Cambridge Tracts in Mathematics, Volume 121This is an up-to-date and comprehensive account of the theory of Lévy processes. This branch of modern probability theory has been developed over recent years and has many applications in such areas as queues, mathematical finance and risk estimation. Professor Bertoin has used the powerful interplay between the probabilistic structure (independence and stationarity of the increments) and analytic tools (especially Fourier and Laplace transforms) to give a quick and concise treatment of the core theory, with the minimum of technical requirements. Special properties of subordinators are developed and then appear as key features in the study of the local times of real-valued Lévy processes and in fluctuation theory. Lévy processes with no positive jumps receive special attention, as do stable processes. In sum, this will become the standard reference on the subject for all working probability theorists. |
Contents
Preliminaries | 1 |
2 Infinitely divisible distributions | 2 |
3 Martingales | 3 |
4 Poisson processes | 4 |
5 Poisson measures and Poisson point processes | 6 |
6 Brownian motion | 8 |
7 Regular variation and Tauberian theorems | 9 |
Levy Processes as Markov Processes | 11 |
5 The cases of holding points and of irregular points | 121 |
6 Exercises | 123 |
7 Comments | 124 |
Local Times of a Levy Process | 125 |
2 Hilbert transform of local times | 134 |
3 Jointly continuous local times | 143 |
4 Exercises | 150 |
5 Comments | 153 |
2 Markov property and related operators | 18 |
3 Absolutely continuous resolvents | 24 |
4 Transience and recurrence | 31 |
5 Exercises | 39 |
6 Comments | 41 |
Elements of Potential Theory | 43 |
2 Capacitary measure | 48 |
3 Essentially polar sets and capacity | 53 |
4 Energy | 56 |
5 The case of a single point | 61 |
6 Exercises | 68 |
7 Comments | 70 |
Subordinators | 71 |
2 Passage across a level | 75 |
3 The arcsine laws | 81 |
4 Rates of growth | 84 |
5 Dimension of the range | 93 |
6 Exercises | 99 |
7 Comments | 100 |
Local Time and Excursions of a Markov Process | 103 |
2 Construction of the local time | 105 |
3 Inverse local time | 112 |
4 Excursion measure and excursion process | 116 |
Fluctuation Theory | 155 |
2 Fluctuation identities | 159 |
3 Some applications of the ladder time process | 166 |
4 Some applications of the ladder height process | 171 |
5 Increase times | 176 |
6 Exercises | 182 |
7 Comments | 184 |
Levy Processes with no Positive Jumps | 187 |
2 The scale function | 194 |
3 The process conditioned to stay positive | 198 |
4 Some path transformations | 206 |
5 Exercises | 212 |
6 Comments | 214 |
Stable Processes and the Scaling Property | 216 |
2 Some sample path properties | 222 |
3 Bridges | 226 |
4 Normalized excursion and meander | 232 |
5 Exercises | 237 |
6 Comments | 240 |
| 242 | |
List of symbols | 261 |
| 264 | |
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Common terms and phrases
absolutely continuous Applying the Markov arcsine law argument assertion Bertoin Blumenthal bounded variation Brownian motion characteristic exponent compact compound Poisson process converges Corollary deduce denote density distribution with parameter Doney drift coefficient dual essentially polar excursion intervals exponential distribution finite follows Fourier Fristedt function f Gaussian Gebiete Getoor identity increasing independent exponential inequality inf{t infimum infinitely divisible integral inverse ladder height process Laplace exponent Lebesgue measure Lemma Lévy measure Lévy process Lévy-Khintchine formula lim inf lim sup Markov processes Markov property martingale Math measurable function passage Poisson point process positive jumps probability measure processes with stationary proof of Theorem prove Pruitt q-excessive Radon measure random variables random walks real number Recall recurrent reflected process regularly varying right-continuous sample paths scaling property semigroup stable process Stochastic Process subordinator Suppose supremum symmetric transient Wahrscheinlichkeitstheorie verw X₁