UvA-DARE (Digital Academic Repository) One-sided Markov additive processes and related exit problems Ivanovs, J. Publication date 2011 Document Version Final published version Link to publication Citation for published version (APA): Ivanovs, J. (2011). One-sided Markov additive processes and related exit problems. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:24 Sep 2021 One-sided Markov Additive Processes and Related Exit Problems Jevgenijs Ivanovs 2011 One-sided Markov Additive Processes and Related Exit Problems Jevgenijs Ivanovs Proefschrift Universiteit van Amsterdam ISBN 978-90-8891-311-2 Cover design INIIII design ideas Printed by Proefschriftmaken.nl || Printyourthesis.com Published by Uitgeverij BOXPress, Oisterwijk One-sided Markov Additive Processes and Related Exit Problems ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op dinsdag 6 september 2011, te 10:00 uur door Jevgenijs Ivanovs geboren te Daugavpils, Letland Promotiecommissie Promotor: Prof. dr. M.R.H. Mandjes Prof. dr. ir. O.J. Boxma Overige leden: Prof. dr. S. Asmussen Prof. dr. O. Kella Prof. dr. C.A.J. Klaassen Prof. dr. R. Núñez-Queija Dr. M.R. Pistorius Dr. P.J.C. Spreij Faculteit der Natuurwetenschappen, Wiskunde en Informatica ‘Experience is not what happens to you. It is what you do with what happens to you.’ Aldous Huxley ‘Consistency is contrary to nature, contrary to life. The only completely consistent people are the dead.’ Aldous Huxley v Acknowledgments This book resulted from the research carried out during the four years of my PhD studies. It will always remind me of this amazing period of my life, great experiences, and, certainly, people I met on my way. I would like to express my gratitude to all of you who were there, who counseled, supported and kept me motivated, who shared my free time, my joy and adventures, and to those who loved. At this very point I feel fortunate, content and happy. Thank you all for that! First of all I would like to thank my advisors, Onno and Michel. I came for the first interview to EURANDOM without really knowing what a Poisson process is. You trusted in me and gave me a chance to learn. Somebody said in the beginning of my PhD: ‘it is a big luck to have one advisor like Onno or Michel - you have two’. I have had uncountably many chances to realize this. Thank you for your trust, guidance, support, and kindness. There are so many things, apart from mathematics, I learned from you. I had a privelledge of sharing my time between two institutes: EURANDOM and KdV. I had two offices, two groups of colleagues, two flows of knowledge and expertise. Sometimes it was a bit confusing. I would like to thank Connie, Marlies, Jonelleke, Evelien and Hanneke for all kind of arrangements, their patience and help. EURANDOM, with its steady stream of seminars, visitors, meetings and cel- ebrations, is a wonderful place for a PhD student. I am so thankfull to all EU- RANDOMers. Peter, I felt cozy and welcome in EURANDOM whenever you were there. Thanks for that and for all your help. Yoav, you always made me cheerful - EPPS will never be the same! Çagdas, Seva, Yoni, Andreas, Josine, Ingrid, Brian, Marko, Paul, Florence, Dimitris, Ahmad, Vika, Henrik, Bala, Maria, Bert, Misha, Matthieu, Liqiang, it was a great pleasure to meet you. I wish I spent more time with you outside of working hours. Instead, I would be often sitting half-asleep in a train back to Amsterdam. vi KdV, which is now located in a new building in Science Park, is another won- derful place. I really enjoyed working in the (used to be) open PhD corner room, which we sometimes called ’KdV plein’. Ricardo and Enno, you are among the nicest people I met in my life. I felt lonely on the days you were not in the depart- ment. Paul, Loek, Shota, Piotr, Kamil, Benjamin, Abdel, Nabi, Naser, Arie, and Masoumeh, it was a pleasure to get to know you. I could not imagine we would have so much fun together before I organized that first dinner at my place. I would also like to thank Peter, whose course in measure theoretic probability helped me realize that I do want to do research in mathematics. You are the best lecturer for me. I am grateful to Thomas, whom I assisted in teaching complex analysis. I was amazed by your enthusiasm and involvment. During my PhD I had a number of foreign visits. I am extremely grateful to Offer. Those ten days in Israel were incredible! Thank you also for your nice and inspiring words. I am thankful to Bernardo. It was a great pleasure to collaborate and to visit you in Madrid. Many thanks to Zbyszek for inviting me to Wroclaw and arranging things. I would also like to thank Erik, Ronnie, Martijn, and Neil, whom I met at foreign conferences, for their friendliness and motivating talks. There are many people outside of mathematics community whose support was essential. My sincere friends, you are the ones who make my life meaningful and interesting. Maria, Vlad, Nelly, Roma V., Aline, Andrey, Oana, Masha, Max, Sunil, Lena, Oleg, Roma K., Laurine, Artem, Leha, Jonas, Eline, Julian, you made me feel at home in Amsterdam. Thank you all for being with me, sharing my adventures and my joy from climbing, mountaineering, snowboarding, and travelling. The last years were so great indeed! Climbing or just walking in mountains is a kind of meditation I use to refresh and stimulate myself. I am also thankful to Sarah, Kuba, Slava, Alona, Ilya, Igor, Ruslan, Jieying, Shurik, and Serega, whom I do not see often anymore. It was simply impossible for me not to mention your names in this thesis. Finally, I am in debt to my parents and my relatives for their support, care and love. Специально для вас, мои родные: огромное вам спасибо за поддержку и понимание, за искренний интерес, за заботу и вашу любовь! Прямо как в ‘Поле Чудес’, хочу передать привет маме, папе, бабушке, Андрею, Саше и (уже совсем немаленькой;) Дарине, т. Лиле, д. Андрею, Оле, Диме и моему крестному Саше! Я вас всех очень люблю! Thank you all, Jevgenij 15 July 2011 vii viii Contents Acknowledgments vi Contents ix 1 Introduction 1 1.1 Outline . .4 1.2 Contribution . .6 2 Basic theory 8 2.1 Lévy processes . .9 2.2 Markov additive processes (MAPs) . 10 2.3 Matrix exponent of a MAP . 13 2.4 Perron-Frobenius eigenvalue . 15 2.5 Killing and time reversal . 18 2.6 First passage process . 19 2.7 Phase-type distributions and MAPs . 20 2.8 Reflection . 22 3 First passage: time-reversible case 25 3.1 Main results . 25 3.2 Computational aspects . 28 4 First passage: general theory 31 4.1 Generalized Jordan chains . 33 4.2 Main results . 35 4.3 The matrix integral equation . 39 ix 4.4 Alternative approaches: the analytic method . 40 4.5 Applications via martingale calculations . 45 4.6 Queues and extremum processes . 46 5 Markov-modulated Brownian motion (MMBM) in a strip 50 5.1 Preliminaries . 51 5.2 Transform of the stationary distribution and the loss vectors . 54 5.3 Two-sided exit . 57 5.4 Stationary distribution revisited . 59 5.5 On alternative approaches . 62 6 MMBM in a strip: inverse of a regulator 65 6.1 Main results . 66 6.2 Proof of Theorem 6.1 . 68 6.3 Back to the loss vectors . 71 6.4 Special cases . 72 7 The scale matrix 74 7.1 Auxiliary process . 76 7.2 Path properties of certain Lévy processes . 77 7.3 Occupation densities of a MAP . 81 7.4 Two-sided exit . 83 7.5 The scale matrix and its transform . 85 7.6 The general case . 87 7.7 First examples . 90 8 Further exit problems 93 8.1 First passage over a negative level . 94 8.2 Arrival of the first excursion exceeding a certain height . 96 8.3 Two-sided reflection . 97 8.4 Exit of the reflected process . 99 Appendix 102 A.1 Location of eigenvalues . 102 A.2 Jordan normal form . 103 A.3 Matrix norms and convergence . 104 A.4 Transition rate matrix . 106 A.5 Analytic functions of a complex variable . 107 A.6 Differentiation under the integral sign . 108 x A.7 The Laplace transform . 109 A.8 Various relations . 110 Bibliography 112 List of symbols 118 Index 121 Summary 123 Samenvatting 125 About the author 127 xi xii Chapter 1 Introduction The Compound Poisson Process (CPP) is one of the most basic and popular models in applied probability.