On the discounted penalty at ruin in a jump-diffusion and the perpetual put option

We consider the jump-diffusion that is obtained if an independent Wiener process is added to the surplus process of classical ruin theory. In this model, we examine the expected discounted value of a penalty at ruin; we show that it satisfies a defective renewal equation which has a probabilistic interpretation. For this purpose, results for the jump-diffusion process are derived concerning the first record low caused by a jump and downcrossings before the first record low caused by a jump. As an application, we determine the optimal exercise boundary for a perpetual put option.