Retrieving information from subordination

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 $τ= (τ_{\ell}, \ell \geq 0)$ a subordinator, then the time-changed process $(Y_{τ_{\ell}}, \ell\geq 0)$ allows to retrieve the information about $(X_{τ_{\ell}}, \ell\geq 0)$ when $τ$ is stable, but not when $τ$ 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.

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