Regularity and variationality of solutions  to Hamilton-Jacobi equations. 
Part I: Regularity

Andrea C.G. Mennucci

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

Bernard Roynette, Pierre Vallois, Agnès Volpi (2007)

ESAIM: Probability and Statistics

Similarity:

Let () be a Lévy process started at , with Lévy measure . We consider the first passage time of () to level , and the overshoot and the undershoot. We first prove that the Laplace transform of the random triple () satisfies some kind of integral equation. Second, assuming that admits exponential moments, we show that $(\tilde{T_{x}}, K_{x}, L_{x})$ converges in distribution as → ∞, where $\tilde{T_{x}}$ denotes a suitable renormalization of . 

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