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  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 , where the same question is studied for a Lévy process drifting to $-\infty $. The connection to  relies on the so-called Lamperti transformation  which links the class of positive self-similar Markov processes with that of Lévy processes. Our approach shows that the results in  for Bessel processes can also be seen as a consequence of self-similarity.
"Optimal prediction for positive self-similar Markov processes." Electron. J. Probab. 21 1 - 24, 2016. https://doi.org/10.1214/16-EJP4280