If $\tau$ is a stopping time wrt $\mathcal{F}=\{\mathcal{F}_t : t \geq 0\}$ and $\alpha \in \mathbb{R}$, then is $\alpha \tau$ also a stopping time?
I know that if $\alpha \geq 1$, then $\mathcal{F_{t / \alpha}} \subseteq \mathcal{F_{t}}$ and it follows that $$\alpha \tau \text{ is a stopping time wrt } \mathcal{F}$$ and if $\alpha < 1$, then $$\alpha \tau \text{ is a stopping time wrt } \{\mathcal{F_{t / \alpha}}\}$$ Question: Is $\alpha \tau$ a stopping time wrt $\mathcal{F}$ for any $\alpha$?