Dynamic risk management with Markov decision processes

154 Pages·2008·3.167 MB·English

by  André Philipp Mundt

André Philipp Mundt
Dynamic risk management with Markov decision processes

Dynamic risk management with
Markov decision processes
von
André Philipp Mundt
Dissertation, Universität Karlsruhe (TH)
Fakultät für Mathematik, 2007
Referenten: Prof. Dr. Nicole Bäuerle, Prof. Dr. Ulrich Rieder
Impressum
Universitätsverlag Karlsruhe
c/o Universitätsbibliothek
Straße am Forum 2
D-76131 Karlsruhe
www.uvka.de
Dieses Werk ist unter folgender Creative Commons-Lizenz
lizenziert: http://creativecommons.org/licenses/by-nc-nd/2.0/de/
Universitätsverlag Karlsruhe 2008
Print on Demand
ISBN: 978-3-86644-200-9
Preface
During almost four years of work at the Kompetenzzentrum Versicherungswissen-
schaften GmbH, Leibniz Universita¨t Hannover, and the Institut fu¨r Stochastik,
Universita¨t Karlsruhe (TH), many people supported me.
First of all, I would like to thank my supervisor Nicole B¨auerle. She draw me to
the subject of dynamic risk management and made many fruitful suggestions for
the research treated in this work. Furthermore, she always took time for me and
encouraged me when I was stuck and came to her with questions and problems.
Secondly, let me thank Ulrich Rieder for being the second advisor and for useful
discussions.
From all the other people, I firstly and most importantly have to mention Anja
Blatter. She read almost the whole manuscript and made many valuable comments
that led to improvements in formulations and the layout. Furthermore, I have to
thank Gunther Amt and Bruno Ebner for helping me improve parts of the thesis
and Mirko K¨otter and Lars Michael Hoffmann for useful discussions.
Altogether, my colleagues and former colleagues from the Kompetenzzentrum
Versicherungswissenschaften GmbH, the Institut fu¨r Mathematische Stochastik in
Hanover and the Institut fu¨r Stochastik in Karlsruhe always provided a friendly
and enjoyable atmosphere at work and during working breaks. When I moved to
Hanover and Karlsruhe they made it very easy for me to settle in there.
Finally, I have to mention my family and in particular my parents for supporting
my studies and their confidence in my ability to complete my diploma and this
thesis.
Karlsruhe,
October 2007 Andr´e Mundt
v

Contents
Introduction ix
1 Static and conditional risk measures 1
1.1 Model and definition . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Further properties and representations . . . . . . . . . . . . . . . . 8
1.4 Remarks on conditional risk measures and measurability . . . . . . 11
2 Portfolio optimization with constraints in a binomial model 15
2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Risk minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 The unconstrained case . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 A constraint on the final value . . . . . . . . . . . . . . . . . 21
2.3 Utility maximization . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 Comparison with the risk minimization . . . . . . . . . . . . 34
2.3.2 The unconstrained case . . . . . . . . . . . . . . . . . . . . . 36
2.3.3 Intermediate constraints . . . . . . . . . . . . . . . . . . . . 37
3 Dynamic risk measures 51
3.1 An overview on the literature . . . . . . . . . . . . . . . . . . . . . 51
3.2 Definitions, axioms and properties . . . . . . . . . . . . . . . . . . . 55
3.3 Stable sets of probability measures . . . . . . . . . . . . . . . . . . 62
4 A risk measure by Pflug and Ruszczyn´ski 67
4.1 Definition of the dynamic risk measure . . . . . . . . . . . . . . . . 67
4.2 Properties of the dynamic risk measure . . . . . . . . . . . . . . . . 70
4.3 Solution via Markov decision processes . . . . . . . . . . . . . . . . 73
4.4 A stable representation result . . . . . . . . . . . . . . . . . . . . . 84
4.5 Martingales & co. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
vii
5 A Bayesian control approach 93
5.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 A Bayesian control approach to dynamic risk measures . . . . . . . 94
5.3 Explicit solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3.1 The case when the parameter ϑ is known . . . . . . . . . . . 101
5.3.2 Beta distributions as initial distribution . . . . . . . . . . . 102
5.4 Comparison of value functions . . . . . . . . . . . . . . . . . . . . . 109
5.4.1 A comparison result for general distributions . . . . . . . . . 109
5.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A Absolutely continuous probability measures 125
B The Beta distribution 129
Bibliography 131