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Stochastic Risk Analysis and Management

ISBN: 978-1-119-38886-9
150 pages
February 2017, Wiley-ISTE
Stochastic Risk Analysis and Management (1119388864) cover image

Description

The 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.

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Table of Contents

Chapter 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

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