Subset and optional times

Question

The below is a well known fact but can anyone help me to prove it?

If I have a right continuous filtration and $\eta$ is an optional time, how can I show that if $\eta\leq t$ then $\mathcal{F}_\eta \subset \mathcal{F}_t$ ?

Answer 1

I recall definition of optional times to avoid any possible confusion as sometimes different authors use some variations around those notions.

So an optional time (Karatzas and Shreve) $\eta$ is a positive random variable over $(\Omega,\mathcal{F}=\vee_{t>0}\mathcal{F}_t)$ such that $\forall t>0$, $\{\eta<t $} $ \in\mathcal{F}_t$.

The $\sigma$-algebra $\mathcal{F}_\eta$ consists of events $A \in \mathcal{F}$ such that $\forall t>0,A\cap\{\eta<t$} $\in \mathcal{F}_t$

Now our problem, we know that the filtration is right continuous and impose that $\eta\leq t$ for a fixed $t>0$.

Let's give ourselves an event $A$ in $\mathcal{F}_\eta$ and let's look at it.

We know from the property of $\eta$ that $\{\eta\leq t $} $=\Omega$ so that $A\cap\{\eta\leq t $}$=A$.

Now we use the right continuity of the filtration, we have :

$A=A\cap\{\eta\leq t $} $=A\cap\cap_{u>t}\{\eta< u $} $=\cap_{u>t}A\cap\{\eta< u $} and it implies that $A=A\cap\{\eta\leq t $} $\in \cap_{u>t}\mathcal{F}_{u}=\mathcal{F}_{t+}=\mathcal{F}_t$ (by hypothesis) so we are done.

Best regards