Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy
measure ν. We consider the first passage time Tx of
(Xt, t ≥ 0) to level x > 0, and Kx := XTx - x the
overshoot and Lx := x- XTx- the undershoot. We first prove
that the Laplace transform of the random triple (Tx,Kx,Lx)
satisfies some kind of integral equation. Second, assuming that
ν admits exponential moments, we show that
$(\widetilde{T_x},K_x,L_x)$ converges in distribution as
x → ∞, where $\widetilde{T_x}$ denotes a suitable
renormalization of Tx.