Non-life insurance mathematics: an introduction with stochastic processes
This book 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. Throughout th...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2006
|
| Ausgabe: | Corr. 2. print. |
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Zusammenfassung: | This book 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. Throughout the book the language of stochastic processes is used for describing the dynamics of an insurance portfolio in claim size space and time. In addition to the standard actuarial notions, the reader learns about the basic models of modern non-life insurance mathematics: the Poisson, compound Poisson and renewal processes in collective risk theory and heterogeneity and Bühlmann models in experience rating. The reader gets to know how the underlying probabilistic structures allow one to determine premiums in a portfolio or in an individual policy. Special emphasis is given to the phenomena which are caused by large claims in these models. What makes this book special are more than 100 figures and tables illustrating and visualizing the theory. Every section ends with extensive exercises. They are an integral part of this course since they support the access to the theory. The book can serve either as a text for an undergraduate/graduate course on non-life insurance mathematics or applied stochastic processes. Its content is in agreement with the European "Groupe Consultatif" standards. An extensive bibliography, annotated by various comments sections with references to more advanced relevant literature, make the book broadly and easiliy accessible. |
| Beschreibung: | XI, 235 S. graph. Darst. |
| ISBN: | 3540406506 |
Internformat
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| 245 | 1 | 0 | |a Non-life insurance mathematics |b an introduction with stochastic processes |c Thomas Mikosch |
| 250 | |a Corr. 2. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XI, 235 S. |b graph. Darst. | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a Universitext | |
| 520 | 3 | |a This book 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. Throughout the book the language of stochastic processes is used for describing the dynamics of an insurance portfolio in claim size space and time. In addition to the standard actuarial notions, the reader learns about the basic models of modern non-life insurance mathematics: the Poisson, compound Poisson and renewal processes in collective risk theory and heterogeneity and Bühlmann models in experience rating. The reader gets to know how the underlying probabilistic structures allow one to determine premiums in a portfolio or in an individual policy. Special emphasis is given to the phenomena which are caused by large claims in these models. What makes this book special are more than 100 figures and tables illustrating and visualizing the theory. Every section ends with extensive exercises. They are an integral part of this course since they support the access to the theory. The book can serve either as a text for an undergraduate/graduate course on non-life insurance mathematics or applied stochastic processes. Its content is in agreement with the European "Groupe Consultatif" standards. An extensive bibliography, annotated by various comments sections with references to more advanced relevant literature, make the book broadly and easiliy accessible. | |
| 650 | 4 | |a Versicherungsmathematik - Stochastisches Modell | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Insurance |x Mathematics | |
| 650 | 4 | |a Stochastic processes | |
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Datensatz im Suchindex
| _version_ | 1804135490696249344 |
|---|---|
| adam_text | Non-Life Insurance Mathematics
This book offers a mathematical introduction to non-life insurance and,
at the same time, to a multitude of applied stochastic processes. It gives de¬
tailed discussions of the fundamental models for claim sizes, claim arrivals,
the total claim amount, and their probabilistic properties. Throughout the
book the language of stochastic processes is used for describing the dynam¬
ics of an insurance portfolio in claim size space and time. In addition to
the standard actuarial notions, the reader learns about the basic models of
modern non-life insurance mathematics: the
Poisson,
compound
Poisson
and renewal processes in collective risk theory and heterogeneity and Biihl-
mann models in experience rating. The reader gets to know how the underly¬
ing probabilistic structures allow one to determine premiums in a portfolio
or in an individual policy. Special emphasis is given to the phenomena which
are caused by large claims in these models.
What makes this book special are more than
100
figures and tables illus¬
trating and visualizing the theory. Every section ends with extensive exer¬
cises. They are an integral part of this course since they support the access
to the theory.
The book can serve either as a text for an undergraduate/graduate course
on non-life insurance mathematics or applied stochastic processes. Its con¬
tent is in agreement with the European Group
Consultatif
standards. An
extensive bibliography, annotated by various comments sections with refer¬
ences to more advanced relevant literature, make the book broadly and easi¬
ly accessible.
Contents
Guidelines to the Reader
...................................... 1
Part I Collective Risk Models
1
The Basic Model
.......................................... 7
2
Models for the Claim Number Process
.................... 13
2.1
The
Poisson
Process
..................................... 13
2.1.1
The Homogeneous
Poisson
Process, the Intensity
Function, the
Cramér-Lundberg
Model
............... 15
2.1.2
The Markov Property
.............................. 18
2.1.3
Relations Between the Homogeneous and the
Inhomogeneous
Poisson
Process
..................... 20
2.1.4
The Homogeneous
Poisson
Process as a Renewal Process
21
2.1.5
The Distribution of the Inter-Arrival Times
........... 26
2.1.6
The Order Statistics Property
....................... 28
2.1.7
A Discussion of the Arrival Times of the Danish Fire
Insurance Data
1980-1990.......................... 38
2.1.8
An Informal Discussion of Transformed and
Generalized
Poisson
Processes
....................... 41
Exercises
............................................... 52
2.2
The Renewal Process
.................................... 59
2.2.1
Basic Properties
................................... 59
2.2.2
An Informal Discussion of Renewal Theory
........... 66
Exercises
............................................... 71
2.3
The Mixed
Poisson
Process
............................... 71
Exercises
............................................... 75
X
Contents
3
The Total Claim Amount
.................................. 77
3.1
The Order of Magnitude of the Total Claim Amount
......... 78
3.1.1
The Mean and the Variance in the Renewal Model
..... 79
3.1.2
The Asymptotic Behavior in the Renewal Model
...... 80
3.1.3
Classical Premium Calculation Principles
............. 84
Exercises
............................................... 86
3.2
Claim Size Distributions
.................................. 88
3.2.1
An Exploratory Statistical Analysis: QQ-Plots
........ 88
3.2.2
A Preliminary Discussion of Heavy- and Light-Tailed
Distributions
..................................... 92
3.2.3
An Exploratory Statistical Analysis: Mean Excess Plots
94
3.2.4
Standard Claim Size Distributions and Their Properties
100
3.2.5
Regularly Varying Claim Sizes and Their Aggregation.
. 105
3.2.6
Subexponential Distributions
.......................109
Exercises
...............................................112
3.3
The Distribution of the Total Claim Amount
................115
3.3.1
Mixture Distributions
.............................115
3.3.2
Space-Time Decomposition of a Compound
Poisson
Process
..........................................121
3.3.3
An Exact Numerical Procedure for Calculating the
Total Claim Amount Distribution
...................126
3.3.4
Approximation to the Distribution of the Total Claim
Amount Using the Central Limit Theorem
............131
3.3.5
Approximation to the Distribution of the Total Claim
Amount by Monte Carlo Techniques
.................135
Exercises
...............................................143
3.4
Reinsurance Treaties
.....................................147
Exercises
...............................................154
4
Ruin Theory
...............................................155
4.1
Risk Process, Ruin Probability and Net Profit Condition
.....155
Exercises
...............................................160
4.2
Bounds for the Ruin Probability
............................161
4.2.1
Lundberg s Inequality
..............................161
4.2.2
Exact Asymptotics for the Ruin Probability: the
Small Claim Case
.................................166
4.2.3
The Representation of the Ruin Probability as a
Compound Geometric Probability
...................176
4.2.4
Exact Asymptotics for the Ruin Probability: the
Large Claim Case
.................................178
Exercises
...............................................181
Contents
XI
Part II Experience Rating
5
Bayes
Estimation
..........................................191
5.1
The Heterogeneity Model
.................................191
5.2
Bayes
Estimation in the Heterogeneity Model
...............193
Exercises
...............................................199
6
Linear
Bayes
Estimation
...................................203
6.1
An Excursion to Minimum Linear Risk Estimation
..........204
6.2
The
Bühlmann
Model
....................................208
6.3
Linear
Bayes
Estimation in the
Bühlmann
Model
............210
6.4
The
Bühlmann-Straub
Model
.............................213
Exercises
...............................................215
References
.....................................................217
Index
..........................................................223
List of Abbreviations and Symbols
.............................233
|
| adam_txt |
Non-Life Insurance Mathematics
This book offers a mathematical introduction to non-life insurance and,
at the same time, to a multitude of applied stochastic processes. It gives de¬
tailed discussions of the fundamental models for claim sizes, claim arrivals,
the total claim amount, and their probabilistic properties. Throughout the
book the language of stochastic processes is used for describing the dynam¬
ics of an insurance portfolio in claim size space and time. In addition to
the standard actuarial notions, the reader learns about the basic models of
modern non-life insurance mathematics: the
Poisson,
compound
Poisson
and renewal processes in collective risk theory and heterogeneity and Biihl-
mann models in experience rating. The reader gets to know how the underly¬
ing probabilistic structures allow one to determine premiums in a portfolio
or in an individual policy. Special emphasis is given to the phenomena which
are caused by large claims in these models.
What makes this book special are more than
100
figures and tables illus¬
trating and visualizing the theory. Every section ends with extensive exer¬
cises. They are an integral part of this course since they support the access
to the theory.
The book can serve either as a text for an undergraduate/graduate course
on non-life insurance mathematics or applied stochastic processes. Its con¬
tent is in agreement with the European Group
Consultatif
standards. An
extensive bibliography, annotated by various comments sections with refer¬
ences to more advanced relevant literature, make the book broadly and easi¬
ly accessible.
Contents
Guidelines to the Reader
. 1
Part I Collective Risk Models
1
The Basic Model
. 7
2
Models for the Claim Number Process
. 13
2.1
The
Poisson
Process
. 13
2.1.1
The Homogeneous
Poisson
Process, the Intensity
Function, the
Cramér-Lundberg
Model
. 15
2.1.2
The Markov Property
. 18
2.1.3
Relations Between the Homogeneous and the
Inhomogeneous
Poisson
Process
. 20
2.1.4
The Homogeneous
Poisson
Process as a Renewal Process
21
2.1.5
The Distribution of the Inter-Arrival Times
. 26
2.1.6
The Order Statistics Property
. 28
2.1.7
A Discussion of the Arrival Times of the Danish Fire
Insurance Data
1980-1990. 38
2.1.8
An Informal Discussion of Transformed and
Generalized
Poisson
Processes
. 41
Exercises
. 52
2.2
The Renewal Process
. 59
2.2.1
Basic Properties
. 59
2.2.2
An Informal Discussion of Renewal Theory
. 66
Exercises
. 71
2.3
The Mixed
Poisson
Process
. 71
Exercises
. 75
X
Contents
3
The Total Claim Amount
. 77
3.1
The Order of Magnitude of the Total Claim Amount
. 78
3.1.1
The Mean and the Variance in the Renewal Model
. 79
3.1.2
The Asymptotic Behavior in the Renewal Model
. 80
3.1.3
Classical Premium Calculation Principles
. 84
Exercises
. 86
3.2
Claim Size Distributions
. 88
3.2.1
An Exploratory Statistical Analysis: QQ-Plots
. 88
3.2.2
A Preliminary Discussion of Heavy- and Light-Tailed
Distributions
. 92
3.2.3
An Exploratory Statistical Analysis: Mean Excess Plots
94
3.2.4
Standard Claim Size Distributions and Their Properties
100
3.2.5
Regularly Varying Claim Sizes and Their Aggregation.
. 105
3.2.6
Subexponential Distributions
.109
Exercises
.112
3.3
The Distribution of the Total Claim Amount
.115
3.3.1
Mixture Distributions
.115
3.3.2
Space-Time Decomposition of a Compound
Poisson
Process
.121
3.3.3
An Exact Numerical Procedure for Calculating the
Total Claim Amount Distribution
.126
3.3.4
Approximation to the Distribution of the Total Claim
Amount Using the Central Limit Theorem
.131
3.3.5
Approximation to the Distribution of the Total Claim
Amount by Monte Carlo Techniques
.135
Exercises
.143
3.4
Reinsurance Treaties
.147
Exercises
.154
4
Ruin Theory
.155
4.1
Risk Process, Ruin Probability and Net Profit Condition
.155
Exercises
.160
4.2
Bounds for the Ruin Probability
.161
4.2.1
Lundberg's Inequality
.161
4.2.2
Exact Asymptotics for the Ruin Probability: the
Small Claim Case
.166
4.2.3
The Representation of the Ruin Probability as a
Compound Geometric Probability
.176
4.2.4
Exact Asymptotics for the Ruin Probability: the
Large Claim Case
.178
Exercises
.181
Contents
XI
Part II Experience Rating
5
Bayes
Estimation
.191
5.1
The Heterogeneity Model
.191
5.2
Bayes
Estimation in the Heterogeneity Model
.193
Exercises
.199
6
Linear
Bayes
Estimation
.203
6.1
An Excursion to Minimum Linear Risk Estimation
.204
6.2
The
Bühlmann
Model
.208
6.3
Linear
Bayes
Estimation in the
Bühlmann
Model
.210
6.4
The
Bühlmann-Straub
Model
.213
Exercises
.215
References
.217
Index
.223
List of Abbreviations and Symbols
.233 |
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| building | Verbundindex |
| bvnumber | BV021673438 |
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| callnumber-label | HG8781 |
| callnumber-raw | HG8781 |
| callnumber-search | HG8781 |
| callnumber-sort | HG 48781 |
| callnumber-subject | HG - Finance |
| classification_rvk | QQ 630 SK 980 |
| classification_tum | MAT 600f WIR 190f MAT 902f |
| ctrlnum | (OCoLC)255192361 (DE-599)BVBBV021673438 |
| dewey-full | 368.00151962 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 368 - Insurance |
| dewey-raw | 368.00151962 |
| dewey-search | 368.00151962 |
| dewey-sort | 3368.00151962 |
| dewey-tens | 360 - Social problems and services; associations |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| edition | Corr. 2. print. |
| format | Book |
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| id | DE-604.BV021673438 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:09:28Z |
| indexdate | 2024-07-09T20:41:20Z |
| institution | BVB |
| isbn | 3540406506 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014887767 |
| oclc_num | 255192361 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-29 DE-706 DE-29T DE-739 DE-634 |
| owner_facet | DE-19 DE-BY-UBM DE-29 DE-706 DE-29T DE-739 DE-634 |
| physical | XI, 235 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Mikosch, Thomas 1955- Verfasser (DE-588)141029412 aut Non-life insurance mathematics an introduction with stochastic processes Thomas Mikosch Corr. 2. print. Berlin [u.a.] Springer 2006 XI, 235 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext This book 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. Throughout the book the language of stochastic processes is used for describing the dynamics of an insurance portfolio in claim size space and time. In addition to the standard actuarial notions, the reader learns about the basic models of modern non-life insurance mathematics: the Poisson, compound Poisson and renewal processes in collective risk theory and heterogeneity and Bühlmann models in experience rating. The reader gets to know how the underlying probabilistic structures allow one to determine premiums in a portfolio or in an individual policy. Special emphasis is given to the phenomena which are caused by large claims in these models. What makes this book special are more than 100 figures and tables illustrating and visualizing the theory. Every section ends with extensive exercises. They are an integral part of this course since they support the access to the theory. The book can serve either as a text for an undergraduate/graduate course on non-life insurance mathematics or applied stochastic processes. Its content is in agreement with the European "Groupe Consultatif" standards. An extensive bibliography, annotated by various comments sections with references to more advanced relevant literature, make the book broadly and easiliy accessible. Versicherungsmathematik - Stochastisches Modell Mathematik Insurance Mathematics Stochastic processes Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 s Stochastisches Modell (DE-588)4057633-4 s DE-604 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014887767&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014887767&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Mikosch, Thomas 1955- Non-life insurance mathematics an introduction with stochastic processes Versicherungsmathematik - Stochastisches Modell Mathematik Insurance Mathematics Stochastic processes Versicherungsmathematik (DE-588)4063194-1 gnd Stochastisches Modell (DE-588)4057633-4 gnd |
| subject_GND | (DE-588)4063194-1 (DE-588)4057633-4 |
| title | Non-life insurance mathematics an introduction with stochastic processes |
| title_auth | Non-life insurance mathematics an introduction with stochastic processes |
| title_exact_search | Non-life insurance mathematics an introduction with stochastic processes |
| title_exact_search_txtP | Non-life insurance mathematics an introduction with stochastic processes |
| title_full | Non-life insurance mathematics an introduction with stochastic processes Thomas Mikosch |
| title_fullStr | Non-life insurance mathematics an introduction with stochastic processes Thomas Mikosch |
| title_full_unstemmed | Non-life insurance mathematics an introduction with stochastic processes Thomas Mikosch |
| title_short | Non-life insurance mathematics |
| title_sort | non life insurance mathematics an introduction with stochastic processes |
| title_sub | an introduction with stochastic processes |
| topic | Versicherungsmathematik - Stochastisches Modell Mathematik Insurance Mathematics Stochastic processes Versicherungsmathematik (DE-588)4063194-1 gnd Stochastisches Modell (DE-588)4057633-4 gnd |
| topic_facet | Versicherungsmathematik - Stochastisches Modell Mathematik Insurance Mathematics Stochastic processes Versicherungsmathematik Stochastisches Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014887767&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014887767&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT mikoschthomas nonlifeinsurancemathematicsanintroductionwithstochasticprocesses |