I have difficulties with a very concret example in stochastic calculus.
Let $B$ and $W$ be two independent Brownian motions on a filtration $(F_t)_{t\geq0}$ and let be $\lambda = 1 + \exp(-B^2_1)$ a stopping time.
Compute $\mathbb{E}[B_\lambda]$, $\mathbb{E}[B_\lambda^2]$, $\mathbb{E}[W_\lambda]$ and $\mathbb{E}[W_\lambda^2]$.
I firsted started to write that $\lambda$ is a stopping time so $\mathbb{E}[W_\lambda]=\mathbb{E}[W_0]=0$ and $\mathbb{E}[W_\lambda^2] = \lambda$ by definition of the brownian motion but I have really no intuition when computing the expectation of the brownian motion B at time $\lambda$ because it depends on $B_1$ so it can't be the same result than for $W$.
$(W_t)_{t\geq0},(W_t^2-t)_{t\geq0},(B_t)_{t\geq0},(B_t^2-t)_{t\geq0}$ are martingales and $\lambda$ is a bounded stopping time, thus optional stopping theorem implies that $$\mathbb EW_\lambda = \mathbb EW_0 = 0$$ $$\mathbb EB_\lambda = \mathbb EB_0 = 0$$ $$\mathbb EW_\lambda^2 - \mathbb E\lambda = \mathbb EW_0^2 = 0$$ $$\mathbb EB_\lambda^2 -\mathbb E\lambda= \mathbb EB_0^2 = 0.$$
Thus you just need to evaluate $\mathbb E\lambda$.