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

2.1 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

3.1 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