Statistical analysis of Lévy processes with application in finance [Elektronische Ressource] / Achim Gegler

Sujets

• Mathematics

Informations

• Publié par: universitat_ulm
• Publié le: 01 janvier 2011
• Nombre de lectures: 10
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Universitat Ulm
Fakultat fur Mathematik und
Wirtschaftswissenschaften
Statistical Analysis of Levy Processes with
Application in Finance
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakultat fur Mathematik und Wirtschaftswissenschaften
der Universitat Ulm
vorgelegt von
Achim Gegler
aus Stuttgart
2011Amtierender Dekan: Prof. Dr. Paul Wentges
1. Gutachter: Prof. Dr. Ulrich Stadtmuller
2. Gutachter: Prof. Dr. Rudiger Kiesel
Tag der Promotion: 17. Oktober 2011Notations
N,R are respectively the set of all natural numbers and the real numbers.R is
dthe conjunction ofR andf 1 ;1g,N is the conjunction ofN andf1g, X is
+the d-fold cartesian product of a set X, it is always assumed that d2N.R is
+
the set (0;1) andR the set [0;1),N is the setN[f0g.00
B( ) is the Borel algebra of a set .
1(:) is the indicator function, with a boolean argument. If the argument is true,
then the indicator function is equal to 1, if it is false then the indicator function
is equal to 0.
a^b := minfa;bg and a_b := maxfa;bg.
The convergence in probability of random variables, random vectors or stochas-
P
tic processes is denoted by! . The convergence in distribution of random
d
variables or random vectors is denoted by! . Weak convergence of a stochas-
D
tic process is denoted by! .
Weak convergence of a probability measure is denoted by).
The identity matrix is denoted by I.
The supremum norm is denoted byjjjj and the Euclidian norm byjj.
The number of elements that are contained in a set A is denoted by #A.
The Landau symbols are denoted byO and o.
w.l.o.g. = without loss of generality
p
T~b := 2h logn and b := (1;:::;d) bn n n
d T 1 2B (D;) :=fx2R :x D xb gn n
CA nonstandard Wiener process is denoted by L .t
JA pure jump Levy process is denoted by L , which can be decomposed intot
CP sJits large jump part L and into its small jump part L .t tContents
Notations iv
1. Introduction 1
1.1. Motivation and summary . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 5
2. Basics 7
2.1. Levy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1. De nition and basic examples . . . . . . . . . . . . . . . . 7
2.1.2. Innite divisibility and Levy Khinchin representation . . . 11
2.1.3. Poisson random measures and Levy-Ito^ decomposition . . 15
2.1.4. Further properties . . . . . . . . . . . . . . . . . . . . . . 19
2.1.5. Parametric models . . . . . . . . . . . . . . . . . . . . . . 22
2.2. Weak convergence in a generalized Skorohod space . . . . . . . . 23
2.2.1. Weak convergence of probability measures . . . . . . . . . 23
2.2.2. Convergence in distribution . . . . . . . . . . . . . . . . . 26
2.2.3. The Skorohod space . . . . . . . . . . . . . . . . . . . . . 28
2.2.4. A generalization of the Skorohod space in higher dimensions 32
3. Estimation method 35
3.1. Estimation setting . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1. Setting in the nite activity case . . . . . . . . . . . . . . 39
3.1.2. Setting in the in nite activity case . . . . . . . . . . . . . 40
3.2. High frequency data of a Wiener process . . . . . . . . . . . . . . 42
3.3. A continuously observed Levy process . . . . . . . . . . . . . . . 44
3.4. Estimation of a nite activity Levy process . . . . . . . . . . . . 48
3.5. Estimation of an in nite activity Levy process . . . . . . . . . . . 55
3.6. Estimation of the Blumenthal-Getoor index . . . . . . . . . . . . 61
3.7. Estimation of the critical region and parameter choice . . . . . . 66
3.7.1. An iteration method for the critical region . . . . . . . . . 66
3.7.2. Parameter choice . . . . . . . . . . . . . . . . . . . . . . . 72
3.8. Correction method in the univariate and nite activity case . . . 74
3.9. Simulations and real data application . . . . . . . . . . . . . . . 79
3.9.1. In the univariate case . . . . . . . . . . . . . . . . . . . . 80
3.9.2. In the multivariate case . . . . . . . . . . . . . . . . . . . 86
3.9.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4. Estimation in the Kou model 91
4.1. Estimation setting . . . . . . . . . . . . . . . . . . . . . . . . . . 91
v