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Ultimate Ruin Probability in the Dependent Claim Sizes and Claim Occurrence Times Models

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Ultimate Ruin Probability in the Dependent Claim Sizes and Claim Occurrence Times Models University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2013-07-10 Ultimate Ruin Probability in the Dependent Claim Sizes and Claim Occurrence Times Models Thompson, Emmanuel Thompson, E. (2013). Ultimate Ruin Probability in the Dependent Claim Sizes and Claim Occurrence Times Models (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/28541 http://hdl.handle.net/11023/801 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY Ultimate Ruin Probability in the Dependent Claim Sizes and Claim Occurrence Times Models by Emmanuel Thompson A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS AND STATISTICS CALGARY, ALBERTA JULY, 2013 c Emmanuel Thompson 2013 Abstract In the literature of risk theory, two risk models have emerged to have been studied extensively. These are the Classical Compound Poisson risk model and the Continuous Time Renewal risk model also referred to as Sparre-Andersen model. The focus of this thesis is the latter and in the Sparre-Andersen risk model, the distribution of claim inter-arrival times (T ) and the claim sizes (amounts) (X) are assumed to be independent. This assumption is not only too restrictive but also inappropriate for some real world situations like collision coverages under automobile insurance in cities with hash weather conditions. In this thesis, the assumption of independence between T and X for different classes of bivariate distributions is relaxed prior to computing the ultimate ruin probability. Also, the effect of correlation on ruin probability is investigated. Correlation is introduced through the use of the Moran and Downton's bivariate Exponential distribution. The underlying method in the entire modeling process is the Wiener-Hopf factorization technique, details of which are covered in chapter 2. The main results are covered in chapters 3, 4, 5, 6 and 7. In all of these chapters, we assume T to follow an Exponential distribution while X is Gamma distributed with shape parameter 2 in chapter 3. In chapter 4, X follows Gamma distribution with shape parameter (m) where m is greater than 2. In chapter 5, a mixture of Exponential (Hyper Exponential) distributions is used to model X. The Pareto distribution is used to model X in chapter 6, in the modeling process however, the cummulative distribution function (CDF) of W = X − cT where (T;X) is Exponential-Pareto distribution is approximated by the CDF of Y = X − cT where (T;X) is Exponential-Hyper Exponential distribution and Exponential- Gamma (m) distribution with the main estimation approach being method of moments estimation. Chapter 7 follows somewhat a different approach by employing asymmetric finite mixture of Double Exponential distributions and the EM algorithm prior to computing i the ruin probability. In some of these situations, an explicit expression for the ultimate ruin probability is obtained and in chapters 3 and 5, we show that increasing correlation between T and X diminishes the impact of ruin probability. Also, in situations where an explicit expression for the ultimate ruin probability is not possible, excellent approximation is obtained. Chapter 8 summarizes the entire thesis, provides concluding remarks and future research. Acknowledgments This thesis would not have been possible without the guidance and the help of several indi- viduals who in one way or another contributed and extended their valuable assistance in the preparation and completion of this study. First and foremost, my utmost gratitude to my su- pervisor, advisor and teacher, Professor Rohana S. Ambagaspitiya whose guidance, sincerity and encouragement I will never forget. I am grateful to my supervisory committee members, particularly Professor David Scollnik and Professor Alexandru Badescu for their concern and deep seated advice. I am highly indebted to Professor Abraham Olatunji Fapojuwo from Faculty of Engineering, University of Calgary who accepted to serve as internal/external examiner and my profound appreciation also goes to Professor Cary Tsai from Simon Fraser university actuarial science program who accepted to serve as an external examiner to this thesis. Professor Renate Scheidler, director of graduate studies in the department of Mathematics and Statistics had kind concern and consideration regarding my academic requirements to whom I say a big thank you. I am also thankful to the many wonderful people I came into contact with during both my Masters and PhD degree. Of particular reference is Niroshan Withanage, Beilei Wu my colleagues, and all my office mates Shuang Lu, Xia Bing and Shiva Gopal Wagle. Last but not the least, my lovely wife Sandra Senam Addison, my parents and the one above all of us, the omnipresent God, for answering my prayers for giving me the strength to plod on despite my constitution wanting to give up and throw in the towel, thank you so much Dear Lord. iii Table of Contents Abstract ........................................ i Acknowledgments .................................. iii Table of Contents . iv List of Tables . vii List of Figures . viii List of Symbols . ix 1 Introduction . 1 1.1 Collective Risk Theory . 1 1.1.1 Historical Perspective . 1 1.2 Risk Models . 3 1.2.1 The Compound Binomial Model . 4 1.2.2 The Compound Poisson Model . 5 1.2.3 The Compound Mixed Poisson Model . 6 1.2.4 The Compound Negative Binomial Model . 6 1.3 Claim Size Distributions . 7 1.3.1 Light-Tailed Claim Distribution . 7 1.3.2 Heavy-Tailed Claim Distribution . 8 1.4 Arrival Process . 8 1.5 Risk Surplus Process and Ruin Probability . 10 1.5.1 Classical Compound Poisson Risk Model . 11 1.5.2 Continuous Time Renewal Process . 11 1.6 Mathematical Preliminaries . 14 1.6.1 Moran and Downton's Bivariate Exponential Distribution . 14 1.6.2 Laplace Transform . 15 1.6.3 Inverse Laplace Transform . 15 1.6.4 Rational Function . 16 1.6.5 Locating Zeros of a Certain Polynomial . 16 2 Wiener-Hopf Factorization Method . 18 2.1 Introduction . 18 2.2 Ladder Processes . 18 2.3 Renewal Functions . 20 2.4 Maximum and Minimum . 24 2.5 Survival Probability . 27 3 Ruin Probability for Correlated Poisson Claim Count and Gamma (2) Risk Process . 30 3.1 Introduction . 30 3.1.1 Poisson Claim . 30 3.1.2 Relationship Between Poisson and Exponential Distribution . 31 3.2 Joint Moment Generating Function of (T;X).................. 31 3.3 Characteristic Function of Y and its Related Transform . 32 3.4 Zeros of the Polynomials g(s) and g(s) − z ................... 33 3.5 Factors and Properties of the Transform of Y . 36 iv 3.6 Survival Probability . 38 3.7 Effect of Correlation on Ultimate Ruin Probability . 40 3.8 Numerical Examples . 40 4 Ruin Probability for Correlated Poisson Claim Count and Gamma (m) Risk Process . 44 4.1 Introduction . 44 4.2 Joint Moment Generating Function of (T;X).................. 44 4.3 Characteristic Function of Y and its Related Transform . 45 4.4 Zeros of the Polynomial g(s) − z ......................... 46 4.4.1 Application of Rouche's Theorem . 50 4.5 Factors and Properties of the Transform of Y . 53 4.6 Survival Probability . 54 4.7 Numerical Examples . 56 5 Ruin Probability for Correlated Poisson Claim Count and Hyper Exponential Risk Process . 59 5.1 Introduction . 59 5.1.1 Hyper Exponential Distribution . 59 5.2 Joint Moment Generating Function of (T;Xj) . 62 5.3 Characteristic Function of Y and Related Transform . 62 5.3.1 Zeros of a Certain m + 1 Degree Polynomial . 65 5.4 Survival Probability . 71 5.5 Numerical Examples . 73 6 Ruin Probability for Correlated Poisson Claim Count and Pareto Risk Process 77 6.1 Introduction . 77 6.1.1 Pareto Distribution . 77 6.1.2 Method of Moments Matching . 78 6.2 The Bivariate Exponential-Pareto Distribution . 79 6.3 Approximating Distributions . 83 6.3.1 Mixture of Exponential Distributions . 83 6.3.2 Gamma (m) Distribution . 84 6.4 Numerical Results . 86 7 Asymmetric Double Exponential Distributions and its Finite Mixtures . 92 7.1 Introduction . 92 7.1.1 Double Exponential Distribution . 92 7.1.2 Finite Mixture Models . 93 7.2 Asymmetric Double Exponential Distribution . 94 7.2.1 MLE of Parameters . 94 7.3 Finite Mixture . 96 7.4 Numerical Results . 97 8 Summary, Concluding Remarks and Future Research . 115 8.1 Summary . 115 8.1.1 Chapter 2: Wiener-Hopf Factorization Method . 115 8.1.2 Chapter 3: Ruin Probability for Correlated Poisson Claim Count and Gamma (2) Risk Process . 116 v 8.1.3 Chapter 4: Ruin Probability for Correlated Poisson Claim Count and Gamma (m) Risk Process . 118 8.1.4 Chapter 5: Ruin Probability for Correlated Poisson Claim Count and Hyper Exponential Risk Process . 119 8.1.5 Chapter 6: Ruin Probability for Correlated Poisson Claim Count and Pareto Risk Process . 120 8.1.6 Chapter 7: Asymmetric Double Exponential Distributions and its Fi- nite Mixtures . 121 8.2 Concluding Remarks . 121 8.3 Future Research . 122 Bibliography . 123 A Maple and Matlab codes . 128 A.1 Maple Codes . ..
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