On the last zero process with an application in corporate bankruptcy

For a spectrally negative Lévy process X, consider and its infinitesimal generator. Moreover, with , the last time X is below the level zero before time the length of a current positive excursion, we derive a general formula that allows us to calculate a functional of the whole path of . We use a perturbation method for Lévy processes to derive an Itô formula for the three-dimensional process in terms of the positive and negative excursions of the process X. As a corollary, we find the joint Laplace transform of , where is an independent exponential time, and the q-potential measure of the process (U, X). Furthermore, using the results mentioned above, we find a solution to a general optimal stopping problem depending on (U, X) with an application in corporate bankruptcy. Lastly, we establish a link between the optimal prediction of and optimal stopping problems in terms of (U, X) as per Baurdoux, E. J. and Pedraza, J. M., optimal prediction of the last zero of a spectrally negative Lévy process, Annals of Applied Probability, 34 (2024), 1350–1402.