Are $1+\exp(-B_{1}^2)$ and $\inf\{t \geq 0 : B_t \geq W_t + \exp(-t)\}$ stopping times?

Question

I have got difficulties with an exercise on stochastic processes.

Let $B$ and $W$ be two independent Brownian motions on filtration $(\mathcal{F}_t)_{t\geq 0}$

Are $\lambda$ = $1+\exp(-B_{1}^2)$ and $\tau$ = $\inf\{t \geq 0 : B_t \geq W_t + \exp(-t)\}$ stopping times ?

For the first one it seems to me that this is a constant so a constant is a stopping time right ? The second implies two brownian motion so I have no idea where to start.

Answer 1

$\lambda$ is a stopping time: $\{\lambda\le t\}$ is empty if $t\le 2$; if $t>2$ then $\{\lambda\le t\} =\{B_1^2\ge -\log(t-1)\}\in\mathcal F_1\subset\mathcal F_t$.