Description
Book SynopsisThe author investigates the Cramer –Lundberg model, collecting the most interesting theorems and methods, which estimate probability of default for a company of insurance business. These offer different kinds of approximate values for probability of default on the base of normal and diffusion approach and some special asymptotic.
Table of ContentsChapter 1. Mathematical Bases 1
1.1. Introduction to stochastic risk analysis 1
1.1.1. About the subject 1
1.1.2. About the ruin model 2
1.2. Basic methods 4
1.2.1. Some concepts of probability theory 4
1.2.2. Markov processes 14
1.2.3. Poisson process 18
1.2.4. Gamma process 21
1.2.5. Inverse gamma process 23
1.2.6. Renewal process 24
Chapter 2. Cramér-Lundberg Model 29
2.1. Infinite horizon 29
2.1.1. Initial probability space 29
2.1.2. Dynamics of a homogeneous insurance company portfolio 30
2.1.3. Ruin time 33
2.1.4. Parameters of the gain process 33
2.1.5. Safety loading 35
2.1.6. Pollaczek-Khinchin formula 36
2.1.7. Sub-probability distribution G+ 38
2.1.8. Consequences from the Pollaczek-Khinchin formula 41
2.1.9. Adjustment coefficient of Lundberg 44
2.1.10. Lundberg inequality 45
2.1.11. Cramér asymptotics 46
2.2. Finite horizon 49
2.2.1. Change of measure 49
2.2.2. Theorem of Gerber 54
2.2.3. Change of measure with parameter gamma 56
2.2.4. Exponential distribution of claim size 57
2.2.5. Normal approximation 64
2.2.6. Diffusion approximation 68
2.2.7. The first exit time for the Wiener process 70
Chapter 3. Models With the Premium Dependent on the Capital 77
3.1. Definitions and examples 77
3.1.1. General properties 78
3.1.2. Accumulation process 81
3.1.3. Two levels 86
3.1.4. Interest rate 90
3.1.5. Shift on space 91
3.1.6. Discounted process 92
3.1.7. Local factor of Lundberg 98
Chapter 4. Heavy Tails 107
4.1. Problem of heavy tails 107
4.1.1. Tail of distribution 107
4.1.2. Subexponential distribution 109
4.1.3. Cramér-Lundberg process 117
4.1.4. Examples 120
4.2. Integro-differential equation 124
Chapter 5. Some Problems of Control 129
5.1. Estimation of probability of ruin on a finite interval 129
5.2. Probability of the credit contract realization 130
5.2.1. Dynamics of the diffusion-type capital 132
5.3. Choosing the moment at which insurance begins 135
5.3.1. Model of voluntary individual insurance 135
5.3.2. Non-decreasing continuous semi-Markov process 139
Bibliography 147
Index 149
