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Connections Between the Extreme Points of Vandermonde determinants and minimizing risk measure in financial mathematics
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Busitema University, P. O Box 236 Tororo, Uganda..ORCID iD: 0000-0002-1288-9471
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.ORCID iD: 0000-0003-3204-617X
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.ORCID iD: 0000-0002-0139-0747
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.ORCID iD: 0000-0003-4554-6528
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(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this study, we show a connection between the extreme points of Vandermonde determinants and minimizing risk measures in financial mathematics. We illustrate it with an applications to option pricing and optimal portfolio selection.

Keywords [en]
Vandermonde determinant, risk measure, option pricing, optimal portfolio selection
National Category
Natural Sciences
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-51522OAI: oai:DiVA.org:mdh-51522DiVA, id: diva2:1476413
Available from: 2020-10-14 Created: 2020-10-14 Last updated: 2020-11-11Bibliographically approved
In thesis
1. Extreme points of the Vandermonde determinant in numerical approximation, random matrix theory and financial mathematics
Open this publication in new window or tab >>Extreme points of the Vandermonde determinant in numerical approximation, random matrix theory and financial mathematics
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis discusses the extreme points of the Vandermonde determinant on various surfaces, their applications in numerical approximation, random matrix theory and financial mathematics. Some mathematical models that employ these extreme points such as curve fitting, data smoothing, experimental design, electrostatics, risk control in finance and method for finding the extreme points on certain surfaces are demonstrated.

The first chapter introduces the theoretical background necessary for later chapters. We review the historical background of the Vandermonde matrix and its determinant, some of its properties that make it more applicable to symmetric polynomials, classical orthogonal polynomials and random matrices.

The second chapter discusses the construction of the generalized Vandermonde interpolation polynomial based on divided differences. We explore further, the concept of weighted Fekete points and their connection to zeros of the classical orthogonal polynomials as stable interpolation points.

The third chapter discusses some extended results on optimizing the Vandermonde determinant on a few different surfaces defined by univariate polynomials. The coordinates of the extreme points are shown to be given as roots of univariate polynomials.

The fourth chapter describes the symmetric group properties of the extreme points of Vandermonde and Schur polynomials as well as application of these extreme points in curve fitting.

The fifth chapter discusses the extreme points of Vandermonde determinant to number of mathematical models in random matrix theory where the joint eigenvalue probability density distribution of a Wishart matrix when optimized over surfaces implicitly defined by univariate polynomials.

The sixth chapter examines some properties of the extreme points of the joint eigenvalue probability density distribution of the Wishart matrix and application of such in computation of the condition numbers of the Vandermonde and Wishart matrices. 

The seventh chapter establishes a connection between the extreme points of Vandermonde determinants and minimizing risk measures in financial mathematics. We illustrate this with an application to optimal portfolio selection.

The eighth chapter discusses the extension of the Wishart probability distributions in higher dimension based on the symmetric cones in Jordan algebras. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart distributions.

The ninth chapter demonstrates the connection between the extreme points of the Vandermonde determinant and Wishart joint eigenvalue probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras that occur in both the discrete and continuous part of the Gindikin set.

Place, publisher, year, edition, pages
Västerås: Mälardalen University, 2020
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 327
National Category
Mathematical Analysis Probability Theory and Statistics Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-51538 (URN)978-91-7485-484-8 (ISBN)
Public defence
2020-12-14, Lambda +(digitalt Zoom), Mälardalens Högskola, Västerås, 15:15 (English)
Opponent
Supervisors
Funder
Sida - Swedish International Development Cooperation Agency, 316
Available from: 2020-10-16 Created: 2020-10-15 Last updated: 2020-11-23Bibliographically approved

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Muhumuza, Asaph KeikaraLundengård, KarlMalyarenko, AnatoliySilvestrov, Sergei
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