Joint law of the hitting time, overshoot and undershoot for a Lévy process

Abstract

Let be $(X_t,t\ge 0)$ be a Lévy 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 $(X_t,t\ge 0)$ 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 $({\tau }_x,K_x,L_x).$

References in corpus (1)

• The hitting time of zero for a stable process