  This paper discusses Parisian ruin problem with capital injection for Levy
insurance risk process. Capital injection takes place at the draw-down time of
the surplus process when it drops below a pre-specified function of its last
record maximum. The capital is continuously paid to keep the surplus above the
draw-down level until either the surplus process goes above the record high or
a Parisian type ruin occurs, which is announced at the first instance the
surplus process has undergone an excursion below the record for an independent
exponential period of time consecutively since the time the capital was first
injected. Some distributional identities concerning the 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 of the
surplus process) of the ruin time, 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 expected present
value of the total discounted capital payments up to the Parisian ruin time.
The results are derived using recent developments in fluctuation and excursion
theory of spectrally negative Levy process and are presented semi explicitly in
terms of the scale function of the Levy process. Some numerical examples are
given to facilitate the analysis of the impact of initial surplus and frequency
of observation period to the ruin probability and to the expected total capital
injection.
