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

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