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 .

Classification : 60E10,  60F05,  60G17,  60G40,  60G51,  60J65,  60J75,  60J80,  60K05
Mots clés : Lévy processes, ruin problem, hitting time, overshoot, undershoot, asymptotic estimates, functional equation
     author = {Roynette, Bernard and Vallois, Pierre and Volpi, Agn\`es},
     title = {Asymptotic behavior of the hitting time, overshoot and undershoot for some {L\'evy} processes},
     journal = {ESAIM: Probability and Statistics},
     pages = {58--93},
     publisher = {EDP-Sciences},
     volume = {12},
     year = {2008},
     doi = {10.1051/ps:2007034},
     mrnumber = {2367994},
     language = {en},
     url = {}
AU  - Roynette, Bernard
AU  - Vallois, Pierre
AU  - Volpi, Agnès
TI  - Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes
JO  - ESAIM: Probability and Statistics
PY  - 2008
DA  - 2008///
SP  - 58
EP  - 93
VL  - 12
PB  - EDP-Sciences
UR  -
UR  -
UR  -
DO  - 10.1051/ps:2007034
LA  - en
ID  - PS_2008__12__58_0
ER  - 
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.

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