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