On the last zero process with an application in corporate bankruptcy

Erik J. Baurdoux and José M. Pedraza

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

Abstract: 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).

Keywords: corporate bankruptcy; Itô's formula; last zero; Lévy processes; optimal stopping; positive excursions (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Date: 2025-05-28
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Published in Advances in Applied Probability, 28, May, 2025. ISSN: 0001-8678

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