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functional equation.

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Asymptotic behavior of the hitting time, overshootandundershoot for some Lévy processes
Bernard Roynette, Pierre Vallois, Agnès Volpi
Journal:

ESAIM: Probability and Statistics / Volume 12 / 2008

Published online by Cambridge University Press:

13 November 2007, pp. 58-93

Print publication:

2008
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Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévymeasure ν. We consider the first passage time T x of(Xt, t ≥ 0) to level x > 0, and Kx := XTx - x theovershoot and Lx := x- XTx- the undershoot. We first provethat the Laplace transform of the random triple (Tx,Kx,Lx )satisfies some kind of integral equation. Second, assuming thatν admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution asx → ∞, where $\widetilde{T_x}$ denotes a suitablerenormalization of T x .