Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 58-93.

Let (X t ,t0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time T x of (X t ,t0) 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 ν admits exponential moments, we show that (T x ˜,K x ,L x ) converges in distribution as x, where T x ˜ denotes a suitable renormalization of T x .

DOI : https://doi.org/10.1051/ps:2007034
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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     title = {Asymptotic behavior of the hitting time, overshoot and undershoot for some {L\'evy} processes},
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     url = {http://www.numdam.org/articles/10.1051/ps:2007034/}
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Roynette, Bernard; Vallois, Pierre; Volpi, Agnès. Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 58-93. doi : 10.1051/ps:2007034. http://www.numdam.org/articles/10.1051/ps:2007034/

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