Polynomial optimization : Applications in finance
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Julkaisu on tekijänoikeussäännösten alainen. Teosta voi lukea ja tulostaa henkilökohtaista käyttöä varten. Käyttö kaupallisiin tarkoituksiin on kielletty.
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This thesis discusses the theory of modern polynomial optimization and its applications
in the field of finance. From a theoretical point of view, special attention
is directed towards examining and proving the finite convergence of the so-called
Lasserre's hierarchy which serves as the backbone of the polynomial optimization
procedure detailed in this thesis.
The first two sections of the thesis mainly deal with the relevant background theory.
(Positive) polynomials, moment problems, and polynomial optimization are introduced.
The culmination of the second section is the proof of finite convergence of
Lasserre's hierarchy.
The third section provides an overview of algorithmic implementation of the polynomial
optimization methodology. Relevant algorithms are described in detail and
various issues pertaining to the implementation are discussed.
The fourth section consists of numerical examples from the field of finance and
beyond. Each example starts with a problem statement and an overview of how
the question tackled can be stated as a polynomial optimization problem. Explicit
numerical examples illustrate the proposed methods. Most importantly, it is acknowledged
that via the polynomial optimization procedure one is able to obtain
numerically verified global optimal solutions to problems which are often solved in
earlier literature using various heuristic (quasi-global) methods. The fifth section
concludes the thesis.