ИСТИНА

The classical ruin theory usually assumed that the capital reserve of an insurance company is invested in a bank account with zero interest rate. Even in this setting, nowadays considered as unrealistic, it developed a number of models: the Lundberg, Cramér-Lundberg, Sparre Andersen etc., with versions for the non-life insurance case and for the annuity payment case. In the modern world insurance companies try to get profits by investing their reserves in risky assets. For the latter mathematical finance suggested quite a lot of models as well where the price processes given by a geometric Brownian motion, geometric Lévy processes, models with local volatility and models with stochastic volatility. The ruin theory with investments deals with situations where any setting of the classical ruin theory can be combined with a model for the price of risky assets. That is we have a huge number of combinations and study of many of them are still challenging problems. Explicit expressions for the ruin probabilities are rarely available and the typical question is about their asymptotic behavior. It happens that extra randomness added by the price of a risky asset changes the asymptotical behavior of the ruin probability in a rather dramatic way. The talk will be devoted to a recent results on the asymptotic of ruin probabilities when the characteristics of the price process are modulated by a Markov process with a finite number of states.