= Time-changed filtration
{title2=$\mathcal G_s=\mathcal F_{\tau_s}$}
A <time-changed filtration> records the information available after a new clock has reached an old time. Let $(\mathcal F_t)$ be a <filtration (probability theory)> and let $(\tau_s)$ be a nondecreasing family of finite <stopping times>. Define
$$
\mathcal G_s=\mathcal F_{\tau_s},\qquad
\mathcal F_\tau=\{A\in\mathcal F:A\cap\{\tau\leq t\}\in\mathcal F_t\text{ for every }t\geq0\}.
$$
The second expression is the <stopping-time sigma-algebra>. If $s\leq u$ and $A\in\mathcal F_{\tau_s}$, then
$$
A\cap\{\tau_u\leq t\}=A\cap\{\tau_s\leq t\}\cap\{\tau_u\leq t\}\in\mathcal F_t.
$$
Thus $\mathcal G_s\subseteq\mathcal G_u$, as required for a <filtration (probability theory)>.
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 (probability theory)>. 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.
Back to article page
