Optimal prediction for positive self-similar Markov processes

TL;DR

This paper extends the prediction of extremal events in positive self-similar Markov processes by linking them to Levy processes via the Lamperti transformation, broadening previous diffusion-based results.

Contribution

It generalizes prior work on diffusion processes to the broader class of self-similar Markov processes using the Lamperti transformation.

Findings

Established a connection between self-similar Markov processes and Levy processes.

Extended extremal prediction results to a wider class of processes.

Showed that results for Bessel processes follow from self-similarity.

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 Glover et al. (2013) 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 Baurdoux and van Schaik (2013), where the same question is studied for a Levy process drifting to minus infinity. The connection to Baurdoux and van Schaik (2013) relies on the so-called Lamperti transformation which links the class of positive self-similar Markov processes with that of Levy processes. Our approach will reveal that the results in Glover et al. (2013) for Bessel processes can also be seen as a consequence of self-similarity.