Collective Risk Models / Part I: |
The Basic Model / 1: |
Models for the Claim Number Process / 2: |
The Poisson Process / 2.1: |
The Homogeneous Poisson Process, the Intensity Function, the Cramer-Lundberg Model / 2.1.1: |
The Markov Property / 2.1.2: |
Relations Between the Homogeneous and the Inhomogeneous Poisson Process / 2.1.3: |
The Homogeneous Poisson Process as a Renewal Process / 2.1.4: |
The Distribution of the Inter-Arrival Times / 2.1.5: |
The Order Statistics Property / 2.1.6: |
A Discussion of the Arrival Times of the Danish Fire Insurance Data 1980-1990 / 2.1.7: |
An Informal Discussion of Transformed and Generalized Poisson Processes / 2.1.8: |
Exercises |
The Renewal Process / 2.2: |
Basic Properties / 2.2.1: |
An Informal Discussion of Renewal Theory / 2.2.2: |
The Mixed Poisson Process / 2.3: |
The Total Claim Amount / 3: |
The Order of Magnitude of the Total Claim Amount / 3.1: |
The Mean and the Variance in the Renewal Model / 3.1.1: |
The Asymptotic Behavior in the Renewal Model / 3.1.2: |
Classical Premium Calculation Principles / 3.1.3: |
Claim Size Distributions / 3.2: |
An Exploratory Statistical Analysis: QQ-Plots / 3.2.1: |
A Preliminary Discussion of Heavy- and Light-Tailed Distributions / 3.2.2: |
An Exploratory Statistical Analysis: Mean Excess Plots / 3.2.3: |
Standard Claim Size Distributions and Their Properties / 3.2.4: |
Regularly Varying Claim Sizes and Their Aggregation / 3.2.5: |
Subexponential Distributions / 3.2.6: |
The Distribution of the Total Claim Amount / 3.3: |
Mixture Distributions / 3.3.1: |
Space-Time Decomposition of a Compound Poisson Process / 3.3.2: |
An Exact Numerical Procedure for Calculating the Total Claim Amount Distribution / 3.3.3: |
Approximation to the Distribution of the Total Claim Amount Using the Central Limit Theorem / 3.3.4: |
Approximation to the Distribution of the Total Claim Amount by Monte Carlo Techniques / 3.3.5: |
Reinsurance Treaties / 3.4: |
Ruin Theory / 4: |
Risk Process, Ruin Probability and Net Profit Condition / 4.1: |
Bounds for the Ruin Probability / 4.2: |
Lundberg's Inequality / 4.2.1: |
Exact Asymptotics for the Ruin Probability: the Small Claim Case / 4.2.2: |
The Representation of the Ruin Probability as a Compound Geometric Probability / 4.2.3: |
Exact Asymptotics for the Ruin Probability: the Large Claim Case / 4.2.4: |
Experience Rating / Part II: |
Bayes Estimation / 5: |
The Heterogeneity Model / 5.1: |
Bayes Estimation in the Heterogeneity Model / 5.2: |
Linear Bayes Estimation / 6: |
An Excursion to Minimum Linear Risk Estimation / 6.1: |
The Buhlmann Model / 6.2: |
Linear Bayes Estimation in the Buhlmann Model / 6.3: |
The Buhlmann-Straub Model / 6.4: |
A Point Process Approach to Collective Risk Theory / Part III: |
The General Poisson Process / 7: |
The Notion of a Point Process / 7.1: |
Definition and First Examples / 7.1.1: |
Distribution and Laplace Functional / 7.1.2: |
Poisson Random Measures / 7.2: |
Laplace Functional and Non-Negative Poisson Integrals / 7.2.1: |
Properties of General Poisson Integrals / 7.2.3: |
Construction of New Poisson Random Measures from Given Poisson Random Measures / 7.3: |
Transformation of the Points of a Poisson Random Measure / 7.3.1: |
Marked Poisson Random Measures / 7.3.2: |
The Cramer-Lundberg and Related Models as Marked Poisson Random Measures / 7.3.3: |
Aggregating Poisson Random Measures / 7.3.4: |
Poisson Random Measures in Collective Risk Theory / 8: |
Decomposition of the Time-Claim Size Space / 8.1: |
Decomposition by Claim Size / 8.1.1: |
Decomposition by Year of Occurrence / 8.1.2: |
Decomposition by Year of Reporting / 8.1.3: |
Effects of Dependence Between Delay in Reporting Time and Claim Size / 8.1.4: |
Effects of Inflation and Interest / 8.1.5: |
A General Model with Delay in Reporting and Settlement of Claim Payments / 8.2: |
The Basic Model and the Basic Decomposition of Time-Claim Size Space / 8.2.1: |
The Basic Decomposition of the Claim Number Process / 8.2.2: |
The Basic Decomposition of the Total Claim Amount / 8.2.3: |
An Excursion to Teletraffic and Long Memory: The Stationary IBNR Claim Number Process / 8.2.4: |
A Critique of the Basic Model / 8.2.5: |
Weak Convergence of Point Processes / 9: |
Definition and Basic Examples / 9.1: |
Convergence of the Finite-Dimensional Distributions / 9.1.1: |
Convergence of Laplace Functionals / 9.1.2: |
Point Processes of Exceedances and Extremes / 9.2: |
Convergence of the Point Processes of Exceedances / 9.2.1: |
Convergence in Distribution of Maxima and Order Statistics Under Affine Transformations / 9.2.2: |
Maximum Domains of Attraction / 9.2.3: |
The Point Process of Exceedances at the Times of a Renewal Process / 9.2.4: |
Asymptotic Theory for the Reinsurance Treaties of Extreme Value Type / 9.3: |
Special Topics / Part IV: |
An Excursion to Levy Processes / 10: |
Definition and First Examples of Levy Processes / 10.1: |
Some Basic Properties of Levy Processes / 10.2: |
Infinite Divisibility: The Levy-Khintchine Formula / 10.3: |
The Levy-Ito Representation of a Levy Process / 10.4: |
Some Special Levy Processes / 10.5: |
Cluster Point Processes / 11: |
The General Cluster Process / 11.1: |
The Chain Ladder Method / 11.2: |
The Chain Ladder Model / 11.2.1: |
Mack's Model / 11.2.2: |
Some Asymptotic Results in the Chain Ladder Model / 11.2.3: |
Moments of the Chain Ladder Estimators / 11.2.4: |
Prediction in Mack's Model / 11.2.5: |
An Informal Discussion of a Cluster Model with Poisson Arrivals / 11.3: |
Specification of the Model / 11.3.1: |
An Analysis of the First and Second Moments / 11.3.2: |
A Model when Clusters are Poisson Processes / 11.3.3: |
References |
Index |
List of Abbreviations and Symbols |
Collective Risk Models / Part I: |
The Basic Model / 1: |
Models for the Claim Number Process / 2: |