Stochastic Risk Analysis and Management

Boris Harlamov, Head of Laboratory of Institute of Problems of Mechanical Engineering (IPME), RAN.

• Chapter 1. Mathematical Bases  11.1. Introduction to stochastic risk analysis  11.1.1. About the subject  11.1.2. About the ruin model  21.2. Basic methods  41.2.1. Some concepts of probability theory 41.2.2. Markov processes  141.2.3. Poisson process 181.2.4. Gamma process 211.2.5. Inverse gamma process 231.2.6. Renewal process 24Chapter 2. Cramér-Lundberg Model 292.1. Infinite horizon 292.1.1. Initial probability space 292.1.2. Dynamics of a homogeneous insurance company portfolio 302.1.3. Ruin time  332.1.4. Parameters of the gain process 332.1.5. Safety loading  352.1.6. Pollaczek-Khinchin formula  362.1.7. Sub-probability distribution G+  382.1.8. Consequences from the Pollaczek-Khinchin formula  412.1.9. Adjustment coefficient of Lundberg  442.1.10. Lundberg inequality  452.1.11. Cramér asymptotics  462.2. Finite horizon  492.2.1. Change of measure 492.2.2. Theorem of Gerber 542.2.3. Change of measure with parameter gamma 562.2.4. Exponential distribution of claim size 572.2.5. Normal approximation 642.2.6. Diffusion approximation  682.2.7. The first exit time for the Wiener process 70Chapter 3. Models With the Premium Dependent on the Capital  773.1. Definitions and examples  773.1.1. General properties  783.1.2. Accumulation process 813.1.3. Two levels  863.1.4. Interest rate 903.1.5. Shift on space  913.1.6. Discounted process 923.1.7. Local factor of Lundberg  98Chapter 4. Heavy Tails  1074.1. Problem of heavy tails 1074.1.1. Tail of distribution 1074.1.2. Subexponential distribution  1094.1.3. Cramér-Lundberg process 1174.1.4. Examples  1204.2. Integro-differential equation  124Chapter 5. Some Problems of Control  1295.1. Estimation of probability of ruin on a finite interval 1295.2. Probability of the credit contract realization 1305.2.1. Dynamics of the diffusion-type capital  1325.3. Choosing the moment at which insurance begins 1355.3.1. Model of voluntary individual insurance 1355.3.2. Non-decreasing continuous semi-Markov process  139Bibliography  147Index 149