Non-life insurance mathematics: an introduction with the poisson process
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
2009
|
| Edition: | 2. ed. |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 432 S. graph. Darst. |
| ISBN: | 9783540882329 |
Staff View
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| 245 | 1 | 0 | |a Non-life insurance mathematics |b an introduction with the poisson process |c Thomas Mikosch |
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| 650 | 4 | |a Insurance |x Mathematics | |
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Record in the Search Index
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| adam_text | Titel: Non-life insurance mathematics
Autor: Mikosch, Thomas
Jahr: 2009
Contents
Part I Collective Risk Models
1 The Basic Model 3
2 Models for the Claim Number Process 7
2.1 The Poisson Process 7
2.1.1 The Homogeneous Poisson Process, the Intensity
Function, the Cramer-Lundberg Model 9
2.1.2 The Markov Property 12
2.1.3 Relations Between the Homogeneous and the
Inhomogeneous Poisson Process 14
2.1.4 The Homogeneous Poisson Process as a Renewal Process 16
2.1.5 The Distribution of the Inter-Arrival Times 20
2.1.6 The Order Statistics Property 22
2.1.7 A Discussion of the Arrival Times of the Danish Fire
Insurance Data 1980-1990 32
2.1.8 An Informal Discussion of Transformed and
Generalized Poisson Processes 35
Exercises 46
2.2 The Renewal Process 53
2.2.1 Basic Properties 53
2.2.2 An Informal Discussion of Renewal Theory 60
Exercises 65
2.3 The Mixed Poisson Process 66
Exercises 69
3 The Total Claim Amount 71
3.1 The Order of Magnitude of the Total Claim Amount 72
3.1.1 The Mean and the Variance in the Renewal Model 73
3.1.2 The Asymptotic Behavior in the Renewal Model 74
3.1.3 Classical Premium Calculation Principles 78
Exercises 80
3.2 Claim Size Distributions 82
3.2.1 An Exploratory Statistical Analysis: QQ-Plots 82
3.2.2 A Preliminary Discussion of Heavy- and Light-Tailed
Distributions 86
3.2.3 An Exploratory Statistical Analysis: Mean Excess Plots 88
3.2.4 Standard Claim Size Distributions and Their Properties 94
3.2.5 Regularly Varying Claim Sizes and Their Aggregation.. 99
3.2.6 Subexponential Distributions 103
Exercises 106
3.3 The Distribution of the Total Claim Amount 109
3.3.1 Mixture Distributions 110
3.3.2 Space-Time Decomposition of a Compound Poisson
Process 115
3.3.3 An Exact Numerical Procedure for Calculating the
Total Claim Amount Distribution 120
3.3.4 Approximation to the Distribution of the Total Claim
Amount Using the Central Limit Theorem 125
3.3.5 Approximation to the Distribution of the Total Claim
Amount by Monte Carlo Techniques 130
Exercises 138
3.4 Reinsurance Treaties 142
Exercises 149
4 Ruin Theory 151
4.1 Risk Process, Ruin Probability and Net Profit Condition 151
Exercises 156
4.2 Bounds for the Ruin Probability 157
4.2.1 Lundberg s Inequality 157
4.2.2 Exact Asymptotics for the Ruin Probability: the
Small Claim Case 162
4.2.3 The Representation of the Ruin Probability as a
Compound Geometric Probability 172
4.2.4 Exact Asymptotics for the Ruin Probability: the
Large Claim Case 174
Exercises yjl
Part II Experience Rating
5 Bayes Estimation 187
5.1 The Heterogeneity Model 187
5.2 Bayes Estimation in the Heterogeneity Model 189
Exercises 195
6 Linear Bayes Estimation 199
6.1 An Excursion to Minimum Linear Risk Estimation 200
6.2 The Biihlmann Model 204
6.3 Linear Bayes Estimation in the Biihlmann Model 206
6.4 The Biihlmann-Straub Model 209
Exercises 211
Part III A Point Process Approach to Collective Risk Theory
7 The General Poisson Process 215
7.1 The Notion of a Point Process 215
7.1.1 Definition and First Examples 215
7.1.2 Distribution and Laplace Functional 222
Exercises 224
7.2 Poisson Random Measures 226
7.2.1 Definition and First Examples 227
7.2.2 Laplace Functional and Non-Negative Poisson Integrals 232
7.2.3 Properties of General Poisson Integrals 236
Exercises 242
7.3 Construction of New Poisson Random Measures from Given
Poisson Random Measures 244
7.3.1 Transformation of the Points of a Poisson Random
Measure 244
7.3.2 Marked Poisson Random Measures 246
7.3.3 The Cramer-Lundberg and Related Models as Marked
Poisson Random Measures 249
7.3.4 Aggregating Poisson Random Measures 254
Exercises 256
8 Poisson Random Measures in Collective Risk Theory 259
8.1 Decomposition of the Time-Claim Size Space 259
8.1.1 Decomposition by Claim Size 259
8.1.2 Decomposition by Year of Occurrence 261
8.1.3 Decomposition by Year of Reporting 263
8.1.4 Effects of Dependence Between Delay in Reporting
Time and Claim Size 264
8.1.5 Effects of Inflation and Interest 266
Exercises 267
8.2 A General Model with Delay in Reporting and Settlement of
Claim Payments 268
8.2.1 The Basic Model and the Basic Decomposition
of Time-Claim Size Space 268
8.2.2 The Basic Decomposition of the Claim Number Process 271
8.2.3 The Basic Decomposition of the Total Claim Amount . . 273
8.2.4 An Excursion to Teletramc and Long Memory:
The Stationary IBNR Claim Number Process 278
8.2.5 A Critique of the Basic Model 284
Exercises 286
9 Weak Convergence of Point Processes 291
9.1 Definition and Basic Examples 292
9.1.1 Convergence of the Finite-Dimensional Distributions ... 292
9.1.2 Convergence of Laplace Functionals 294
Exercises 299
9.2 Point Processes of Exceedances and Extremes 300
9.2.1 Convergence of the Point Processes of Exceedances .... 300
9.2.2 Convergence in Distribution of Maxima and Order
Statistics Under Afrine Transformations 305
9.2.3 Maximum Domains of Attraction 309
9.2.4 The Point Process of Exceedances at the Times of a
Renewal Process 316
Exercises 321
9.3 Asymptotic Theory for the Reinsurance Treaties of Extreme
Value Type 324
Exercises 331
Part IV Special Topics
10 An Excursion to Levy Processes 335
10.1 Definition and First Examples of Levy Processes 335
Exercises 338
10.2 Some Basic Properties of Levy Processes 338
Exercises 340
10.3 Infinite Divisibility. The Levy-Khintchine Formula 341
Exercises 347
10.4 The Levy-Ito Representation of a Levy Process 348
Exercises 355
10.5 Some Special Levy Processes 355
Exercises 3gl
11 Cluster Point Processes 363
11.1 The General Cluster Process 363
11.2 The Chain Ladder Method 365
11.2.1 The Chain Ladder Model 365
11.2.2 Mack s Model 366
11.2.3 Some Asymptotic Results in the Chain Ladder Model.. 369
11.2.4 Moments of the Chain Ladder Estimators 372
11.2.5 Prediction in Mack s Model 376
Exercises 381
11.3 An Informal Discussion of a Cluster Model with Poisson
Arrivals 386
11.3.1 Specification of the Model 386
11.3.2 An Analysis of the First and Second Moments 389
11.3.3 A Model when Clusters are Poisson Processes 394
Exercises 402
References 405
Index 413
List of Abbreviations and Symbols 429
|
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| discipline | Soziologie Mathematik Wirtschaftswissenschaften |
| edition | 2. ed. |
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| isbn | 9783540882329 |
| language | German |
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| spelling | Mikosch, Thomas 1955- Verfasser (DE-588)141029412 aut Non-life insurance mathematics an introduction with the poisson process Thomas Mikosch 2. ed. Berlin [u.a.] Springer 2009 XV, 432 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Stochastischer Prozess stw Theorie stw Versicherungsmathematik - Stochastisches Modell Versicherungstechnik stw Mathematik Insurance Mathematics Stochastic processes Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 s Stochastisches Modell (DE-588)4057633-4 s DE-604 Erscheint auch als Online-Ausgabe 978-3-540-88233-6 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017047974&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mikosch, Thomas 1955- Non-life insurance mathematics an introduction with the poisson process Stochastischer Prozess stw Theorie stw Versicherungsmathematik - Stochastisches Modell Versicherungstechnik stw Mathematik Insurance Mathematics Stochastic processes Stochastisches Modell (DE-588)4057633-4 gnd Versicherungsmathematik (DE-588)4063194-1 gnd |
| subject_GND | (DE-588)4057633-4 (DE-588)4063194-1 |
| title | Non-life insurance mathematics an introduction with the poisson process |
| title_auth | Non-life insurance mathematics an introduction with the poisson process |
| title_exact_search | Non-life insurance mathematics an introduction with the poisson process |
| title_full | Non-life insurance mathematics an introduction with the poisson process Thomas Mikosch |
| title_fullStr | Non-life insurance mathematics an introduction with the poisson process Thomas Mikosch |
| title_full_unstemmed | Non-life insurance mathematics an introduction with the poisson process Thomas Mikosch |
| title_short | Non-life insurance mathematics |
| title_sort | non life insurance mathematics an introduction with the poisson process |
| title_sub | an introduction with the poisson process |
| topic | Stochastischer Prozess stw Theorie stw Versicherungsmathematik - Stochastisches Modell Versicherungstechnik stw Mathematik Insurance Mathematics Stochastic processes Stochastisches Modell (DE-588)4057633-4 gnd Versicherungsmathematik (DE-588)4063194-1 gnd |
| topic_facet | Stochastischer Prozess Theorie Versicherungsmathematik - Stochastisches Modell Versicherungstechnik Mathematik Insurance Mathematics Stochastic processes Stochastisches Modell Versicherungsmathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017047974&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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