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

Bernard Roynette; Pierre Vallois; Agnès Volpi

ESAIM: Probability and Statistics (2008)

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

Abstract

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Let ( X t , t ≥ 0 ) be a Lévy process started at 0 , with Lévy measure ν . We consider the first passage time T x of ( X t , t ≥ 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 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 .

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 (2008): 58-93. <http://eudml.org/doc/245499>.

@article{Roynette2008,
abstract = {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 $x &gt; 0$, and $ K_x:=X_\{T_x\}-\{\it x\}$ the overshoot and $L_x:=x-X_\{T_\{\{\it x\}^-\}\}$ 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\rightarrow \infty $, where $\widetilde\{T_x\}$ denotes a suitable renormalization of $T_x$.},
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},
language = {eng},
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/245499},
volume = {12},
year = {2008},
}

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
PY - 2008
PB - EDP-Sciences
VL - 12
SP - 58
EP - 93
AB - 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 $x &gt; 0$, and $ K_x:=X_{T_x}-{\it x}$ the overshoot and $L_x:=x-X_{T_{{\it x}^-}}$ 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\rightarrow \infty $, where $\widetilde{T_x}$ denotes a suitable renormalization of $T_x$.
LA - eng
KW - Lévy processes; ruin problem; hitting time; overshoot; undershoot; asymptotic estimates; functional equation
UR - http://eudml.org/doc/245499
ER -

References

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