Moments of passage times and asymptotic behavior of increasing self-similar markov processes

By using Lamperti's bijection between self-similar Markov processes and Lévy processes, we prove finiteness of moments and asymptotic behavior of passage times for increasing self-similar Markov processes valued in (0, ∞). We also investigate the behavior of the process when it crosses a level. A limit theorem concerning the distribution of the process immediately before it crosses some level is proved. Some useful examples are given.