Suprema and sojourn times of Lévy processes with exponential tails

We study the tail behaviour of the supremum of sample paths of Lévy process with exponential tail of the Lévy measure. Our approach is based on the theory of sojourn times developed by S. Berman. It allows us to compute the value of the limit of the ratio P(sup0〈t〈1 X(t) > x)ϱ(x, ∞) as x → ∞, where ϱ is the Lévy measure of the process.