Boris (State University, St. Petersburg, Russia) Harlamov,
B Harlamov
John Wiley & Sons
e druk, 2017
9781786300089
Stochastic Risk Analysis and Management
Specificaties
Gebonden, 164 blz.
|
Engels
John Wiley & Sons |
e druk, 2017
ISBN13: 9781786300089
Rubricering
Verwachte levertijd ongeveer 16 werkdagen
Samenvatting
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.
Specificaties
Inhoudsopgave
<p>Chapter 1. Mathematical Bases 1</p>
<p>1.1. Introduction to stochastic risk analysis 1</p>
<p>1.1.1. About the subject 1</p>
<p>1.1.2. About the ruin model 2</p>
<p>1.2. Basic methods 4</p>
<p>1.2.1. Some concepts of probability theory 4</p>
<p>1.2.2. Markov processes 14</p>
<p>1.2.3. Poisson process 18</p>
<p>1.2.4. Gamma process 21</p>
<p>1.2.5. Inverse gamma process 23</p>
<p>1.2.6. Renewal process 24</p>
<p>Chapter 2. Cramér–Lundberg Model 29</p>
<p>2.1. Infinite horizon 29</p>
<p>2.1.1. Initial probability space 29</p>
<p>2.1.2. Dynamics of a homogeneous insurance company portfolio 30</p>
<p>2.1.3. Ruin time 33</p>
<p>2.1.4. Parameters of the gain process 33</p>
<p>2.1.5. Safety loading 35</p>
<p>2.1.6. Pollaczek–Khinchin formula 36</p>
<p>2.1.7. Sub–probability distribution G+ 38</p>
<p>2.1.8. Consequences from the Pollaczek–Khinchin formula 41</p>
<p>2.1.9. Adjustment coefficient of Lundberg 44</p>
<p>2.1.10. Lundberg inequality 45</p>
<p>2.1.11. Cramér asymptotics 46</p>
<p>2.2. Finite horizon 49</p>
<p>2.2.1. Change of measure 49</p>
<p>2.2.2. Theorem of Gerber 54</p>
<p>2.2.3. Change of measure with parameter gamma 56</p>
<p>2.2.4. Exponential distribution of claim size 57</p>
<p>2.2.5. Normal approximation 64</p>
<p>2.2.6. Diffusion approximation 68</p>
<p>2.2.7. The first exit time for the Wiener process 70</p>
<p>Chapter 3. Models With the Premium Dependent on the Capital 77</p>
<p>3.1. Definitions and examples 77</p>
<p>3.1.1. General properties 78</p>
<p>3.1.2. Accumulation process 81</p>
<p>3.1.3. Two levels 86</p>
<p>3.1.4. Interest rate 90</p>
<p>3.1.5. Shift on space 91</p>
<p>3.1.6. Discounted process 92</p>
<p>3.1.7. Local factor of Lundberg 98</p>
<p>Chapter 4. Heavy Tails 107</p>
<p>4.1. Problem of heavy tails 107</p>
<p>4.1.1. Tail of distribution 107</p>
<p>4.1.2. Subexponential distribution 109</p>
<p>4.1.3. Cramér–Lundberg process 117</p>
<p>4.1.4. Examples 120</p>
<p>4.2. Integro–differential equation 124</p>
<p>Chapter 5. Some Problems of Control 129</p>
<p>5.1. Estimation of probability of ruin on a finite interval 129</p>
<p>5.2. Probability of the credit contract realization 130</p>
<p>5.2.1. Dynamics of the diffusion–type capital 132</p>
<p>5.3. Choosing the moment at which insurance begins 135</p>
<p>5.3.1. Model of voluntary individual insurance 135</p>
<p>5.3.2. Non–decreasing continuous semi–Markov process 139</p>
<p>Bibliography 147</p>
<p>Index 149</p>
<p>1.1. Introduction to stochastic risk analysis 1</p>
<p>1.1.1. About the subject 1</p>
<p>1.1.2. About the ruin model 2</p>
<p>1.2. Basic methods 4</p>
<p>1.2.1. Some concepts of probability theory 4</p>
<p>1.2.2. Markov processes 14</p>
<p>1.2.3. Poisson process 18</p>
<p>1.2.4. Gamma process 21</p>
<p>1.2.5. Inverse gamma process 23</p>
<p>1.2.6. Renewal process 24</p>
<p>Chapter 2. Cramér–Lundberg Model 29</p>
<p>2.1. Infinite horizon 29</p>
<p>2.1.1. Initial probability space 29</p>
<p>2.1.2. Dynamics of a homogeneous insurance company portfolio 30</p>
<p>2.1.3. Ruin time 33</p>
<p>2.1.4. Parameters of the gain process 33</p>
<p>2.1.5. Safety loading 35</p>
<p>2.1.6. Pollaczek–Khinchin formula 36</p>
<p>2.1.7. Sub–probability distribution G+ 38</p>
<p>2.1.8. Consequences from the Pollaczek–Khinchin formula 41</p>
<p>2.1.9. Adjustment coefficient of Lundberg 44</p>
<p>2.1.10. Lundberg inequality 45</p>
<p>2.1.11. Cramér asymptotics 46</p>
<p>2.2. Finite horizon 49</p>
<p>2.2.1. Change of measure 49</p>
<p>2.2.2. Theorem of Gerber 54</p>
<p>2.2.3. Change of measure with parameter gamma 56</p>
<p>2.2.4. Exponential distribution of claim size 57</p>
<p>2.2.5. Normal approximation 64</p>
<p>2.2.6. Diffusion approximation 68</p>
<p>2.2.7. The first exit time for the Wiener process 70</p>
<p>Chapter 3. Models With the Premium Dependent on the Capital 77</p>
<p>3.1. Definitions and examples 77</p>
<p>3.1.1. General properties 78</p>
<p>3.1.2. Accumulation process 81</p>
<p>3.1.3. Two levels 86</p>
<p>3.1.4. Interest rate 90</p>
<p>3.1.5. Shift on space 91</p>
<p>3.1.6. Discounted process 92</p>
<p>3.1.7. Local factor of Lundberg 98</p>
<p>Chapter 4. Heavy Tails 107</p>
<p>4.1. Problem of heavy tails 107</p>
<p>4.1.1. Tail of distribution 107</p>
<p>4.1.2. Subexponential distribution 109</p>
<p>4.1.3. Cramér–Lundberg process 117</p>
<p>4.1.4. Examples 120</p>
<p>4.2. Integro–differential equation 124</p>
<p>Chapter 5. Some Problems of Control 129</p>
<p>5.1. Estimation of probability of ruin on a finite interval 129</p>
<p>5.2. Probability of the credit contract realization 130</p>
<p>5.2.1. Dynamics of the diffusion–type capital 132</p>
<p>5.3. Choosing the moment at which insurance begins 135</p>
<p>5.3.1. Model of voluntary individual insurance 135</p>
<p>5.3.2. Non–decreasing continuous semi–Markov process 139</p>
<p>Bibliography 147</p>
<p>Index 149</p>