Retrieving Information from Subordination

Abstract

We recall some instances of the recovery problem of a signal process hidden in an observation process. Our main focus is then to show that if $(X_{s},\,s\,\geq \,0)$ is a right-continuous process, $Y _{t} = \int \limits _{0}^{t}X_{s}\mathrm{d}s$ its integral process and $\tau = (\tau _{u},u \geq 0)$ a subordinator, then the time-changed process $(Y _{\tau _{u}},\,u\,\geq \,0)$ allows to retrieve the information about $(X_{\tau _{v}},\,v\,\geq \,0)$ 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.

Mathematics Subject Classification (2010): 60G35, 60G51