Abstract. 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 := XT x − x the overshoot and Lx := x − XT x − 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 ( Tx, Kx, Lx) converges in distribution as x → ∞, where Tx denotes a suitable renormalization of Tx.

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