Suppose $\{W(t) \mid t \geqslant 0\}$ is a Wiener process with $W(t) \sim N(0, σ^2 t)$ and $a > 0$ a constant. Define$$ Z(t) = \begin{cases} W(t); & \sup\limits_{0 \leqslant s \leqslant t} W(s) < a\\ a; & \text{otherwise} \end{cases}. $$ Prove that $\{Z(t) \mid t \geqslant 0\}$ is a Markov process.
Basically the definition of $\{Z(t) \mid t \geqslant 0\}$ means that once $W(t)$ hits the point $a$, all $Z(t)$ from this instant on becomes $a$, namely "attracted" by this point.
The hint is to prove that $\{Z(t) \mid t \geqslant 0\}$ has independent increments. I know that $\{W(t) \mid t \geqslant 0\}$ has independent increments, but in the definition of $\{Z(t) \mid t \geqslant 0\}$, it seems that each $Z(t)$ is "related" to all $W(s)$ before $t$, thus all $Z(s)$ before $t$. So how to prove the independence of the increments of $\{Z(t) \mid t \geqslant 0\}$?