2020/03/15 by Erik J. Baurdoux, Baurdoux, Erik J., J. M. Pedraza +1
Business, Management and Accounting · Decision Sciences · #60G40 #60G51 #60J35 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Supply Chain and Inventory Management
paper · pdf · doi:10.48550/arxiv.2003.06871
openalex publication_date 2020/03/15 · openalex created_date 2022/11/16 · openalex updated_date 2026/07/28
For a spectrally negative Lévy process X, consider gt, the last time X is below the level zero before time t≥ 0. We use a perturbation method for Lévy processes to derive an Itô formula for the three-dimensional 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).