= Solution
For a <stopping time> $T$ of a <Brownian filtration>, define the reflected <stochastic process>
$$
\boxed{\widehat B_t=\begin{cases}B_t,&t\leq T,\\2B_T-B_t,&t>T.\end{cases}}
$$
The <Brownian reflection at a stopping time> form of the <Brownian reflection principle> states that \b[$\widehat B$ is again standard <Brownian motion>]. If $T=\infty$, leave the path unchanged; the second branch is used only when $T<t$.
For an <almost surely> finite $T$, the <Strong Markov property> says that $(B_{T+s}-B_T)_{s\geq0}$ is a fresh <Brownian motion> <independence>[independent] of $\mathcal F_T$. Its negative has the same <probability distribution>, so reflecting the future preserves the full path <probability distribution>. Allowing $T=\infty$ follows by applying this argument to $T\wedge n$ and restricting to each fixed finite time interval.
Back to article page
