Optimal prediction for positive self-similar Markov processes

Abstract

This paper addresses the question of predicting when a positive self-similar Markov process $X$ attains its pathwise global supremum or infimum before hitting zero for the first time (if it does at all). This problem has been studied in [9] under the assumption that $X$ is a positive transient diffusion. We extend their result to the class of positive self-similar Markov processes by establishing a link to [3], where the same question is studied for a Lévy process drifting to $-\infty $. The connection to [3] relies on the so-called Lamperti transformation [15] which links the class of positive self-similar Markov processes with that of Lévy processes. Our approach shows that the results in [9] for Bessel processes can also be seen as a consequence of self-similarity.