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In the book "Brownian Motion and Stochastic Calculus" by Karatzas and Shreve I encountered Problem 6.1 which the authors refer to as "not be hard to show":

Let $\{B_t, \cal{F}_t;t\geq 0\}$ 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 $W=\{W_t, \cal{F}^W_t;t\geq 0\}$ is not a Brownian motion.

Surely, if S being a stopping time $(W_t)_t$ would still be a Brownian motion. I tried out some distributions for $S$ but none of them gave a sufficient answer.

Most likely I am missing a crucial point here and appreciate any kind of help.

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marked as duplicate by Davide Giraudo probability-theory Nov 16 '18 at 15:23

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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.

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