Delayed and rushed motions through time change

TL;DR

This paper explores how different types of time changes, specifically subordinators and their inverses, affect the delay or rush of base stochastic processes, revealing surprising non-implications.

Contribution

It introduces a new definition of delayed and rushed processes based on lifetimes and analyzes the effects of subordinators and inverse subordinators on these properties.

Findings

Inverse subordinators do not necessarily delay the base process.

Subordinators do not necessarily rush the base process.

The relationship between time change and process delay or rush is more complex than expected.

Abstract

We introduce a definition of delayed and rushed processes in terms of lifetimes of base processes and time-changed base processes. Then, we consider time changes given by subordinators and their inverse processes. Our analysis shows that, quite surprisingly, time-changing with inverse subordinators does not necessarily imply delay of the base process. Moreover, time-changing with subordinators does not necessarily imply rushed base process.