  This paper studies small-time behavior at the supremum of a diffusion
process. For a solution to the SDE $\mathrm{d} X_t=\mu(X_t)\mathrm{d}
t+\sigma(X_t)\mathrm{d} W_t$ (where $W$ is a standard Brownian motion) we
consider $(\epsilon^{-1/2}(X_{m^X+\epsilon t}-\overline{X}))_{t\in\mathbb{R}}$
as $\epsilon\downarrow0$, where $\overline{X}$ is the supremum of $X$ on the
time interval $[0,1]$ and $m^X$ is the time of the supremum. It is shown that
this process converges in law to a process $\hat{\xi}$, where
$(\hat\xi_t)_{t\geq0}$ and $(\hat\xi_{-t})_{t\geq0}$ arise as independent
Bessel-3 processes multiplied by $-\sigma(\overline{X})$. The proof is based on
the fact that a continuous local martingale can be represented as a
time-changed Brownian motion. This representation is also used to prove a limit
theorem for zooming in on $X$ at a fixed time. As an application of the
zooming-in result at the supremum we consider estimation of the supremum
$\overline{X}$ based on observations at equidistant times.
