Market completion and robust utility maximization [Elektronische Ressource] / von Matthias Müller

Matthias Müller

Sujets

• Mathematics

Informations

• Publié par: humboldt-universitat_zu_berlin
• Publié le: 01 janvier 2005
• Nombre de lectures: 44
• Langue: English

Extrait

Market Completion and Robust Utility
Maximization
DISSERTATION
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat.)
im Fach Mathematik
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultat II
Humboldt-Universitat zu Berlin
von
Dipl. math. oec. Matthias Muller
geboren am 12.12.1972 in Dresden
Prasident der Humboldt-Universitat zu Berlin:
Prof. Dr. Jurgen Mlynek
Dekan der Mathematisch-Naturwissenschaftlichen Fakultat II:
Prof. Dr. Uwe Kuchler
Gutachter:
1. Prof. Dr. Peter Imkeller
2. Prof. Dr. Alexander Schied
3. Prof. Dr. Said Hamadene
Tag der mundlichen Prufung: 26. Mai 2005 iiAbstract
Inthisthesiswestudytwoproblemsofnancialmathematicsthatareclosely
related. The rst part proposes a method to nd prices and hedging strate-
gies for risky claims exposed to a risk factor that is not hedgeable on a
nancial market. In the second part we calculate the maximal utility and
optimal trading strategies on incomplete markets using Backward Stochastic
Di erential Equations.
We consider agents with incomes exposed to a non–hedgeable external
source of risk who complete the market by creating either a bond or by
signing contracts. Another possibility is a risk bond issued by an insurance
company. The sources of risk we think of may be insurance, weather or
climate risk. Stock prices are seen as exogenuosly given. We calculate prices
for the additional securities such that supply is equal to demand, the market
clears partially. The preferences of the agents are described by expected
utility. In Chapter 2 through Chapter 4 the agents use exponential utility
functions, the model is placed in a Brownian ltration. In order to nd
the equilibrium price, we use Backward Stochastic Di erential Equations.
Chapter5providesaone–periodmodelwheretheagentsuseutilityfunctions
satisfying the Inada condition.
The second part of this thesis considers the robust utility maximization
problem of a small agent on a incomplete nancial market. The model is
placed in a Brownian