On the asymptotic behaviour of L\\u27evy processes, Part I: Subexponential and exponential processes

We study tail probabilities of suprema of L\\u27evy processes with subexponential or exponential marginal distributions over compact intervals. Several of the processes for which the asymptotics are studied here for the first time have recently become important to model financial time series. Hence our results should be important, for example, in the assessment of financial risk