If $\hat{t}$ is a stopping time, provide an example illustrating how $(\hat{t}-1)\wedge 0$ is indeed not a stopping time

Question

I know that a stopping time is when $\{\hat{t} ≤ t\} ∈ \mathcal{F}_t$ for every $t ∈ T$. But I'm trying to think of a counterexample to prove the above but cannot think of one! Any idea?

Answer 1

Let $\hat t$ be a stopping time , hence $\{\hat t\leq t\}\in \mathcal F_t$ for all $t\geq 0$, any constant defines a stopping time, but the difference between stopping times, is not in general a stopping time.

$$\{\hat{t}-1\leq t\}=\{\hat{t} \leq t+1\}$$ EDIT:

Let $f(t,\omega)$ be a measurable continuous function hence if you define $$\hat t:=\inf\{ t\geq 0: X(t,\omega)=\int_0^t f(s,\omega)^2ds\geq N\}$$

Since $\hat t$ is a stopping time we have that $\{\hat{t}-1\leq t\}\color{grey}{(=\{\hat{t} \leq t+1\})}=\{N\leq X(t+1)\}$ belongs to $\mathcal F_{t+1}$, but not necessarily to $\mathcal F_t$.