Joint law of the hitting time, overshoot and undershoot for a L\'evy process

TL;DR

This paper investigates the joint distribution of the first passage time, overshoot, and undershoot for a Lévy process composed of Brownian motion with drift and a compound Poisson process, providing new insights into their regularity and joint law.

Contribution

It derives the joint law of the first passage time, overshoot, and undershoot for a specific Lévy process, including regularity properties of the first passage time density.

Findings

Established the regularity of the first passage time density.

Derived the explicit joint law of the first passage time, overshoot, and undershoot.

Provided analytical tools for studying passage times in Lévy processes.

Abstract

Let be $(X_t, t\geq 0)$ be a L\'evy process which is the sum of a Brownian motion with drift and a compound Poisson process. We consider the first passage time $\tau_x$ at a fixed level $x>0$ by $(X_t, t\geq 0)$ and $K_x:= X_{\tau_x}-x$ the overshoot and $L_x:= x-X_{\tau_x^-}$ the undershoot. We first study the regularity of the density of the first passage time. Secondly, we calculate the joint law of $(\tau_x, K_x, L_x).$