Ruin probabilities in the Cramér–Lundberg model with temporarily negative capital

We study the asymptotics of the ruin probability in the Cramér–Lundberg model with a modified notion of ruin. The modification is as follows. If the portfolio becomes negative, the asset is not immediately declared ruined but may survive due to certain mechanisms. Under a rather general assumption on the mechanism — satisfied by most such modified models from the literature — we study the relation of the asymptotics of the modified ruin probability to the classical ruin probability. This is done under the Cramér condition as well as for subexponential integrated claim sizes.