On the last zero process with applications in corporate bankruptcy

For a spectrally negative Lévy process \(X\), consider \(g_t\), the last time \(X\) is below the level zero before time \(t\geq 0\). We use a perturbation method for Lévy processes to derive an Itô formula for the three-dimensional process \(\{(g_t,t, X_t), t\geq 0 \}\) and its infinitesimal generat...

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Bibliographic Details
Published in:arXiv.org 2022-11
Main Authors: Baurdoux, Erik J, Pedraza, J M
Format: Article
Language:English
Subjects:
Laplace transforms
Perturbation methods
Stochastic processes
Online Access:Get full text
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Summary:For a spectrally negative Lévy process \(X\), consider \(g_t\), the last time \(X\) is below the level zero before time \(t\geq 0\). We use a perturbation method for Lévy processes to derive an Itô formula for the three-dimensional process \(\{(g_t,t, X_t), t\geq 0 \}\) and its infinitesimal generator. Moreover, with \(U_t:=t-g_t\), the length of a current positive excursion, we derive a general formula that allows us to calculate a functional of the whole path of \( (U, X)=\{(U_t, X_t),t\geq 0\}\) in terms of the positive and negative excursions of the process \(X\). As a corollary, we find the joint Laplace transform of \((U_{\mathbf{e}_q}, X_{\mathbf{e}_q})\), where \(\mathbf{e}_q\) 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 some applications for corporate bankruptcy. Lastly, we establish a link between the optimal prediction of \(g_{\infty}\) and optimal stopping problems in terms of \((U, X)\) as per Baurdoux and Pedraza (2020).
ISSN:2331-8422