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.

thanks to Erik Baurdoux who ironically pointed out that, in the penultimate draft of this manuscript, even the original version of the sentence referred to by this footnote was a grammatical mess. 2 Erik also took issue with the wording in footnote 1 above. ix x Preface to the Second Edition since the last edition, and which I believe are accessible at the level that I originally pitched this book. Within existing chapters, I have included new material on the theory of special subordinators

∞. (2.9) S (ii) When condition (2.9) holds, then (with E as expectation with respect to P ) E eiβX = exp − 1 − eiβf (x) η(dx) (2.10) S for any β ∈ R. (iii) Further E(X) = f (x) η(dx) < ∞ when f (x)η(dx) S (2.11) S and 2 E X2 = f (x)2 η(dx) + f (x)η(dx) S S when f (x)2 η(dx) < ∞ f (x) η(dx) < ∞. and S (2.12) S Proof (i) We begin by defining simple functions to be those of the form n f (x) = fi 1Ai (x), i=1 where fi is constant and {Ai : i = 1, . . . , n} are disjoint

. . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 91 92 94 102 104 . . . . . . . . . . 106 110 . . . . . . . . . . xv xvi Contents 5 Subordinators at First Passage and Renewal Measures 5.1 Killed Subordinators and Renewal Measures . . . . 5.2 Overshoots and Undershoots . . . . . . . . . . . . 5.3 Creeping . . . . . . . . . . . . . . . . . . . . . . . 5.4 Regular Variation and

Wiener–Hopf factorisation. Proof of Theorem 6.15 (i) The crux of the first part of the Wiener–Hopf factorisation lies with the following important observation. Consider the Poisson point process of marked excursions on [0, ∞) × E × [0, ∞), B[0, ∞) × Σ × B[0, ∞), dt × dn × dη , where η(dx) = pe−px dx for x ≥ 0. That is to say, consider a Poisson point process (t) (t) whose points are described by {(t, t , ep ) : t ≤ L∞ and t = ∂}, where ep is an independent copy of an exponentially distributed

e−ρi x 1{x>0} + π(x) = i=1 ∞ aˆ i ρˆi eρˆi x 1{x<0} . (6.38) i=1 Here, the constants ai , aˆ i , ρi , ρˆi are non-negative, ρi and ρˆi are arranged in increasing order with limn↑∞ ρn = limn↑∞ ρˆn = ∞ and they satisfy the summability condition ∞ i=1 ai ρi−2 + ∞ aˆ i ρˆi−2 < ∞. (6.39) i=1 Let us pursue a number of remarks concerning this definition. First note that the summability condition (6.39) is sufficient to ensure the integrability condition 2 R (1 ∧ x )π(x)dx < ∞ is satisfied.