= Brownian motion shifted at its finite-horizon maximum
{c}
{title2=$Y_t=B_{\tau+t}-B_\tau$}
Let $\tau$ be the first attainment of the maximum of standard <Brownian motion> on $[0,T]$, with $T>0$. Time reversal shows $\tau<T$ with probability one. Then $Y_t\leq0$ for $0\leq t\leq T-\tau$, an initial interval of positive length. Standard <Brownian motion> has probability zero of this property by the <Brownian reflection principle>, so $Y$ is not Brownian. Moreover $\tau$ is not a <stopping time> for the <natural Brownian filtration>: for $0<t<T$, $\mathbb P(\tau\leq t\mid\mathcal F_t)=2\Phi((M_t-B_t)/\sqrt{T-t})-1$, which is strictly between zero and one on $\{B_t<0\}$. The <Strong Markov property> is therefore inapplicable at this random time.
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