On the last zero process with an application in corporate bankruptcy

For a spectrally negative L´evy process X, consider gt, the last time X is below the level zero before time t ≥ 0. We use a perturbation method for L´evy processes to derive an Itˆo formula for the threedimensional process {(gt, t,Xt), t ≥ 0} and its infinitesimal generator. Moreover, with Ut := t − gt, the length of a current positive excursion, we derive a general formula that allows us to calculate a functional of the whole path of (U,X) = {(Ut,Xt), t ≥ 0} in terms of the positive and negative excursions of the process X. As a corollary, we find the joint Laplace transform of (Ueq ,Xeq ), where eq 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 g∞ and optimal stopping problems in terms of (U,X) as per Baurdoux and Pedraza (2024).