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

Let be $(X_t, t\geq 0)$ 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 $τ_x$ at a fixed level $x>0$ by $(X_t, t\geq 0)$ and $K_x:= X_{τ_x}-x$ the overshoot and $L_x:= x-X_{τ_x^-}$ the undershoot. We first study the regularity of the density of the first passage time. Secondly, we calculate the joint law of $(τ_x, K_x, L_x).$

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