On the last zero process with an application in corporate bankruptcy

For a spectrally negative L\'evy 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 L\'evy processes to derive an It\^o 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 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).