Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes

Abstract

Let (Xt(δ), t ≥ 0) be the BESQ δ process starting at δx. We are interested in large deviations as δ → ∞ for the family {δ-1 X t(δ), t ≤ T}δ - or, more generally, for the family of squared radial OUδ process. The main properties of this family allow us to develop three different approaches: an exponential martingale method, a Cramér-type theorem, thanks to a remarkable additivity property, and a Wentzell-Freidlin method, with the help of McKean results on the controlled equation. We also derive large deviations for Bessel bridges.