FUNCTIONALS OF INFINITELY DIVISIBLE STOCHASTIC-PROCESSES WITH EXPONENTIAL TAILS

We investigate the tail behavior of the distributions of subadditive f unctionals of the sample paths of infinitely divisible stochastic proc esses when the Levy measure of the process has suitably defined expone ntially decreasing tails. It is shown that the probability tails of su ch functionals are of the same order of magnitude as the tails of the same functionals with respect to the Levy measure, and it turns out th at the results of this kind cannot, in general, be improved. In certai n situations we can further obtain both lower and upper bounds on the asymptotic ratio of the two tails. In the second part of the paper we consider the particular case of Levy processes with exponentially deca ying Levy measures. Here we show that the tail of the maximum of the p rocess is, up to a multiplicative constant, asymptotic to the tail of the Levy measure. Most of the previously published work in the area co nsidered heavier than exponential probability tails.