Mathematical methods in risk theory:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1970 [erschienen] 1996
|
| Edition: | 2. print. |
| Series: | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen
172 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 210 S. |
| ISBN: | 3540617035 |
Staff View
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Record in the Search Index
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Table of Contents
Part I. The 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 15
1.1.5. Chebyshev'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
1.2.5. 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
2.2.1. Mathematical description 41
2.2.2. 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 111
5.1. The retention problem 111
5.1.1. General 111
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 aQ 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
6.5.2. 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.I. 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 |
| adam_txt |
Table of Contents
Part I. The 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 15
1.1.5. Chebyshev'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
1.2.5. 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
2.2.1. Mathematical description 41
2.2.2. 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 111
5.1. The retention problem 111
5.1.1. General 111
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 aQ 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
6.5.2. 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.I. 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 |
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| id | DE-604.BV022118437 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:16:08Z |
| indexdate | 2024-08-08T00:11:07Z |
| institution | BVB |
| isbn | 3540617035 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015333303 |
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| owner | DE-706 |
| owner_facet | DE-706 |
| physical | XII, 210 S. |
| publishDate | 1970 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Springer |
| record_format | marc |
| series | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen |
| series2 | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen |
| spelling | Bühlmann, Hans 1930- Verfasser (DE-588)107824221 aut Mathematical methods in risk theory Hans Bühlmann 2. print. Berlin [u.a.] Springer 1970 [erschienen] 1996 XII, 210 S. txt rdacontent n rdamedia nc rdacarrier Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 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 Mathematische Methode (DE-588)4155620-3 s DE-604 Risikotheorie (DE-588)4135592-1 s Versicherungsmathematik (DE-588)4063194-1 s 1\p DE-604 Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 172 (DE-604)BV000000395 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015333303&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 Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 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_exact_search_txtP | 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 |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015333303&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT buhlmannhans mathematicalmethodsinrisktheory |