Parisian excursion with capital injection for drawdown reflected Lévy insurance risk process

This paper discusses Parisian ruin problem under drawdown with capital injection when the underlying source of randomness of the surplus is modeled by a general Lévy insurance risk process. Capital injection is provided at the first instance the surplus drops below the drawdown level that is a pre-specified function of its current maximum. The capital is continuously injected to keep the surplus above the drawdown level until either it goes above its current maximum or a Parisian type ruin occurs, which is announced at the first time the surplus process has undergone an excursion below its current maximum for an independent exponential period of time consecutively since the most recent drawdown time. Some distributional identities concerning this excursion are presented. Firstly, we give the Parisian ruin probability and the joint Laplace transform (possibly killed at the first passage time above a fixed level for the surplus process) of the ruin time, the surplus position at ruin, and the total capital injection at ruin. Secondly, we obtain the q-potential measure of the surplus process killed at Parisian ruin. Finally, we give the expected present value of the total discounted capitals injected up to the Parisian ruin time. The results are derived using recent developments in fluctuation and excursion theory of spectrally negative Lévy process and are presented semi-explicitly in terms of the scale function of the Lévy process. Some numerical examples are given to facilitate the analysis of the impact of initial surplus and frequency of observation periods to the ruin probability and to the expected total discounted capital injection.