Modelling extremal events for insurance and finance:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2003
|
| Ausgabe: | Corr. 4. print. |
| Schriftenreihe: | Applications of mathematics
33 |
| Schlagworte: | |
| Online-Zugang: | Cover Inhaltsverzeichnis http://scans.hebis.de/HEBCGI/show.pl?23574169_toc.pdf http://scans.hebis.de/HEBCGI/show.pl?23574169_vlg.html http://scans.hebis.de/HEBCGI/show.pl?23574169_cov.jpg http://scans.hebis.de/HEBCGI/show.pl?23574169_kla.pdf Inhaltsverzeichnis Klappentext |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XV, 648 S. graph. Darst. |
| ISBN: | 3540609318 9783540609315 9783642082429 |
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| 100 | 1 | |a Embrechts, Paul |d 1953- |e Verfasser |0 (DE-588)115254447 |4 aut | |
| 245 | 1 | 0 | |a Modelling extremal events for insurance and finance |c Paul Embrechts ; Claudia Klüppelberg ; Thomas Mikosch |
| 250 | |a Corr. 4. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2003 | |
| 300 | |a XV, 648 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Applications of mathematics |v 33 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Finanzmathematik - Extremwertstatistik | |
| 650 | 4 | |a Versicherungsmathematik - Extremwertstatistik | |
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Datensatz im Suchindex
| DE-BY-862_location | 2000 |
|---|---|
| DE-BY-FWS_call_number | 2000/SK 820 E53 |
| DE-BY-FWS_katkey | 734369 |
| DE-BY-FWS_media_number | 083000508908 |
| _version_ | 1806195850092740608 |
| adam_text | Table
of
Contents
Reader Guidelines
............................................ 1
1
Risk Theory
................................................ 21
1.1
The Ruin Problem
...................................... 22
1.2
The
Cramér-Lundberg
Estimate
......................... 28
1.3
Ruin Theory for Heavy-Tailed Distributions
............... 36
1.3.1
Some Preliminary Results
......................... 37
1.3.2
Cramér-Lundberg
Theory for Subexponential
Distributions
.................................... 39
1.3.3
The Total Claim Amount in the Subexponential Case
. 44
1.4
Cramér-Lundberg
Theory for Large Claims: a Discussion
---- 49
1.4.1
Some Related Classes of Heavy-Tailed Distributions
.. 49
1.4.2
The Heavy-Tailed
Cramér-Lundberg
Case Revisited
.. 53
2
Fluctuations of Sums
....................................... 59
2.1
The Laws of Large Numbers
............................. 60
2.2
The Central Limit Problem
.............................. 70
2.3
Refinements of the CLT
................................. 82
2.4
The Functional CLT: Brownian Motion Appears
............ 88
2.5
Random Sums
......................................... 96
2.5.1
General Randomly Indexed Sequences
............... 96
2.5.2
Renewal Counting Processes
.......................103
2.5.3
Random Sums Driven by Renewal Counting Processes
106
3
Fluctuations of Maxima
....................................113
3.1
Limit Probabilities for Maxima
...........................114
3.2
Weak Convergence of Maxima Under
Affine
Transformations
. 120
3.3
Maximum Domains of Attraction and Norming Constants
... 128
3.3.1
The Maximum Domain of Attraction of the
Eréchet
Distribution
Фа(х)
=
exp
{-аГ*}
..................130
XII Table of
Contents
3.3.2
The Maximum
Domain
of
Attraction
of the Weibull
Distribution
Фа(х)
=
exp {-(-x)01}
.................134
3.3.3
The Maximum
Domain of Attraction of the Gumbel
Distribution A{x)
=
exp
{—
exp-f—x}}
...............138
3.4
The Generalised Extreme Value Distribution and the
Generalised Pareto Distribution
..........................152
3.5
Almost Sure Behaviour of Maxima
........................168
4
Fluctuations of Upper Order Statistics
......................181
4.1
Order Statistics
........................................182
4.2
The Limit Distribution of Upper Order Statistics
...........196
4.3
The Limit Distribution of Randomly Indexed
Upper Order Statistics
..................................204
4.4
Some Extreme Value Theory for Stationary Sequences
......209
5
An Approach to Extremes via Point Processes
..............219
5.1
Basic Facts About Point Processes
........................220
5.1.1
Definition and Examples
..........................220
5.1.2
Distribution and Laplace Functional
................225
5.1.3
Poisson
Random Measures
.........................226
5.2
Weak Convergence of Point Processes
.....................232
5.3
Point Processes of Exceedances
...........................237
5.3.1
The
ІШ
Case
....................................238
5.3.2
The Stationary Case
..............................242
5.4
Applications of Point Process Methods to IID Sequences
___247
5.4.1
Records and Record Times
........................248
5.4.2
Embedding Maxima in Extremal Processes
..........250
5.4.3
The Frequency of Records and the Growth
of Record Times
.................................254
5.4.4
Invariance
Principle for Maxima
....................260
5.5
Some Extreme Value Theory for Linear Processes
...........263
5.5.1
Noise in the Maximum Domain of Attraction
of the
Fréchet
Distribution
Фа
.....................264
5.5.2
Subexponential Noise in the Maximum Domain
of Attraction of the Gumbel Distribution
Λ
..........277
β
Statistical Methods for Extremal Events
....................283
6.1
Introduction
...........................................283
6.2
Exploratory Data Analysis for Extremes
...................290
6.2.1
Probability and Quantile Plots
.....................290
6.2.2
The Mean Excess Function
........................294
Table
of
Contents
XIII
6.2.3
Gumbeľs
Method of Exceedances
..................303
6.2.4
The Return Period
...............................305
6.2.5
Records as an Exploratory Tool
....................307
6.2.6
The Ratio of Maximum and Sum
...................309
6.3
Parameter Estimation for the Generalised Extreme
Value Distribution
......................................316
6.3.1
Maximum Likelihood Estimation
...................317
6.3.2
Method of Probability-Weighted Moments
...........321
6.3.3
Tail and Quantile Estimation, a First Go
............323
6.4
Estimating Under Maximum Domain
of Attraction Conditions
................................325
6.4.1
Introduction
.....................................325
6.4.2
Estimating the Shape Parameter
f
..................327
6.4.3
Estimating the Norming Constants
.................345
6.4.4
Tail and Quantile Estimation
......................348
6.5
Fitting Excesses Over a Threshold
........................352
6.5.1
Fitting the GPD
.................................352
6.5.2
An Application to Real Data
......................358
7
Time Series Analysis for Heavy-Tailed Processes
...........371
7.1
A Short Introduction to Classical Time Series Analysis
......372
7.2
Heavy-Tailed Time Series
...............................378
7.3
Estimation of the Autocorrelation Function
................381
7.4
Estimation of the Power Transfer Function
................386
7.5
Parameter Estimation for
ARMA
Processes
................393
7.6
Some Remarks About Non-Linear Heavy-Tailed Models
.....403
8
Special Topics
..............................................413
8.1
The Extremal Index
....................................413
8.1.1
Definition and Elementary Properties
...............413
8.1.2
Interpretation and Estimation of the Extremal Index
. 418
8.1.3
Estimating the Extremal Index from Data
...........424
8.2
A Large Claim Index
...................................430
8.2.1
The Problem
....................................430
8.2.2
The Index
.......................................431
8.2.3
Some Examples
..................................433
8.2.4
On Sums and Extremes
...........................436
8.3
When and How Ruin Occurs
.............................439
8.3.1
Introduction
.....................................439
8.3.2
The Cramer-Lundberg Case
.......................444
8.3.3
The Large Claim Case
............................449
XIV Table of
Contents
8.4
Perpetuities and ARCH Processes
........................454
8.4.1
Stochastic Recurrence Equations and Perpetuities
----455
8.4.2
Basic Properties of ARCH Processes
................461
8.4.3
Extremes of ARCH Processes
......................473
8.5
On the Longest Success-Run
.............................481
8.5.1
The Total Variation Distance to a
Poisson
Distribution
..............................483
8.5.2
The Almost Sure Behaviour
.......................486
8.5.3
The Distributional Behaviour
......................493
8.6
Some Results on Large Deviations
........................498
8.7
Reinsurance Treaties
....................................503
8.7.1
Introduction
.....................................503
8.7.2
Probabilistic Analysis
.............................507
8.8
Stable Processes
........................................521
8.8.1
Stable Random Vectors
...........................522
8.8.2
Symmetric Stable Processes
.......................526
8.8.3
Stable Integrals
..................................527
8.8.4
Examples
.......................................532
8.9
Self-Similarity
.........................................541
Appendix
.....................................................551
Al
Modes of Convergence
..................................551
Al.l Convergence in Distribution
.......................551
A1.2 Convergence in Probability
........................552
A1.3 Almost Sure Convergence
.........................553
A1.4 /¿ -Convergence
..................................553
A1.5 Convergence to Types
.............................554
A
1.6
Convergence of Generalised Inverse Functions
........554
A2 Weak Convergence in Metric Spaces
......................555
A2.1 Preliminaries about Stochastic Processes
............555
A2.2 The Spaces
С
[0,1]
and
D
[0,1].....................557
A2.3 The Skorokhod Space
D
(0,
oo)
.....................559
A2.4 Weak Convergence
...............................559
A2.5 The Continuous Mapping Theorem
.................561
A2.6 Weak Convergence of Point Processes
...............562
A3
Regular Variation and Subexponentiality
..................564
A3.1 Basic Results on Regular Variation
.................564
A3.2 Properties of Subexponential Distributions
..........571
A3.3 The Tail Behaviour of Weighted Sums
of Heavy-Tailed Random Variables
.................583
A4
Some Renewal Theory
..................................587
Table
of Contents XV
References
....................................................591
Index
.........................................................627
List of Abbreviations and Symbols
............................643
P.EMBRECHTS
·
C.KLÜPPELBERG
·
TH.MIKOSCH
Modelling Extremal Events
,
oth in insurance and finance applications, questions involving
_-
extremal events (such as large insurance claims, large fluctua¬
tions, in financial data, stock market shocks, risk management,
reinsurance products
...)
play an increasingly important role.
This book presents a comprehensive treatment of extreme value
methodology for random walk models, time series, certain types
of continuous-time stochastic processes and compound
Poisson
processes, all standard models that occur in applications in insur¬
ance mathematics and mathematical finance. Both probabilistic
and statistical methods are discussed in detail, with such topics as
ruin theory for large claim models, fluctuation theory of sums and
extremes, extremes in time series models, point process methods,
statistical estimation of quantiles and tail probabilities. Besides
summarising and bringing together known results, the book also
features topics which appear for the first time in textbook form,
including the theory of subexponential
distributions and the spectral theory of heavy-tailed time series.
A typical chapter will introduce the new methodology in a
rather intuitive (though always mathematically correct) way,
stressing the understanding of new techniques rather than
following the usual theorem-proof** format. Various examples,
mainly from the realm of insurance and finance, help to convey
the usefulness of the new material. A final chapter on more
extensive applications and/or related fields broadens the scope
further.
The book can serve either as a text for a graduate course on
applied probability and statistics, insurance and/or finance, or as
a basic reference source. Its reference quality is enhanced by a very
extensive bibliography, annotated by various notes and comments
sections making the book broadly and easily accessible.
|
| any_adam_object | 1 |
| author | Embrechts, Paul 1953- Klüppelberg, Claudia 1953- Mikosch, Thomas 1955- |
| author_GND | (DE-588)115254447 (DE-588)17219010X (DE-588)141029412 |
| author_facet | Embrechts, Paul 1953- Klüppelberg, Claudia 1953- Mikosch, Thomas 1955- |
| author_role | aut aut aut |
| author_sort | Embrechts, Paul 1953- |
| author_variant | p e pe c k ck t m tm |
| building | Verbundindex |
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| classification_tum | MAT 902f WIR 160f WIR 190f |
| ctrlnum | (OCoLC)249457335 (DE-599)BVBBV017861857 |
| dewey-full | 650/.01/513 |
| dewey-hundreds | 600 - Technology (Applied sciences) |
| dewey-ones | 650 - Management and auxiliary services |
| dewey-raw | 650/.01/513 |
| dewey-search | 650/.01/513 |
| dewey-sort | 3650 11 3513 |
| dewey-tens | 650 - Management and auxiliary services |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | Corr. 4. print. |
| format | Book |
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| id | DE-604.BV017861857 |
| illustrated | Illustrated |
| indexdate | 2024-08-01T16:29:48Z |
| institution | BVB |
| isbn | 3540609318 9783540609315 9783642082429 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010717484 |
| oclc_num | 249457335 |
| open_access_boolean | |
| owner | DE-91 DE-BY-TUM DE-824 DE-19 DE-BY-UBM DE-20 DE-703 DE-355 DE-BY-UBR DE-11 DE-91G DE-BY-TUM DE-1049 DE-29T DE-739 DE-634 DE-N2 DE-384 DE-523 DE-83 DE-945 DE-706 DE-862 DE-BY-FWS |
| owner_facet | DE-91 DE-BY-TUM DE-824 DE-19 DE-BY-UBM DE-20 DE-703 DE-355 DE-BY-UBR DE-11 DE-91G DE-BY-TUM DE-1049 DE-29T DE-739 DE-634 DE-N2 DE-384 DE-523 DE-83 DE-945 DE-706 DE-862 DE-BY-FWS |
| physical | XV, 648 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series | Applications of mathematics |
| series2 | Applications of mathematics |
| spellingShingle | Embrechts, Paul 1953- Klüppelberg, Claudia 1953- Mikosch, Thomas 1955- Modelling extremal events for insurance and finance Applications of mathematics Finanzmathematik - Extremwertstatistik Versicherungsmathematik - Extremwertstatistik Versicherungsmathematik (DE-588)4063194-1 gnd Finanzmathematik (DE-588)4017195-4 gnd Extremwertstatistik (DE-588)4153429-3 gnd Wirtschaftsmathematik (DE-588)4066472-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| subject_GND | (DE-588)4063194-1 (DE-588)4017195-4 (DE-588)4153429-3 (DE-588)4066472-7 (DE-588)4064324-4 |
| title | Modelling extremal events for insurance and finance |
| title_auth | Modelling extremal events for insurance and finance |
| title_exact_search | Modelling extremal events for insurance and finance |
| title_full | Modelling extremal events for insurance and finance Paul Embrechts ; Claudia Klüppelberg ; Thomas Mikosch |
| title_fullStr | Modelling extremal events for insurance and finance Paul Embrechts ; Claudia Klüppelberg ; Thomas Mikosch |
| title_full_unstemmed | Modelling extremal events for insurance and finance Paul Embrechts ; Claudia Klüppelberg ; Thomas Mikosch |
| title_short | Modelling extremal events for insurance and finance |
| title_sort | modelling extremal events for insurance and finance |
| topic | Finanzmathematik - Extremwertstatistik Versicherungsmathematik - Extremwertstatistik Versicherungsmathematik (DE-588)4063194-1 gnd Finanzmathematik (DE-588)4017195-4 gnd Extremwertstatistik (DE-588)4153429-3 gnd Wirtschaftsmathematik (DE-588)4066472-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Finanzmathematik - Extremwertstatistik Versicherungsmathematik - Extremwertstatistik Versicherungsmathematik Finanzmathematik Extremwertstatistik Wirtschaftsmathematik Wahrscheinlichkeitsrechnung |
| url | http://swbplus.bsz-bw.de/bsz30489124xcov.htm http://bvbr.bib-bvb.de:8991/F?func=service&doc%5Flibrary=BVB01&doc%5Fnumber=016673161&line%5Fnumber=0001&func%5Fcode=DB%5FRECORDS&service%5Ftype=MEDIA http://scans.hebis.de/HEBCGI/show.pl?23574169_toc.pdf http://scans.hebis.de/HEBCGI/show.pl?23574169_vlg.html http://scans.hebis.de/HEBCGI/show.pl?23574169_cov.jpg http://scans.hebis.de/HEBCGI/show.pl?23574169_kla.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010717484&sequence=000005&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=010717484&sequence=000006&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000895226 |
| work_keys_str_mv | AT embrechtspaul modellingextremaleventsforinsuranceandfinance AT kluppelbergclaudia modellingextremaleventsforinsuranceandfinance AT mikoschthomas modellingextremaleventsforinsuranceandfinance |
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2000 SK 820 E53 |
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