We consider insurance risk models where the claim sizes are heavy-tailed subexponential, say Pareto, lognormal or DFR Weibull. This contrast with the classical Cramer conditions which requires tail decaying at least exponentially fast.

In the Cramer case, the ruin probability decays exponentially fast as the initial reserve grows and the most likely way in which ruin can occur is as a consequence of more frequent and somewhat larger claims than typical. In the heavy-tailed case, the folklore states that ruin occurs as consequence of a single huge claim. We present various results making this precise and discuss the asymptotics of the ruin function in a variety of models. We also discuss computational aspects including simulation.

The presentation is in part based upon joint work with K. Binswanger, B. Hø jgaard, C. Klüppelberg, H. Schmidli and V. Schmidt.