Time-changed filtration (${\mathcal{G}}_s={\mathcal{F}}_{{\tau }_s}$)

A time-changed filtration records the information available after a new clock has reached an old time. Let $({\mathcal{F}}_t)$ be a filtration and let $({\tau }_s)$ be a nondecreasing family of finite stopping times. DefineThe second expression is the stopping-time sigma-algebra. If $s\le u$ and $A\in {\mathcal{F}}_{{\tau }_s}$, thenThus ${\mathcal{G}}_s\subseteq {\mathcal{G}}_u$, as required for a filtration.

For the deterministic clock ${\tau }_s=2s$, the new information is ${\mathcal{F}}_{2s}$, not ${\mathcal{F}}_s$. A continuous adapted process sampled at these stopping times is adapted to the new filtration. Preservation of the local martingale property requires further hypotheses such as those in the optional time-change theorem; merely substituting arbitrary random times does not establish it.