On suprema of Lévy processes with light tails

Let X ( t ) , t ≥ 0 , X ( 0 ) = 0 , be a Lévy process with a spectral Lévy measure ρ . Assuming that ∫ − 1 1 | x | ρ ( d x ) < ∞ and the right tail of ρ is light, we show that in the presence of the Brownian component P ( sup 0 ≤ t ≤ 1 X ( t ) > u ) ∼ P ( X ( 1 ) > u ) as u → ∞ , while in the absence of a Brownian component these tails are not always comparable.