= Solution
{c}
<Lévy's characterization of Brownian motion> states that a real continuous <local martingale> $M$ with $M_0=0$ and <quadratic variation> $\langle M\rangle_t=t$ is a standard <Brownian motion> with respect to its <filtration (probability theory)>. Equivalently, its increments are independent of the past and have centered <normal distribution> with variances equal to elapsed times.
Let $H_s=\operatorname{sgn}(\sup_{r\leq s}B_r-1)$. The running maximum is continuous and adapted, so $H$ is bounded and <progressively measurable process>. It has a predictable version for integration; changing its value at the single switching time does not change its <stochastic integral>. Thus $X=\int H\,dB$ is a continuous <local martingale> starting at zero. Since $H_s^2=1$,
$$
\langle X\rangle_t=\int_0^tH_s^2ds=t.
$$
<Lévy's characterization of Brownian motion> yields \b[$X$ is standard <Brownian motion>]. The sign depends on the entire past through the running maximum, but the unit quadratic-variation rate is exactly what the characterization needs.
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