An example of almost sure finite random time S such that $W_t=B_{S+t}-B_S$ is not a brownian motion

Question

Let $\{B_t,\mathcal{F}_t,t \geq 0\}$ be a standard one dimensional Brownian motion. Give an example of a random time $S$ with $P[0\leq S< \infty]=1$ such that with $W_t:=B_{S+t}-B_S$, the process $W=\{W_t, \mathcal{F}^w_t;t\geq0\}$ is not a brownian motion.

How could I go about constructing such an example. I have no idea of how to proceed. I tried to invent an almost sure finite random time using the law of large numbers for Brownian motion $T_{\epsilon}=\{\sup_{s \geq0}: B_s\geq s\epsilon\}$ but I am not sure on how to handle the object $B_{T_{\epsilon}+t}-B_{T_{\epsilon}}$.

Any help would be appreciated

Answer 1

Let $S = \inf \{s\ge 0:B_{s+1}-B_s\ge 0\}$. Since $W_1\ge0$ almost surely, $W_t$ cannot be a Brownian motion.