Insurance risk theory 1000-M1TRU

Risk theory has been created mostly for insurance, but it can be applied in virtually all situations where random losses occur. Risk theory assumed its mature form in the middle of XX century. Among the founders of this theory , there are such prominenent mathematicians as de Finetti and Cramer. From the mathematical viewpoint, the risk theory is a particularly intersting branch of probability. It has clos connections with limit theorems, random walks (theory of ruin) and Bayesian statistics (credibility theory). The programme of the course covers:

1. Individual risk model. Sums of independent random variables. Application of the Central Limit Theorem. Moment generating functions.

2. Collective risk model. Random sums and compound Poisson distributions. Approximation of the individual risk model by the collective one.

3. Ruin theory.

Discrete time models - random walks. Lundberg inequality. Ralations to the renewal theory. Khinchine-Pollaczek formula. Cramer asymptotic approximation.

Continuous time models. Compound Poisson process. Discretization of time.

Numerical iterative methods for ruin probabilities. Monte Carlo approximations.

4. Credibility.

Basics of Bayesian statistics. Empirical Bayes.

Linear prediction and the Buhlmann-Straub model.

Mixed linear models. Estimation of variance components.