Last Exit Time until First Exit Time for Spectrally Negative Lévy Processes

This paper studies the last exit time that a spectrally negative L\'evy process is below zero until it reaches a positive level $b$, denoted by $g_{\tau_b^+}$. It generalizes the results of the infinite-horizon last exit time explored by Chiu and Yin (2005) by incorporating a random horizon $\tau_b^+$, which represents the first passage time above $b$. We derive an explicit expression for the joint Laplace transform of $g_{\tau_b^+}$ and $\tau_b^+$ by utilizing a hybrid observation scheme approach proposed by Li, Willmot, and Wong (2018). We further study the optimal prediction of $g_{\tau_b^+}$ in $L_1$ sense, and find that the optimal stopping time is the first passage time above a level $y_b^{\ast}$, with an explicit characterization of the stopping boundary $y_b^{\ast}$. As examples, Brownian motion with drift and the Cram\'er-Lundberg model with exponential jumps are considered.