Ruin theory concerns the study of stochastic processes that represent the time evolution of the surplus of a stylized non-life insurance company. The initial goal of early researchers of the field, Lundberg (1903) and Cram\'er (1930), was to determine the probability for the surplus to become negative. In those pioneer works, the authors show that the ruin probability decreases exponentially fast to zero with initial reserve tending to infinity when the net profit condition is satisfied and claim sizes are light-tailed.
During lecture we explain when and why we can observe this phenomenon and discuss also the heavy-tailed case.
We demonstrate main techniques and results related with the asymptotics of the ruin probabilities:
Pollaczek-Khinchin formula, Lundberg bounds, change of measure, Wiener-Hopf factorization, principle of one big jump and theory of scale functions of L\'evy processes.