# Fluctuations of Lévy Processes with Applications: Introductory Lectures (2nd Edition) (Universitext)

## Andreas E. Kyprianou

Language: English

Pages: 461

ISBN: B010WFBA56

Format: PDF / Kindle (mobi) / ePub

Lévy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their application appears in the theory of many areas of classical and modern stochastic processes including storage models, renewal processes, insurance risk models, optimal stopping problems, mathematical finance, continuous-state branching processes and positive self-similar Markov processes.

This textbook is based on a series of graduate courses concerning the theory and application of Lévy processes from the perspective of their path fluctuations. Central to the presentation is the decomposition of paths in terms of excursions from the running maximum as well as an understanding of short- and long-term behaviour.

The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Lévy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical tractability.

The second edition additionally addresses recent developments in the potential analysis of subordinators, Wiener-Hopf theory, the theory of scale functions and their application to ruin theory, as well as including an extensive overview of the classical and modern theory of positive self-similar Markov processes. Each chapter has a comprehensive set of exercises.

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(x) be the smallest j for which αj = 1 in the tertiary expansion of k≥1 αk /3k of x. If x ∈ C, then j (x) = ∞ and otherwise, if x ∈ [0, 1]\C, then 1 ≤ j (x) < ∞. The Cantor function is defined as follows f (x) = 1 2j (x) j (x)−1 + i=1 αi 2i+1 for x ∈ [0, 1]. Now consider the function g : [0, 1] → R, given by g(x) = f −1 (x) − ax for a ∈ R. Here, we understand f −1 (x) = inf{θ : f (θ ) > x}. Note that g is monotone if and only if a ≤ 0. Show that g has only positive jumps and the values of x

to −Xt . As was noted in the proof of Theorem 4.10, when 0 < ρ < 1, the limit in distribution of W is independent of w and equal to the distributional limit of Xt − Xt and hence by the previous remarks, is also equal to the distribution of −X∞ . Noting further that P(−X∞ ≤ x) = Px τ0− = ∞ , where τ0− = inf{t > 0 : Xt < 0}, we see that Theorem 4.10 also reads: For all x > 0, Px τ0− ∞ = ∞ = (1 − ρ) ρ k η∗k (x), (4.20) k=0 where, now, η∗0 (x) = 1. However, this is precisely the combined

These potential measures will play an important role in the study of how subordinators cross fixed levels. For this reason, we will devote the remainder of this section to studying some of their analytical properties. One of the most important facts about q-potential measures is that they are closely related to renewal measures. We recall briefly that a renewal process, N = {Nx : x ≥ 0}, counts the number of points in [0, x], for x ≥ 0, of an arrival process on [0, ∞) in which points are laid

Borovkov (1976), Percheskii and Rogozin (1969), Gusak and Korolyuk (1969), Greenwood and Pitman (1980b), Fristedt (1974) and many others. The analytical roots of the so-called Wiener–Hopf method go much further back than these probabilistic references (see Sect. 6.7). The importance of the Wiener– Hopf factorisation is that it gives us information concerning the characteristics of the ascending and descending ladder processes. As indicated earlier, we shall use this knowledge in later chapters to

Queen Elizabeth II, Clément Foucart, Hans Gerber, Sasha Gnedin, Martin Herdegen, Friedrich Hubalek, Lyn Imeson, Robert Knobloch, Takis Konstantopoulos, Alexey Kuznetsov, Eos Kyprianou, Ronnie Loeffen, Juan Carlos Pardo, Pierre Patie, José-Luis Tripitaka Garmendia Pérez, Victor Rivero, Antonio Elbegdorj Murillo Salas, Paavo Salminen, Uwe Schmock, Renming Song, Matija Vidmar, Zoran Vondraˇcek, Long Zhao and Xiaowen Zhou. I must give exceptional thanks to my four current Ph.D. students, Maren