Andreas E. Kyprianou Introductory Lectures ´ onFluctuations of Levy Processes with Applications With 22Figures 123 Andreas E. Kyprianou Departmentof Mathematical Sciences University o f B ath Bath BA2 7AY UK e-mail:[email protected] The background text on the front cover is written in old (pre-1941) Mongolian scripture. It is a translation of the words ‘stochastic processes with stationary and independent increments’ and should be readf rom the top left hand corner of the back cover to the bottom right hand corner of the front cover. MathematicsSubjectClassification(2000):60G50, 60G51, 60G52 LibraryofCongressControlNumber:2006924567 ISBN-10 3-540-31342-7 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-31342-7 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerial isconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Dupli- cationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGerman CopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbe obtainedfromSpringer.ViolationsareliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com ©Springer-VerlagBerlinHeidelberg2006 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Coverdesign:ErichKirchner,Heidelberg Typesettingbytheauthor and SPiusingaSpringerLATEXmacropackage Printedonacid-freepaper SPIN 11338987 41/3100/SPi-543210 Preface In 2003 I began teaching a course entitled L´evy processes on the Amsterdam- Utrecht masters programme in stochastics and financial mathematics. Quite naturally, I wanted to expose my students to my own interests in L´evy processes;thatis,therolethatcertainsubtlebehaviourconcerningtheirfluc- tuations play in explaining different types of phenomena appearing in a num- ber of classical models of applied probability. Indeed, recent developments in the theory of L´evy processes, in particular concerning path fluctuation, have offered the clarity required to revisit classical applied probability models and improve on well established and fundamental results. Whilst teaching the course I wrote some lecture notes which have now matured into this text. Given the audience of students, who were either en- gagedintheir‘afstudeerfase’1orjuststartingaPh.D.,theselecturenoteswere originally written with the restriction that the mathematics used would not surpassthelevelthattheyshouldinprinciplehavereached.Roughlyspeaking that means the following: experience to the level of third year or fourth year university courses delivered by a mathematics department on - foundational real and complex analysis, - basic facts about Lp spaces, - measuretheory,integrationtheoryandmeasuretheoreticprobabilitytheory, - elementsoftheclassicaltheoryofMarkovprocesses,stoppingtimesandthe Strong Markov Property. - Poisson processes and renewal processes, - BrownianmotionasaMarkovprocessandelementarymartingaletheoryin continuous time. For the most part this affected the way in which the material was handled compared to the classical texts and research papers from which almost all of the results and arguments in this text originate. A good example of this is 1The afstudeerfase is equivalent to at least a European masters-level programme. VIII Preface the conscious exclusion of calculations involving the master formula for the Poisson point process of excursions of a L´evy process from its maximum. There are approximately 80 exercises and likewise these are pitched at a level appropriate to the aforementioned audience. Indeed several of the exer- cises have been included in response to some of the questions that have been asked by students themselves concerning curiosities of the arguments given in class. Arguably the exercises are at times quite long. Such exercises reflect some of the other ways in which I have used preliminary versions of this text. A small number of students in Utrecht also used the text as an individual reading/self-study programme contributing to their ‘kleinescripite’ (extended mathematical essay) or ‘onderzoekopdracht’ (research option); in addition, some exercises were used as (take-home) examination questions. The exer- cises in the first chapter in particular are designed to show the reader that the basics of the material presented thereafter is already accessible assuming basic knowledge of Poisson processes and Brownian motion. There can be no doubt, particularly to the more experienced reader, that the current text has been heavily influenced by the outstanding books of Bertoin (1996) and Sato (1999), and especially the former which also takes a predominantly pathwise approach to its content. It should be reiterated how- ever that, unlike the latter two books, this text is not intended as a research monograph nor as a reference manual for the researcher. Writing of this text began whilst I was employed at Utrecht University, The Netherlands. In early 2005 I moved to a new position at Heriot Watt University in Edinburgh, Scotland, and in the final stages of completion of the book to The University of Bath. Over a period of several months my presenceinUtrechtwasphasedoutandmypresenceinEdinburghwasphased in. Along the way I passed through the Technical University of Munich and TheUniversityofManchester.Ishouldliketothankthesefourinstitutesand my hosts for giving me the facilities necessary to write this text (mostly time and awarm, dry,quiet roomwith an ethernet connection). Iwould especially like to thank my colleagues at Utrecht for giving me the opportunity and environmentinwhichtodevelopthiscourse,RonDoneyduringhistwo-month absence for lending me the key to his office and book collection whilst mine was in storage and Andrew Cairns for arranging to push my teaching duties into 2006 allowing me the focus to finalise this text. Let me now thank the many, including several of the students who took the course, who have made a number of remarks, corrections and suggestions (minorandmajor)whichhavehelpedtoshapethistext.Inalphabeticalorder these are: Larbi Alili, David Applebaum, Johnathan Bagley, Erik Baurdoux, M.S. Bratiychuk, Catriona Byrne, Zhen-Qing Chen, Gunther Cornelissen, IrmingardErder,AbdelghafourEs-Saghouani,SergueiFoss,UweFranz,Shota Gugushvili, Thorsten Kleinow, Pawel(cid:3) Kliber, Claudia Klu¨ppelberg, V.S. Korolyuk, Ronnie Loeffen, Alexander Novikov, Zbigniew Palmowski, Goran Peskir, Kees van Schaik, Sonja Scheer, Wim Schoutens, Budhi Arta Surya, Enno Veerman, Maaike Verloop, Zoran Vondraˇcek. In particular I would also Preface IX liketothank,PeterAndrew,JeanBertoin,RonDoney,NielFarricker,Alexan- der Gnedin, Amaury Lambert, Antonis Papapantoleon and Martijn Pistorius rootedoutmanyerrorsfromextensivesectionsofthetextandprovidedvalu- ablecriticism.AntonisPapapantoleonverykindlyproducedsomesimulations ofthepathsofL´evyprocesseswhichhavebeenincludedinChap.1.Iammost grateful to Takis Konstantopoulos who read through earlier drafts of the en- tire text in considerable detail, taking the time to discuss with me at length many of the issues that arose. The front cover was produced in consultation withHurleeGonchigdanzanandJargalmaaMagsarjav.Allfurthercomments, corrections and suggestions on the current text are welcome. Finally, the deepest gratitude of all goes to Jagaa, Sophia and Sanaa for whom the special inscription is written. Edinburgh Andreas E. Kyprianou June 2006 Contents 1 L´evy Processes and Applications .......................... 1 1.1 L´evy Processes and Infinite Divisibility..................... 1 1.2 Some Examples of L´evy Processes ......................... 5 1.3 L´evy Processes and Some Applied Probability Models........ 14 Exercises ................................................... 26 2 The L´evy–Itˆo Decomposition and Path Structure.......... 33 2.1 The L´evy–Itˆo Decomposition.............................. 33 2.2 Poisson Random Measures................................ 35 2.3 Functionals of Poisson Random Measures................... 41 2.4 Square Integrable Martingales............................. 44 2.5 Proof of the L´evy–Itoˆ Decomposition ...................... 51 2.6 L´evy Processes Distinguished by Their Path Type ........... 53 2.7 Interpretations of the L´evy–Itˆo Decomposition .............. 56 Exercises ................................................... 62 3 More Distributional and Path-Related Properties.......... 67 3.1 The Strong Markov Property ............................. 67 3.2 Duality ................................................ 73 3.3 Exponential Moments and Martingales ..................... 75 Exercises ................................................... 83 4 General Storage Models and Paths of Bounded Variation.. 87 4.1 General Storage Models .................................. 87 4.2 Idle Times.............................................. 88 4.3 Change of Variable and Compensation Formulae ............ 90 4.4 The Kella–Whitt Martingale .............................. 97 4.5 Stationary Distribution of the Workload....................100 4.6 Small-Time Behaviour and the Pollaczek–Khintchine Formula.102 Exercises ...................................................105 XII Contents 5 Subordinators at First Passage and Renewal Measures ....111 5.1 Killed Subordinators and Renewal Measures ................111 5.2 Overshoots and Undershoots..............................119 5.3 Creeping ...............................................121 5.4 Regular Variation and Tauberian Theorems.................126 5.5 Dynkin–Lamperti Asymptotics ............................130 Exercises ...................................................133 6 The Wiener–Hopf Factorisation............................139 6.1 Local Time at the Maximum..............................140 6.2 The Ladder Process......................................147 6.3 Excursions..............................................154 6.4 The Wiener–Hopf Factorisation ...........................157 6.5 Examples of the Wiener–Hopf Factorisation.................168 6.6 Brief Remarks on the Term “Wiener–Hopf”.................174 Exercises ...................................................174 7 L´evy Processes at First Passage and Insurance Risk ......179 7.1 Drifting and Oscillating ..................................179 7.2 Cram´er’s Estimate of Ruin ...............................185 7.3 A Quintuple Law at First Passage .........................189 7.4 The Jump Measure of the Ascending Ladder Height Process ..195 7.5 Creeping ...............................................197 7.6 Regular Variation and Infinite Divisibility ..................200 7.7 Asymptotic Ruinous Behaviour with Regular Variation.......203 Exercises ...................................................206 8 Exit Problems for Spectrally Negative Processes...........211 8.1 Basic Properties Reviewed................................211 8.2 The One-Sided and Two-Sided Exit Problems...............214 8.3 The Scale Functions W(q) and Z(q) ........................220 8.4 Potential Measures ......................................223 8.5 Identities for Reflected Processes ..........................227 8.6 Brief Remarks on Spectrally Negative GOUs................231 Exercises ...................................................233 9 Applications to Optimal Stopping Problems ...............239 9.1 Sufficient Conditions for Optimality........................240 9.2 The McKean Optimal Stopping Problem ...................241 9.3 Smooth Fit versus Continuous Fit .........................245 9.4 The Novikov–Shiryaev Optimal Stopping Problem ...........249 9.5 The Shepp–Shiryaev Optimal Stopping Problem.............255 9.6 Stochastic Games........................................260 Exercises ...................................................269 Contents XIII 10 Continuous-State Branching Processes.....................271 10.1 The Lamperti Transform .................................271 10.2 Long-term Behaviour ....................................274 10.3 Conditioned Processes and Immigration ....................280 10.4 Concluding Remarks.....................................291 Exercises ...................................................293 Epilogue.......................................................295 Solutions ......................................................299 References.....................................................361 Index..........................................................371