévy processes and the Feynman path integral introduce readers

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Infinitesimal Calculus, Consistently and Accessibly,nternal Set Theory [59], motivated by the groundbreaking work of Abraham Robinson [66, 67]. One decade on, Nelson [60] introduced an even more elementary, yet still very powerful, formal system, which we shall review presently.

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Excursion to Mathematical Physics: A Radically Elementary Definition of Feynman Path Integrals,grals in Minimal Internal Set Theory. A summary of these ideas—combined with a brief introduction to radically elementary mathematics for mathematical physicists and some references to previous attempts at formalizing the Feynman path integral by means of nonstandard analysis—can be found in [35].