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 dXt = µ(Xt)dt + σ(Xt)dWt (where W is a standard Brownian motion) we consider (ǫ −1/2 (X m X +ǫt − X)) t∈R as ǫ ↓ 0, where 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 ξ, where ( ξt)t≥0 and ( ξ−t ) t≥0 arise as independent Bessel-3 processes multiplied by −σ(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 X based on observations at equidistant times.