Last passage times for generalized drawdown processes with applications

In recent years, there has been a significant amount of work dedicated to the study of the generalized drawdown process with its extensive applications in insurance and finance. While existing studies have primarily focused on analyzing the associated first passage times, which signal early warnings, the investigation of last passage times should not be overlooked. Last passage times involve knowledge of the future and can thus offer additional insights. This paper aims to fill this gap in the literature by studying the last passage times for the generalized drawdown process with an independent exponential killing and discussing their applications to insurance risk. Our analysis focuses on the Lévy insurance risk processes, for which we derive the Laplace transforms for these random times. Additionally, we obtain new results on the joint distribution of the duration of the drawdown and the surplus level at killing. As applications, we implement our results in the loss-carry-forward tax and dividend models and investigate the valuation of an European digital drawdown option. Detailed numerical examples are presented for illustrative purposes.