On the last zero process with an application in corporate bankruptcy

TL;DR

This paper develops new stochastic calculus tools for spectrally negative Lévy processes, deriving formulas for last zero times, excursions, and optimal stopping problems with applications in corporate bankruptcy modeling.

Contribution

It introduces a perturbation-based Itô formula for the process involving last zero times and excursions, and applies it to solve optimal stopping problems in bankruptcy risk.

Findings

Derived an Itô formula for the process involving last zero times.

Calculated the joint Laplace transform of excursion-related variables.

Provided solutions to optimal stopping problems with bankruptcy applications.

Abstract

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).