Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Roynette, Bernard; Vallois, Pierre; Volpi, Agnès

ESAIM: Probability and Statistics, Tome 12 (2008), pp. 58-93.

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 $T_x$.

Classification : 60E10, 60F05, 60G17, 60G40, 60G51, 60J65, 60J75, 60J80, 60K05
Mots clés : Lévy processes, ruin problem, hitting time, overshoot, undershoot, asymptotic estimates, functional equation

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