Inequality between stopping times

Question

I have a problem with the following question:

Find an example of a stochastic process $X=(X_t)$ and a measurable subset $A$ of the state space such that the hitting times $\tau_0=\inf\{t\geq 0:X_t\in A\}$ and $\tau_1=\inf\{t>0:X_t\in A\}$ satisfy $$\tau_0<\tau_1.$$

So, my idea is to set $X_t=e^t$ and A=R, but i am not sure

Answer 1

Consider the set $A:=\{0\}$ and the deterministic process $X_t(\omega) := t$. Then

$$\tau_0 = \inf\{t \geq 0; X_t \in A\} = 0$$

but

$$\tau_1 = \inf\{t \geq 0; X_t \in A\} = \infty.$$