Retrieving information from subordination

TL;DR

The paper investigates how time-changing processes with subordinators affects the retrieval of original process information, highlighting differences between stable and gamma subordinators, motivated by identities involving Bessel clocks and inverse Gaussian variables.

Contribution

It demonstrates that stable subordinators enable retrieval of original process information after time change, unlike gamma subordinators, revealing a key distinction in stochastic process transformations.

Findings

Stable subordinators preserve information about the original process.

Gamma subordinators do not allow such information retrieval.

The results are motivated by identities involving Bessel clocks and inverse Gaussian variables.

Abstract

We show that if $(X_s, s\geq 0)$ is a right-continuous process, $Y_t=\int_0^t\d s X_s$ its integral process and $\tau = (\tau_{\ell}, \ell \geq 0)$ a subordinator, then the time-changed process $(Y_{\tau_{\ell}}, \ell\geq 0)$ allows to retrieve the information about $(X_{\tau_{\ell}}, \ell\geq 0)$ when $\tau$ is stable, but not when $\tau$ is a gamma subordinator. This question has been motivated by a striking identity in law involving the Bessel clock taken at an independent inverse Gaussian variable.