Author(s)

Michael Grabinski, Galiya Klinkova

Department of Business and Economics, Neu-Ulm University, Neu-Ulm, Germany.

It is well known that an integral is nothing but a continuous form of a sum. Is it possible to do the same thing with a product? The answer is yes and done for the first time in this publication. The new operator is called inteduct. As an integral is a proper tool to calculate the arithmetic mean of a function, the inteduct gives the geometric mean of a function. This defines a new branch of mathematics. Most applications may lay way ahead. Only some are discussed here. One is applying the inteduct to probability theory. There it is possible e.g., to determine a function for a life expectation rather than just a numerical value. Another application is to distinguish chaos from randomness within numerically given values. At least for the logistic map there exists a direct connection between Lyapunov exponent and inteduct. To distinguish between chaos and randomness is particularly important in finance. While randomness implies ergodicity, chaos is non-ergodic. And many fundamental financial theories from portfolio theory to market efficiency require ergodicity.

Geometric Mean, Chaos, Finance, Ergodicity

Grabinski, M. and Klinkova, G. (2023) Like a Sum Is Generalized into an Integral, a Product May Be Generalized into an Inteduct. Applied Mathematics, 14, 279-289. doi: 10.4236/am.2023.145017.