Michael Braverman

Suprema and sojourn times of Lévy processes with exponential tails

We study the tail behaviour of the supremum of sample paths of Levy process with exponential tail of the Levy 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 x)/[varrho](x, [infinity]) as x --> [infinity], where [varrho] is the Levy measure of the process.

Symmetric infinitely divisible processes with sample paths in Orlicz spaces and absolute continuity of infinitely divisible processes

We give necessary and sufficient conditions under which a symmetric measurable infinitely divisible process has sample paths in an Orlicz space L[psi] with a function [psi] satisfying the [Delta]2 condition and, as an application, obtain necessary and sufficient conditions for a symmetric infinitely divisible process to have a version with absolutely continuous paths.

About boundedness of stable sequences

A sufficient condition under which a symmetric [alpha]-stable process {X(n),n[set membership, variant]N} is a.s. bounded is given. We also show that in some sense this condition is optimal.

A REMARK ON STOCHASTIC MONOTONICITY

We show that the stochastic domination of an infinitely divisible random variable X by another infinitely divisible random variable Y does not imply, generally speaking, a comparison between their corresponding Levy measures.