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edit approved Jan 18, 2023 at 8:04

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I just came across a 2013 book by F. Herzberg entitled "Stochastic Calculus with Infinitesimals", see http://link.springer.com/book/10.1007/978-3-642-33149-7/page/1 "Stochastic Calculus with Infinitesimals" where probability and stochastic analysis are done without having to develop the complexities of measure and integration theory first. Ever since E.Nelson Nelson, such an approach is called "radically elementary" and it really is. What this proves is the new result that stochastic calculus can be done without measure theory.

To give a historical parallel, recall that Leibniz's mentor in mathematics was Huygens. When Huygens first learned of Leibniz's invention of infinitesimal calculus, Huygens was sceptical, and wrote to Leibniz that he is merely doing what Fermat and others have done before him in a different language. What Huygens failed to recognize immediately (but did recognize later) was the generality of the methods and the lucidity of the presentation of Leibniz's new approach. The Nelson-HerzbergNelson–Herzberg approach to stochastic calculus is in a way more significant than merely a new "result", since it provides a new methodology.

I just came across a 2013 book by F. Herzberg entitled "Stochastic Calculus with Infinitesimals", see http://link.springer.com/book/10.1007/978-3-642-33149-7/page/1 where probability and stochastic analysis are done without having to develop the complexities of measure and integration theory first. Ever since E.Nelson, such an approach is called "radically elementary" and it really is. What this proves is the new result that stochastic calculus can be done without measure theory.

To give a historical parallel, recall that Leibniz's mentor in mathematics was Huygens. When Huygens first learned of Leibniz's invention of infinitesimal calculus, Huygens was sceptical, and wrote to Leibniz that he is merely doing what Fermat and others have done before him in a different language. What Huygens failed to recognize immediately (but did recognize later) was the generality of the methods and the lucidity of the presentation of Leibniz's new approach. The Nelson-Herzberg approach to stochastic calculus is in a way more significant than merely a new "result", since it provides a new methodology.

I just came across a 2013 book by F. Herzberg entitled "Stochastic Calculus with Infinitesimals" where probability and stochastic analysis are done without having to develop the complexities of measure and integration theory first. Ever since E. Nelson, such an approach is called "radically elementary" and it really is. What this proves is the new result that stochastic calculus can be done without measure theory.

To give a historical parallel, recall that Leibniz's mentor in mathematics was Huygens. When Huygens first learned of Leibniz's invention of infinitesimal calculus, Huygens was sceptical, and wrote to Leibniz that he is merely doing what Fermat and others have done before him in a different language. What Huygens failed to recognize immediately (but did recognize later) was the generality of the methods and the lucidity of the presentation of Leibniz's new approach. The Nelson–Herzberg approach to stochastic calculus is in a way more significant than merely a new "result", since it provides a new methodology.

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created Apr 11, 2013 at 8:19

Mikhail Katz

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To give a historical parallel, recall that Leibniz's mentor in mathematics was Huygens. When Huygens first learned of Leibniz's invention of infinitesimal calculus, Huygens was sceptical, and wrote to Leibniz that he is merely doing what Fermat and others have done before him in a different language. What Huygens failed to recognize immediately (but did recognize later) was the generality of the methods and the lucidity of the presentation of Leibniz's new approach. The Nelson-Herzberg approach to stochastic calculus is in a way more significant than merely a new "result", since it provides a new methodology.