New Proofs of the Basel Problem using Stochastic Processes



Uwe Hassler1, Mehdi H. Kouchack1, *
1 Department of Business Administratio and Economics, Goethe University, Frankfurt, Germany

Abstract

Background:

The variance of the standard Gumbel (or extreme value) distribution is equal to . This number also solves the so-called Basel problem which is equal to This latter problem was first solved by Leonhard Euler. A more detailed historical exposition is provided in Section 2.

Objective:

Our first contribution is to classify the multitude of earlier proofs in Section 3. The second contribution consists of a new class of proofs.

Methods:

Our method of proofs is rooted in the theory of stochastic processes. It relies on the Karhunen-Loève expansion of a Gaussian process and the related eigenstructure of the covariance kernel.

Results:

Using the cases of a (standard) Wiener process, of a demeaned or a detrended Wiener process, we provide new simple proofs of

Conclusion:

We outlined the eigenstructure of different Gaussian processes that could be used to solve the Basel problem and give, as an example, the so-called Brownian bridge.

Keywords: Basel Problem, Stochastic Processes, Variance, Karhunen-Loève expansion, Mathematical Bernovlli dynasty, Taylor expansions.


Abstract Information


Identifiers and Pagination:

Year: 2019
Volume: 10
Publisher Item Identifier: EA-TOSPJ-2017-2447

Article History:

Received Date: 02/02/2019
Revision Received Date: 20/01/2019
Acceptance Date: 05/03/2019
Electronic publication date: 25/10/2019
Collection year: 2019

© 2019 Hassler and H. Kouchack

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: https://creativecommons.org/licenses/by/4.0/legalcode. This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.


* Address correspondence to this author at Business Administratio and Economics, Goethe University Frankfurt, Theodor-W.-Adorno-Platz 4, Frankfurt, Germany; Tel: 00496979834771;
E-mail: hosseinkouchack@wiwi.uni-frankfurt.de