Mathematical methods in risk theory:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2005
|
| Edition: | 2. print. |
| Series: | Grundlehren der mathematischen Wissenschaften
172 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 210 S. graph. Darst. |
| ISBN: | 9783540051176 3540051171 3540617035 |
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Record in the Search Index
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|---|---|
| adam_text |
Titel: Mathematical methods in risk theory
Autor: Bühlmann, Hans
Jahr: 2005
Table of Contents
Part I. Hie Theoretical Model
Chapter 1: Probability Aspects of Risk. 3
1.1. Random variables explained by the example of claim amount. 3
1.1.1. Definition. 3
1.1.2. Classification and examples of distribution functions. 4
1.1.3. Expected values. 12
1.1.4. Characteristics of a probability distribution and auxiliary functions IS
1.1.5. Chebysbev's Inequality. 21
1.2. Sequences of random variables explained by the example of claim amount
reproductions. 22
1.2.1. Multi-dimensional distributions and auxiliary functions. 22
1.2.2. Conditional distribution functions and conditional expectation . . 25
1.2.3. Independence. 28
1.2.4. Covariance and correlation. 31
1.2.3. The law of large numbers. 32
Chapter 2: The Risk Process. 35
2.1. Fundamentals. 35
2.1.1. Definitions and intuitive description of risk. 35
2.1.2. Stochastic processes with independent increments. 37
2.1.3. Markov processes. 39
2.2. The claim number process. 41
2.Z1. Mathematical description. 41
2.2.1 The claim interoccurrence time. 47
2.2.3. The homogeneous claim number process?operational time . 49
2.2.4. The case of time-independent intensities of claim frequency: conta-
gion models. 51
2.3. The accumulated claim process. 54
2.3.1. Definition as random sum and basic representation. 54
2.3.2. Proof of the basic representation of the accumulated claim distribution 56
2.3.3. The reduced basic representation: time-independent claim amounts 57
2.3.4. The reduced basic representation: time-dependent claim amounts 58
2.3.5. An example. 60
Chapter 3: The Risk in the Collective. 63
3.1. Risk-theoretical definitions. 63
3.1.1. Risk and collective. 63
3.1.2. The structure function. 65
3.2. The weighted risk process as description of the risk in the collective . 65
3.2.1. Weighted laws of probability. 65
3.2.2. The risk pattern in the collective. 67
X Table of Contents
3.2.3. The number of claims process in the collective. 68
3.2.4. The weighted Poisson and negative binomial distributions . 69
3.2.5. The accumulated claim process in the collective. 73
3.3. Portfolios in the collective. 76
3.3.1. Some definitions. 76
3.3.2. Stabilizing in time (Theorem of Ove Lundberg). 77
3.3.3. Stabilizing in size. 80
Part II. Consequences of the Theoretical Model
Chapter 4: Premium Calculation. 85
4.1. Principles of premium calculation. 85
4.1.1. General. 85
4.1.2. Some principles of premium calculation. 86
4.1.3. Discussion of the principles of premium calculation. 86
4.2. The risk premium and the collective premium. 87
4.2.1. The risk premium. 87
4.2.2. The collective premium. 88
4.2.3. Statistics and collective premium. 89
4.2.4. The dilemma and the connection between risk and collective premium 90
4.3. The credibility premium. 93
4.3.1. The credibility premium as sequential approximation to the risk
premium. 93
4.3.2. A new interpretation of the variance principle for calculation of
premiums. 94
4.3.3. Construction of the credibility premium. 96
4.3.4. Assumptions for our further investigations. 98
4.3.5. Properties of the credibility premium. 98
4.3.6. The credibility formulae for the three components of the credibility
premium. 100
4.3.7. Determining the weights in the credibility formulae. 103
4.4. A practical example: risk, collective and credibility premium in automobile
liability insurance. 106
Chapter 5: Retentions and Reserves. Ill
5.1. The retention problem. Ill
5.1.1. General. Ill
5.1.2. The retention under proportional and non-proportional reinsurance 112
5.2. The relative retention problem. 113
5.2.1. Proportional reinsurance. 114
5.2.2. Non-proportional reinsurance. 116
5.2.3. The risk with given risk parameter and the risk in the collective under
non-proportional reinsurance. 119
5.2.4. Credibility approximation for the relative retention. 121
5.3. The absolute retention problem. 124
5.3.1. Exact statement of the problem. 124
5.3.2. The random walk of the risk carrier's free reserves generated by the
risk mass. 126
5.4. Reserves. 129
Table of Contents XI
Chapter 6: The Insurance Carrier's Stability Criteria. 131
6.1. The stability problem. 131
6.1.1. Decision variables. 131
6.1.2. Stability problem and stability criteria. 132
6.2. The probability of ruin as stability criterion. 133
6.2.1. Planning horizon and ruin probability. 133
6.2.2. Admissible stability policies. 135
6.2.3. Hypotheses about the model variables in calculating the probability
of ruin. 135
6.2.4. Calculating the probability of ruin in the discrete case with finite
planning horizon. 137
6.2.5. Calculating the probability of ruin with an infinite planning horizon
using the Wiener-Hopf method. 141
6.2.6. Calculating the probability of ruin in the continuous case with
infinite planning horizon using renewal theory methods. 144
6.3. The absolute retention when the probability of ruin is chosen as the
stability criterion. 152
6.3.1. Restatement of the problem and assumptions. 152
6.3.2. The optimal gradation of retentions. 154
6.3.3. The stability condition. 155
6.3.4. Determining the absolute retention when the risk parameter is known 156
6.3.5. Determining the absolute retention when the risk parameters are
drawn from one or more collectives. 159
6.3.6. Practical remark on the probability of ruin as stability criterion . . 163
6.4. Dividend policy as criterion of stability. 164
6.4.1. General description of the criterion. 164
6.4.2. Hypotheses about the model variables when the dividend policy is
used as stability criterion. 165
6.4.3. Dividend policy in the discrete case. 165
6.4.4. Results in the discrete case. 166
6.4.5. Barrier strategies in the discrete case. 168
6.4.6. Dividend policy in the continuous case. 168
6.4.7. The integro-differential equation of the barrier strategy in the
continuous case. 171
6.4.8. Solving the integro-differential equation for V(Q, a). 172
6.4.9. Asymptotic formula for fl?. 174
6.4.10. Optimum dividend policy for Q a0 and other evaluations. . . . 177
6.5. Utility as criterion of stability. 178
6.5.1. Evaluating the random walk of free reserves. 178
6.5.Z Equivalent evaluations; definition of utility. 179
6.5.3. Axioms about utility. 182
6.5.4. Existence theorem for an equivalent utility. 184
6.5.5. Integral evaluation. 188
6.5.6. The problem of risk exchange. 190
6.5.7. The theorem of Borch. 191
6.5.8. A consequence of Borch's theorem. 195
6.5.9. Price structures with quadratic utility kernels. 197
XII Table of Contents
Appendix: The Generalized Riemann-Stieltjes Integral.201
A.l. Preliminary. 201
A.2. Definition of the generalized Riemann-Stieltjes integral in two special
cases. 201
A.3. Definition in the general case. 203
A.4. Integrable functions. 203
A.5. Properties of the generalized Riemann-Stieltjes integral. 204
Bibliography.206
Index.209 |
| any_adam_object | 1 |
| author | Bühlmann, Hans 1930- |
| author_GND | (DE-588)107824221 |
| author_facet | Bühlmann, Hans 1930- |
| author_role | aut |
| author_sort | Bühlmann, Hans 1930- |
| author_variant | h b hb |
| building | Verbundindex |
| bvnumber | BV025397876 |
| classification_rvk | QH 170 SK 850 |
| ctrlnum | (OCoLC)255614956 (DE-599)BVBBV025397876 |
| dewey-full | 368.01 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 368 - Insurance |
| dewey-raw | 368.01 |
| dewey-search | 368.01 |
| dewey-sort | 3368.01 |
| dewey-tens | 360 - Social problems and services; associations |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 2. print. |
| format | Book |
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| isbn | 9783540051176 3540051171 3540617035 |
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| spelling | Bühlmann, Hans 1930- Verfasser (DE-588)107824221 aut Mathematical methods in risk theory Hans Bühlmann 2. print. Berlin [u.a.] Springer 2005 XII, 210 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 172 Risikotheorie (DE-588)4135592-1 gnd rswk-swf Mathematische Methode (DE-588)4155620-3 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Risikotheorie (DE-588)4135592-1 s Versicherungsmathematik (DE-588)4063194-1 s DE-604 Mathematische Methode (DE-588)4155620-3 s 1\p DE-604 Grundlehren der mathematischen Wissenschaften 172 (DE-604)BV000000395 172 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020020639&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Bühlmann, Hans 1930- Mathematical methods in risk theory Grundlehren der mathematischen Wissenschaften Risikotheorie (DE-588)4135592-1 gnd Mathematische Methode (DE-588)4155620-3 gnd Versicherungsmathematik (DE-588)4063194-1 gnd |
| subject_GND | (DE-588)4135592-1 (DE-588)4155620-3 (DE-588)4063194-1 |
| title | Mathematical methods in risk theory |
| title_auth | Mathematical methods in risk theory |
| title_exact_search | Mathematical methods in risk theory |
| title_full | Mathematical methods in risk theory Hans Bühlmann |
| title_fullStr | Mathematical methods in risk theory Hans Bühlmann |
| title_full_unstemmed | Mathematical methods in risk theory Hans Bühlmann |
| title_short | Mathematical methods in risk theory |
| title_sort | mathematical methods in risk theory |
| topic | Risikotheorie (DE-588)4135592-1 gnd Mathematische Methode (DE-588)4155620-3 gnd Versicherungsmathematik (DE-588)4063194-1 gnd |
| topic_facet | Risikotheorie Mathematische Methode Versicherungsmathematik |
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