Parisian ruin with exponential 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. Working with a classical surplus process with exponential jump size, we obtain the Laplace transform of the time of ruin and the probability of ruin in the infinite horizon. We also consider a diffusion approximation and use it to obtain similar results for the Brownian motion with drift.