MATRIX ANALYTIC METHODS APPLIED TO VARIOUS RISK PROCESSES (Thesis format: monograph) by Kaiqi Yu Graduate Program in Statistics Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario August 2008 © Kaiqi Yu 2008 Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-43106-1 Our file Notre reference ISBN: 978-0-494-43106-1 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. 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Canada THE UNIVERSITY OF WESTERN ONTARIO SCHOOL OF GRADUATE AND POSTDOCTORAL STUDIES CERTIFICATE OF EXAMINATION Supervisor Examiners Dr. David A. Stanford Dr. Kristina Sendova Co-Supervisor Dr. Bruce Jones Dr. Jiandong Ren Dr. Henning Rasmussen Supervisory Committee Dr. Qi-ming He The thesis by Kaiqi Yu entitled: MATRIX ANALYTIC METHODS APPLIED TO VARIOUS RISK PROCESSES is accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Date Chair of Thesis Examination Board ii ABSTRACT Finding the ruin probability has been a popular topic in the literature of actuarial science. Since the finding of the connection between fluid flows and risk processes, the analysis technique of matrix analytic methods has been successfully applied to analyze a large class of risk processes. In this thesis, we continue along this line by applying matrix analytic methods to obtain ruin time related quantities arising in risk theory and the busy period in stochastic fluid flows as well. The first issue of the thesis develops a unified approach for computing ruin probabilities for both infinite and finite time horizons for a large class of perturbed risk models. Specific members of this class include, among others, the perturbed Sparre-Andersen model when distributions of inter-claim time and claim sizes are both phase-type, the perturbed Markov-modulated risk processes, and the per­ turbed risk processes with inter-claim times distributed according to a Markovian Arrival process (MAP) and claim sizes being phase-type. From a ruin-theoretic point of view, the thesis establishes a unified, tractable phase-type structure for all variants of the class, and presents explicit formulas of manageable size for the ruin probability for models in this class. The key to the solution method is the ability to identify a sample-path-equivalent continuous-time Markov-modulated process with the risk process, and this is the sole limiting factor to the class of models that can be considered. iii Whereas the analysis is exact in the case of the ultimate ruin probability, for finite time ruin probability, the Erlangization method is applied to the risk process to develop an approximation. An efficient recursive algorithm based on the block structure of the underlying generator matrices is given to perform the Erlangiza­ tion. The second issue in the thesis provides a method to calculate the moments of the risk processes, which are the non-perturbed versions of the risk processes in the previous class. Based on the Laplace transform of the time of ruin, a recursive formula is established for the moments of the time of ruin by taking derivatives of the relevant Laplace transform. Numerical examples show that the initial procedure is unstable, although theoretically the result is valid. To overcome this problem, the uniformization technique is applied first to the underlying Markov process, which ensures that the algorithms obtained are stable and efficient. Finally, we present similar procedures to computing the mean busy period of the perturbed fluid flow models. For all cases, numerical examples are given throughout the thesis to illustrate the methodologies we developed. Keywords: matrix analytic method, risk process, Markov-modulated risk model, continuous-time Markov-modulated process, time of ruin, ruin probability, fluid flow, Laplace-Stieltjes transform, moment, uniformization, busy period IV CO-AUTHORSHIP STATEMENT The materials presented in the thesis have been supervised by Dr. David A. Stanford and Dr. Jiandong Ren. Dr. Stanford and Dr. Ren proposed the study of the materials in Chapter 3. The goal is to obtain the ultimate ruin probability of perturbed risk models in a unified, tractable way in terms of the phase-type structures. My idea to ex­ ploit Asmussen [10] achieved this goal and we jointly determined the computation algorithm. The idea of using Erlangization to approximate the finite time ruin probability in perturbed risk models in Chapter 4 was initiated by Dr. Stanford and we jointly formed the generator of the composite underlying Markov process for the perturbed risk model. Based on this idea, I developed the algorithm to approximate the finite time ruin probability. The algorithm is brute-force. Dr. Stanford and Dr. Ren then suggested there is a repeating block structure for the generator of the composite underlying Markov chain. I verified this through numerical examples, and proposed the proof of it. I also derived recursive formulas for calculating the generator of the composite Markov chain. Chapter 5 is related to the calculation of the moments of risk processes. Dr. Ren and I jointly came up with the idea. The approach to calculating the moments of the time of ruin is based on the LST of the time of ruin. I found the way to v obtain the moments by taking derivatives of the LST of the time of ruin (as given in Badescu et al. [16]) and derived the expressions of the moments in terms of the infinite matrix summation. Then I observed the algorithm was not stable in some situations. The idea of using uniformization to solve the stability problem belongs to Dr. Stanford. The algorithms for calculating the moments are mine. In addition, I established theorems and gave the proofs in this chapter: for equations for ^(0) and M/(s) for s > 0; smoothness of the LST of the time of ruin; convergence of the algorithm; and so on. In Chapter 5, we also considered the mean busy period of the perturbed fluid flow models. The idea to study this is mine. I derived the LST of the busy pe­ riod based on Dr. Ren's suggestion. The algorithm for the mean busy period and the proof of convergence of the algorithm belong to me. During the implementa­ tion, Dr. Ren suggested using reflection to obtain the correct probability of the upcrossing first passage time of the flow. VI Dedicated to Jane, Sherry and my parents Vll ACKNOWLEDGEMENTS First of all, I want to thank my supervisors Dr. David A. Stanford and Dr. Jian- dong Ren whose help, stimulating suggestions and encouragement helped me in the time of research for and writing of this thesis. In particular, I would like to thank Dr. Stanford for introducing me to the technique of matrix analytic methods. I sincerely thank him for all his advice over these years and I believe it will benefit me in my further work. My sincere thanks to Dr. Qiming He, Dr. Bruce Jones, Dr. Kristina Sendova and Dr. Henning Rasmussen for acting as examiners. Furthermore, I would like to express my heart-felt thanks to all the faculty, staff and graduate students in the Department of Statistical and Actuarial Sciences who have made my stay here both enjoyable and memorable. I would never have reached this point without the selfless love, devotion and encouragement of my parents. Many thanks to my Mom and Dad for supporting me mentally so many years. Above all, I am grateful to my wife Jane and my daughter Sherry for their patience, support and companionship. vm Contents CERTIFICATE OF EXAMINATION ii ABSTRACT iii CO-AUTHORSHIP STATEMENT v DEDICATION viii ACKNOWLEDGEMENTS viii TABLE OF CONTENTS xi LIST OF TABLES xii LIST OF FIGURES xiv 1 INTRODUCTION 1 2 MATHEMATICAL PRELIMINARIES 8 2.1 Notations and Conventions 8 2.2 Risk Processes 10 2.2.1 Definitions 10 2.2.2 Assumptions