免費 Non-Life Insurance Mathematics - A Primer by Mikosch

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Non-Life Insurance Mathematics - A Primer by Mikosch.pdf (5.36 MB)

2013-11-12 12:41:46 上传

Non-Life Insurance Mathematics - A Primer by Mikosch

Contents

I Collective risk models 1

1 Models for the claim number process 2

1.1 The Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 The homogeneous Poisson process, the intensity function, the Cramer-Lundberg

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 The Markov property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Relations between the homogeneous and the inhomogeneous Poisson process 6

1.1.4 The homogeneous Poisson process as a renewal process . . . . . . . . . . . . 7

1.1.5 The distribution of the inter-arrival times . . . . . . . . . . . . . . . . . . . . 9

1.1.6 The order statistics property . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.7 Transformed and generalized Poisson processes . . . . . . . . . . . . . . . . . 17

1.2 The renewal process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.2 An informal discussion of renewal theory . . . . . . . . . . . . . . . . . . . . . 24

1.3 The mixed Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 The total claim amount 31

2.1 The order of magnitude of the total claim amount . . . . . . . . . . . . . . . . . . . 31

2.1.1 The mean and the variance in the renewal model . . . . . . . . . . . . . . . . 31

2.1.2 The asymptotic behavior in the renewal model . . . . . . . . . . . . . . . . . 33

2.1.3 Classical premium calculation principles . . . . . . . . . . . . . . . . . . . . . 34

2.2 Claim size distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.1 Exploratory statistical tools: QQ-plots . . . . . . . . . . . . . . . . . . . . . . 37

2.2.2 A preliminary discussion of heavy- and light-tailed distributions . . . . . . . 39

2.2.3 Exploratory statistical tools: mean excess plots . . . . . . . . . . . . . . . . . 41

2.2.4 Particular claim size distributions and their properties . . . . . . . . . . . . . 45

2.2.5 Regularly varying claim sizes and their aggregation . . . . . . . . . . . . . . . 49

2.2.6 Subexponential distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3 The distribution of the total claim amount . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3.1 Mixture distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3.2 An exact numerical procedure for calculating the total claim amount distribution. . . . . . . 59

2.3.3 Approximation to the distribution of the total claim amount using the central limit theorem . . . .. 62

2.3.4 Approximation to the distribution of the total claim amount by Monte-Carlo techniques . . . . . .  . 65

2.4 Reinsurance treaties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3 Risk theory 72

3.1 Risk process, ruin probability and net pro t condition . . . . . . . . . . . . . . . . . 72

3.2 Bounds for the ruin probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2.1 Lundberg's inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2.2 Exact asymptotics for the ruin probability: the small claim case . . . . . . . 79

3.2.3 The representation of the ruin probability as a compound geometric probability 83

3.2.4 Exact asymptotics for the ruin probability: the large claim case . . . . . . . . 85

I Credibility theory 90

4 Bayes estimation 90

4.1 The heterogeneity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2 Bayes estimation in the heterogeneity model . . . . . . . . . . . . . . . . . . . . . . . 91

5 Linear Bayes estimation 95

5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 The Buhlmann model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3 Linear Bayes estimation in the Buhlmann model . . . . . . . . . . . . . . . . . . . . 100

5.4 The Buhlmann-Straub model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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关键词：mathematics Mathematic Insurance Thematic Mikosch Insurance function property process number

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