Stopping times algebra

Question

Let $W$ be a Brownian motion and let two stopping times: $\sigma_1, \sigma_2$. Then is the following a stopping time with respect to its filtration and why? $$τ=\sigma_1-\sigma_2$$ for $b∈\mathbb{R}$.

Answer 1

I will just ignore $b$, but in general, the difference of two stopping times is not a stopping time.

Let $\sigma_{1}=\inf_{t>0}\{|W_{t}|=1\}$ and $\sigma_{2}=\inf_{t>0}\{|W_{t}|=2\}$.

$\sigma_{1}-\sigma_{2}$ is always negative, which doesn't make any sense.

And even if we take $\sigma':= \sigma_{2}-\sigma_{1}$, then $\{\sigma'\le 1\}= \{\sigma_{2}\le\sigma_{1}+1\}$ is not in $F_{1}$. Can you figure out why?