Let's consider a probability space $(\Omega,\mathcal A,P)$ and a filtration for this space $\Bbb F:=(\mathcal F_t)_{t\ge0}$.
Let's consider a Brownian Motion $B=(B_t)_{t\ge0}$ and the stopping time defined by $\tau_a:=\inf\{s\ge0\;:\;B_s=a\}$ where $a>0$, both with respect to $\Bbb F$.
Let's then set $\mathcal G_t:=\mathcal F_{t+\tau_a}$, where $\mathcal G_{t+\tau_a}:=\{A\in\mathcal A\;:\; A\cap\{t+\tau_a\le s\}\in\mathcal F_s\;\;\forall s\ge0\}$ ; thus $\Bbb G:=(\mathcal G_t)_{t\ge0}$ is another filtration. : Now it is well known (by a theorem of Markov) that $Z_t:=B_{t+\tau_a}-B_{\tau_a}$ is a Brownian Motion with respect to $\Bbb G$.
Now I would like to prove or disprove the independence of $B$ and $Z$, but I don't know how to proceed.
Can someone give me an hint please?
Many thanks.