Non-Life Insurance Mathematics: An Introduction With the Poisson Process

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Non-Life Insurance Mathematics: An Introduction With the Poisson Process

Format:  Paperback,

432 pages

Publisher: Springer Verlag

Publish Date: May 2009

ISBN-13: 9783540882329

ISBN-10: 3540882324

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Book Information

The following content was provided by the publisher.

"Offers a mathematical introduction to non-life insurance and, at the same time, to a multitude of applied stochastic processes. It gives detailed discussions of the fundamental models for claim sizes, claim arrivals, the total claim amount, and their probabilistic properties....The reader gets to know how the underlying probabilistic structures allow one to determine premiums in a portfolio or in an individual policy." --Zentralblatt fur Didaktik der Mathematik

Specifications

Publisher: Springer Verlag
Publish Date: May 2009
ISBN-13: 9783540882329
ISBN-10: 3540882324
Format: Paperback
Number of Pages: 432
Shipping Weight (in pounds): 1.46
Product in Inches (L x W x H): 6.0 x 9.0 x 0.75
Walmart No.: 9783540882329

Chapter outline

Collective Risk Models
The Basic Model
Models for the Claim Number Process
The Poisson Process
The Homogeneous Poisson Process, the Intensity Function, the Cramer-Lundberg Model
The Markov Property
Relations Between the Homogeneous and the Inhomogeneous Poisson Process
The Homogeneous Poisson Process as a Renewal Process
The Distribution of the Inter-Arrival Times
The Order Statistics Property
A Discussion of the Arrival Times of the Danish Fire Insurance Data 1980-1990
An Informal Discussion of Transformed and Generalized Poisson Processes
Exercises
The Renewal Process
Basic Properties
An Informal Discussion of Renewal Theory
Exercises
The Mixed Poisson Process
Exercises
The Total Claim Amount
The Order of Magnitude of the Total Claim Amount
The Mean and the Variance in the Renewal Model
The Asymptotic Behavior in the Renewal Model
Classical Premium Calculation Principles
Exercises
Claim Size Distributions
An Exploratory Statistical Analysis: QQ-Plots
A Preliminary Discussion of Heavy- and Light-Tailed Distributions
An Exploratory Statistical Analysis: Mean Excess Plots
Standard Claim Size Distributions and Their Properties
Regularly Varying Claim Sizes and Their Aggregation
Subexponential Distributions
Exercises
The Distribution of the Total Claim Amount
Mixture Distributions
Space-Time Decomposition of a Compound Poisson Process
An Exact Numerical Procedure for Calculating the Total Claim Amount Distribution
Approximation to the Distribution of the Total Claim Amount Using the Central Limit Theorem
Approximation to the Distribution of the Total Claim Amount by Monte Carlo Techniques
Exercises
Reinsurance Treaties
Exercises
Ruin Theory
Risk Process, Ruin Probability and Net Profit Condition
Exercises
Bounds for the Ruin Probability
Lundberg's Inequality
Exact Asymptotics for the Ruin Probability: the Small Claim Case
The Representation of the Ruin Probability as a Compound Geometric Probability
Exact Asymptotics for the Ruin Probability: the Large Claim Case
Exercises
Experience Rating
Bayes Estimation
The Heterogeneity Model
Bayes Estimation in the Heterogeneity Model
Exercises
Linear Bayes Estimation
An Excursion to Minimum Linear Risk Estimation
The Buhlmann Model
Linear Bayes Estimation in the Buhlmann Model
The Buhlmann-Straub Model
Exercises
A Point Process Approach to Collective Risk Theory
The General Poisson Process
The Notion of a Point Process
Definition and First Examples
Distribution and Laplace Functional
Exercises
Poisson Random Measures
Definition and First Examples
Laplace Functional and Non-Negative Poisson Integrals
Properties of General Poisson Integrals
Exercises
Construction of New Poisson Random Measures from Given Poisson Random Measures
Transformation of the Points of a Poisson Random Measure
Marked Poisson Random Measures
The Cramer-Lundberg and Related Models as Marked Poisson Random Measures
Aggregating Poisson Random Measures
Exercises
Poisson Random Measures in Collective Risk Theory
Decomposition of the Time-Claim Size Space
Decomposition by Claim Size
Decomposition by Year of Occurrence
Decomposition by Year of Reporting
Effects of Dependence Between Delay in Reporting Time and Claim Size
Effects of Inflation and Interest
Exercises
A General Model with Delay in Reporting and Settlement of Claim Payments
The Basic Model and the Basic Decomposition of Time-Claim Size Space
The Basic Decomposition of the Claim Number Process
The Basic Decomposition of the Total Claim Amount
An Excursion to Teletraffic and Long Memory: The Stationary IBNR Claim Number Process
A Critique of the Basic Model
Exercises
Weak Convergence of Point Processes
Definition and Basic Examples
Convergence of the Finite-Dimensional Distributions
Convergence of Laplace Functionals
Exercises
Point Processes of Exceedances and Extremes
Convergence of the Point Processes of Exceedances
Convergence in Distribution of Maxima and Order Statistics Under Affine Transformations
Maximum Domains of Attraction
The Point Process of Exceedances at the Times of a Renewal Process
Exercises
Asymptotic Theory for the Reinsurance Treaties of Extreme Value Type
Exercises
Special Topics
An Excursion to Levy Processes
Definition and First Examples of Levy Processes
Exercises
Some Basic Properties of Levy Processes
Exercises
Infinite Divisibility: The Levy-Khintchine Formula
Exercises
The Levy-Ito Representation of a Levy Process
Exercises
Some Special Levy Processes
Exercises
Cluster Point Processes
The General Cluster Process
The Chain Ladder Method
The Chain Ladder Model
Mack's Model
Some Asymptotic Results in the Chain Ladder Model
Moments of the Chain Ladder Estimators
Prediction in Mack's Model
Exercises
An Informal Discussion of a Cluster Model with Poisson Arrivals
Specification of the Model
An Analysis of the First and Second Moments
A Model when Clusters are Poisson Processes
Exercises
References
Index
List of Abbreviations and Symbols

Book description

"Offers a mathematical introduction to non-life insurance and, at the same time, to a multitude of applied stochastic processes. It gives detailed discussions of the fundamental models for claim sizes, claim arrivals, the total claim amount, and their probabilistic properties.... The reader gets to know how the underlying probabilistic structures allow one to determine premiums in a portfolio or in an individual policy". --Zentralblatt fur Didaktik der Mathematik

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