"Radically elementary " stochastic analysis for Wiener and Lévy processes via Internal Set Theory

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One of the most thriving areas of modern probability theory, with applications of paramount importance in physics and economics, is stochastic analysis, even in its most basic form of stochastic analysis on the Wiener space. There is a remarkable contrast, however, between the beauty and simplicity of the conceptual ideas underlying stochastic analysis and the technical sophistication of the current mathematical theory used to formulate these notions. In a pioneering monograph entitled Radically Elementary Probability Theory, Nelson (1987) has developed a potentially revolutionary, radically simplified approach (via infinitesimals) to the theory of continuous-time stochastic processes, based on the axioms of Internal Set Theory (IST). IST, also due to Nelson (1977), is an axiomatization of Robinson’s (1961) nonstandard analysis, which in turn is a consistent (relative to Zermelo- Fraenkel set theory), mathematically rigorous framework for analysis with infinitesimals. Whilst Robinson’s (1961) original account of nonstandard analysis is based on a delicate construction from mathematical logic, Nelson’s (1987) axioms indeed deserve the predicate “elementary”. We shall continue Nelson’s work through developing a radically elementary framework for stochastic analysis. First, we plan to derive “radically elementary” versions of Itô’s formula, of the Feynman-Kac formula and Girsanov’s theorem (on the Wiener space). Thereafter, we plan to lay the foundations of a radically elementary stochastic calculus with respect to more general jump-diffusions, starting with Lévy processes.