Ruin probabilities of the Parisian type for small claims

In this paper, we extend the concept of ruin in risk theory to the Parisian type of ruin. For this to occur, the surplus process must fall below zero and stay negative for a continuous time interval of specified length. We obtain the probability of ruin in the infinite horizon for the case when the process starts from zero and the asymptotic form of the probability of ruin in the infinite horizon for the case when the process starts from the point far above zero. We see that in the small claim case an asymptotic formula similar to Cramér’s formula is true.