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

Bernard Roynette; Pierre Vallois; Agnès Volpi

ESAIM: Probability and Statistics (2007)

- Volume: 12, page 58-93
- ISSN: 1292-8100

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topRoynette, Bernard, Vallois, Pierre, and Volpi, Agnès. "Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes." ESAIM: Probability and Statistics 12 (2007): 58-93. <http://eudml.org/doc/104411>.

@article{Roynette2007,

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 := XTx - x the
overshoot and Lx := x- XTx- 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
$(\widetilde\{T_x\},K_x,L_x)$ converges in distribution as
x → ∞, where $\widetilde\{T_x\}$ denotes a suitable
renormalization of Tx.
},

author = {Roynette, Bernard, Vallois, Pierre, Volpi, Agnès},

journal = {ESAIM: Probability and Statistics},

keywords = {Lévy processes; ruin problem; hitting time; overshoot;
undershoot; asymptotic estimates; functional equation.; undershoot; functional equation},

language = {eng},

month = {11},

pages = {58-93},

publisher = {EDP Sciences},

title = {Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes},

url = {http://eudml.org/doc/104411},

volume = {12},

year = {2007},

}

TY - JOUR

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

DA - 2007/11//

PB - EDP Sciences

VL - 12

SP - 58

EP - 93

AB - 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 := XTx - x the
overshoot and Lx := x- XTx- 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
$(\widetilde{T_x},K_x,L_x)$ converges in distribution as
x → ∞, where $\widetilde{T_x}$ denotes a suitable
renormalization of Tx.

LA - eng

KW - Lévy processes; ruin problem; hitting time; overshoot;
undershoot; asymptotic estimates; functional equation.; undershoot; functional equation

UR - http://eudml.org/doc/104411

ER -

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