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 $\epsilon \downarrow 0$, 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 $({\hat{\xi }}_t{)}_{t\ge 0}$ and $({\hat{\xi }}_{-t}{)}_{t\ge 0}$ 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.

10 pages

References in corpus (2)

• Zooming-in on a Lévy process: Failure to observe threshold exceedance over a dense grid
• Optimal estimation of some random quantities of a Lévy process