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

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.

• Usually, you cannot tell at time $\theta$ whether you'll come back to $B$ in the future, so you cannot expect $\theta$ to be a stopping time. What kind of specific example are you looking for and what do you expect to learn from it? – Mars Plastic Feb 28 at 12:38
• That makes sense to me, for it would not be measurable in 𝑋𝑘. However, I'm failing to see an example where this would happen. – Felipe Bpm Feb 28 at 12:40

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 $$(2)$$.