Local behavior of diffusions at the supremum

Abstract

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.