This is the second volume in a subseries of the Lecture Notes in Mathematics called Lévy Matters, which is published at irregular intervals over the years. Each volume examines a number of key topics in the theory or applications of Lévy processes and pays tribute to the state of the art of this rapidly evolving subject with special emphasis on the non-Brownian world. The expository articles in this second volume cover two important topics in the area of Lévy processes. The first article by Serge Cohen reviews the most important findings on fractional Lévy fields to date in a self-contained piece, offering a theoretical introduction as well as possible applications and simulation techniques. The second article, by Alexey Kuznetsov, Andreas E. Kyprianou, and Victor Rivero, presents an up to date account of the theory and application of scale functions for spectrally negative Lévy processes, including an extensive numerical overview.
Author(s): Serge Cohen, Alexey Kuznetsov, Andreas Kyprianou, Victor Rivero
Series: Lecture Notes in Mathematics 2061
Publisher: Springer
Year: 2012
Language: English
Pages: 200
Cover......Page 1
Lévy Matters II......Page 4
Preface......Page 6
From the Preface to L´evy Matters I......Page 8
Contents......Page 10
A Short Biography of Paul Lévy......Page 12
The Theory of Scale Functions for Spectrally Negative Lévy Processes......Page 14
2.1 Poisson Random Measure......Page 16
2.2 Lévy Random Measure......Page 17
2.3 Real Stable Random Measure......Page 19
2.4 Complex Isotropic Random Measure......Page 21
3.1 Gaussian Fields......Page 26
3.2.1 Moving Average Fractional Stable Fields......Page 29
3.2.2 Real Harmonizable Fractional Stable Fields......Page 31
4.1 Moving Average Fractional Lévy Fields......Page 33
4.1.1 Regularity of the Sample Paths......Page 36
4.1.2 Local Asymptotic Self-similarity......Page 37
4.2 Real Harmonizable Fractional Lévy Fields......Page 40
4.2.1 Asymptotic Self-similarity......Page 41
4.2.2 Regularity of the Sample Paths of the rhfLf......Page 49
4.4 Real Harmonizable Multifractional Lévy Fields......Page 51
5.1 Estimation for Real Harmonizable Fractional Lévy Fields......Page 53
5.2 Identification of mafLf......Page 58
6 Simulation......Page 66
6.1 Rate of Almost Sure Convergence for Shot Noise Series......Page 68
6.2 Stochastic Integrals Revisited......Page 69
6.3 Generalized Shot Noise Series......Page 71
6.4 Normal Approximation......Page 77
6.6.1 Moving Average Fractional Lévy Fields......Page 82
6.6.2 Case of Finite Control Measures......Page 83
6.6.3 Case of Infinite Control Measures......Page 84
6.6.4 Moving Average Fractional Stable Fields......Page 85
6.6.5 Linear Fractional Stable Motions......Page 87
6.6.6 Log-Fractional Stable Motion......Page 89
6.6.7 Linear Multifractional Stable Motions......Page 90
6.7 Simulation of Harmonizable Fields......Page 92
Appendix......Page 96
References......Page 107
1.1 Spectrally Negative Lévy Processes......Page 110
1.2 Scale Functions and Applied Probability......Page 111
2.1 Some Additional Facts About Spectrally Negative Lévy Processes......Page 120
2.2 Existence of Scale Functions......Page 123
2.3 Scale Functions and the Excursion Measure......Page 127
2.4 Scale Functions and the Descending Ladder Height Process......Page 131
2.5.1 First Passage Problems......Page 132
2.5.2 First Passage Problems for Reflected Processes......Page 137
3.1 Behaviour at 0 and +......Page 138
3.2 Concave–Convex Properties......Page 142
3.3 Analyticity in q......Page 143
3.4 Spectral Gap......Page 146
3.5 General Smoothness and Doney's Conjecture......Page 147
4.1 Construction Through the Wiener–Hopf Factorization......Page 153
4.2 Special and Conjugate Scale Functions......Page 158
4.3 Tilting and Parent Processes Drifting to -......Page 160
4.4 Complete Scale Functions......Page 162
4.5 Generating Scale Functions via an Analytical Transformation......Page 166
5.1 Introduction......Page 169
5.2 Filon's Method and Fractional Fast Fourier Transform......Page 174
5.3.1 The Gaver-Stehfest Algorithm......Page 178
5.3.2 The Euler Algorithm......Page 179
5.3.3 The Fixed Talbot Algorithm......Page 180
5.4 Processes with Jumps of Rational Transform......Page 181
5.5 Meromorphic Lévy Processes......Page 183
5.6 Numerical Examples......Page 186
5.7 Conclusion......Page 193
References......Page 194
BookmarkTitle:......Page 199
