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

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

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## Abstract

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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 $\left(\stackrel{˜}{{T}_{x}},{K}_{x},{L}_{x}\right)$ converges in distribution as x → ∞, where $\stackrel{˜}{{T}_{x}}$ denotes a suitable renormalization of Tx.

## How to cite

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Roynette, 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 -

## References

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