Notes on Simple Interval Calculus: Universal Definitions and Applications to the Exponential Moving Average
Year: 2026
Type: preprint
Abstract: Many functions and signals have no classical derivative: the Weierstrass function and the Kochsnowflake are continuous yet nowhere differentiable, and market prices are commonly modelled the sameway. We read the calculus of finite differences as a single operational definition: on a sampledsignal the derivative is the increment divided by the interval, and the integral is the step-weightedrunning sum. The two are exact inverses, up to the initial value, at any spacing, with no limit, sothe fundamental theorem holds at every step; on a sequence of points this is the nabla calculus oftime scales. As the grid becomes infinitesimal, the derivative, the integral and the fundamentaltheorem pass to their classical forms wherever the limits exist, while the interval calculus remainsdefined where they do not. Because only division is used, the fundamental theorem also holds forcomplex, quaternion and octonion intervals. As an application, the exponential moving average (EMA) isa leaky integrator with an exact two-term inverse: the price is recovered from the current andprevious EMA values. Given the seed, the EMA history carries exactly the same information as the pricehistory. The results are checked on daily E-mini S&P 500 data.
Author: Ariel, Viktor
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214454133
Cites:
Cited by:
Related to: