Two-side exit problems for taxed Lévy risk process involving the general draw-down time

The literature has been witnessing an aroused interest in the study of the two-side exit problems for various models. Motivated by Kyprianou and Zhou (2009) and Li et al. (2018), the present paper concerns the two-side exit problems of the taxed spectrally negative Lévy risk process involving the general draw-down time. Our two-side exit problem is separated into two sub-problems: one being the Laplace transform of the up-exiting time of a certain level on the event that the taxed risk process up-crosses that level before the general draw-down time; the other being the Laplace transform of the draw-down time on the event that draw-down of the taxed risk process occurs before it up-crosses a certain level. Using a modified approximating method of Li et al. (2018) together with the excursion theory, solutions for the aforementioned two-side exit problems are obtained.