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



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.


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


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.


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


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: 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;