On the last zero process with an application in corporate bankruptcy

Abstract

For a spectrally negative Lévy process $X$, consider $g_t$, the last time $X$ is below the level zero before time $t\ge 0$. We use a perturbation method for Lévy processes to derive an Itô formula for the three-dimensional process $\{(g_t,t,X_t),t\ge 0\}$ 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 $(U,X)=\{(U_t,X_t),t\ge 0\}$ 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 an application in 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 (2024).