This carefully written monograph covers the Sparre Andersen process in an actuarial context using the renewal process as the model for claim counts.
A unified reference on Sparre Andersen (renewal risk) processes is included, often missing from existing literature. The authors explore recent results and analyse various risk theoretic quantities associated with the event of ruin, including the time of ruin and the deficit of ruin. Particular attention is given to the explicit identification of defective renewal equation components, which are needed to analyse various risk theoretic quantities and are also relevant in other subject areas of applied probability such as dams and storage processes, as well as queuing theory.
Aimed at researchers interested in risk/ruin theory and related areas, this work will also appeal to graduate students in classical and modern risk theory and Gerber-Shiu analysis.
Recenzijas
The monograph is devoted to the surplus analysis of the Sparre-Andersen process using the renewal process as the model for claim counts. This book is intended for researchers interested in ruin/risk theory, and will also be useful for graduate students specialized in classical and modern risk theory. (Anatoliy Swishchuk, zbMATH 1391.91006, 2018)
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1 | (10) |
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11 | (34) |
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11 | (1) |
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2.2 Dickson--Hipp Operators and Equilibrium Distributions |
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12 | (6) |
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2.3 Defective Renewal Equations |
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18 | (9) |
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2.4 Mixed Erlang Distributions |
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27 | (11) |
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38 | (7) |
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3 Gerber--Shiu Analysis in the Classical Poisson Risk Model |
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45 | (16) |
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3.1 The Classical Poisson Risk Model |
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45 | (1) |
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3.2 The Time of Ruin and Related Quantities |
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46 | (3) |
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3.3 Derivation of the Classical Poisson Gerber--Shiu Function |
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49 | (3) |
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3.4 Analysis of the Classical Poisson Gerber--Shiu Function |
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52 | (9) |
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4 Gerber--Shiu Analysis in the Dependent Sparre Andersen Model |
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61 | (18) |
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4.1 The Dependent Sparre Andersen Model |
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61 | (1) |
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4.2 Conditioning on the Time and Amount of the First Claim |
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62 | (2) |
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64 | (2) |
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4.4 Conditioning on the First Drop in Surplus |
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66 | (6) |
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4.5 The Distribution and Moments of the Deficit |
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72 | (7) |
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5 Models Involving Erlang Components |
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79 | (48) |
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5.1 A Dependent Coxian Interclaim Time Model |
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80 | (10) |
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5.2 The Independent Exponential Claim Size Model |
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90 | (14) |
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5.3 A Dependent Coxian Claim Size Model |
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104 | (16) |
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5.4 A Dependent Mixed Erlang Claim Size Model |
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120 | (7) |
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6 The Time of Ruin in the Classical Poisson Risk Model |
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127 | (24) |
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6.1 Moments of the Time of Ruin |
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127 | (6) |
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6.2 Finite Time Ruin and a Partial Integrodifferential Equation |
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133 | (9) |
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6.3 Finite Time Ruin Probabilities for Mixed Erlang Claim Amounts |
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142 | (5) |
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6.4 The Joint Distribution of the Time of Ruin and the Deficit |
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147 | (1) |
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6.5 Further Remarks on the Density of the Time of Ruin |
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148 | (3) |
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151 | (28) |
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7.1 Delayed and Stationary Renewal Risk Models |
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151 | (11) |
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7.2 Discrete Renewal Risk Model |
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162 | (17) |
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179 | (38) |
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8.1 Additional Variables in the Penalty Function |
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179 | (22) |
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8.1.1 The Surplus Immediately After the Second Last Claim and the Minimum Surplus Before Ruin |
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180 | (5) |
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8.1.2 The Maximum and the Minimum Surplus Levels Before Ruin |
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185 | (4) |
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8.1.3 The Discounted Aggregate Claim Costs Until Ruin |
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189 | (6) |
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8.1.4 The Number of Claims Until Ruin |
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195 | (6) |
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8.2 Ordering Properties of Some Ruin-Related Quantities |
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201 | (3) |
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8.3 Bounds on Solutions to Renewal Equation |
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204 | (13) |
| References |
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217 | (6) |
| Index |
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223 | |
Gordon E. Willmot is Munich Re Chair in Insurance and professor in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. His research interests are in stochastic modelling in insurance. Willmot has (co-)authored over one hundred research papers in leading actuarial and statistical journals. He is also co-author of Lundberg Approximations for Compound Distributions with Insurance Applications (Springer), Loss Models - From Data to Decisions and Loss Models - Further Topics (Wiley), and Insurance Risk Models (Society of Actuaries). He is editor of Insurance: Mathematics and Economics.
Jae-Kyung Woo is Associate Professor in the School of Risk and Actuarial Studies at the University of New South Wales, Sydney. She has worked at the University of Hong Kong and Columbia University prior to joining UNSW. Her research interests are focused on risk theory, reliability theory, aggregate claim analysis,and queueing theory. She has published about twenty papers dealing with the subject of the present monograph and related topics.
