Approximation of Brownian Motion on Simple Graphs

George A. Anastassiou () and Dimitra Kouloumpou
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George A. Anastassiou: Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
Dimitra Kouloumpou: Section of Mathematics, Hellenic Naval Academy, 18539 Piraeus, Greece

Mathematics, 2023, vol. 11, issue 20, 1-27

Abstract: This article is based on chapters 9 and 19 of the new neural network approximation monograph written by the first author. We use the approximation properties coming from the parametrized and deformed neural networks based on the parametrized error and q -deformed and β -parametrized half-hyperbolic tangent activation functions. So, we implement a univariate theory on a compact interval that is ordinary and fractional. The result is the quantitative approximation of Brownian motion on simple graphs: in particular over a system S of semiaxes emanating from a common origin radially arranged and a particle moving randomly on S . We produce a large variety of Jackson-type inequalities, calculating the degree of approximation of the engaged neural network operators to a general expectation function of this kind of Brownian motion. We finish with a detailed list of approximation applications related to the expectation of important functions of this Brownian motion. The differentiability of our functions is taken into account, producing higher speeds of approximation.

Keywords: neural network operators; Brownian motion on simple graphs; expectation; quantitative approximation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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