Example of last hitting time failing to be a stopping time?

Question

I'm trying to find an example where the last hitting time $\theta = \sup\{k \ge 0 : X_k \in B\}$ of a set $B$ by the stochastic process $(X_k)_{k\ge 0}$ is not a stopping time.

Answer 1

Define a stochastic process $(X_k)_{k \geq 0}$ by

$$X_0 := 0 \qquad X_1 := Y, \quad X_k := 1, \quad k \geq 2,$$

for a random variable $Y$ such that $\mathbb{P}(Y=1)=\mathbb{P}(Y=0)=1/2$. The associated canonical $\sigma$-algebra is given by $$\mathcal{F}_0 = \{\emptyset,\Omega\}, \qquad \qquad \mathcal{F}_k = \sigma(Y), \quad k \geq 1. \tag{2}$$ If we consider $$\tau := \sup\{k \geq 0; X_k = 0\}$$ then $\tau$ is not a stopping time. Indeed: Since $$\{\tau=0\} = \{Y=1\}$$ we have $\{\tau=0\} \notin \mathcal{F}_0$ because of $(2)$.