Limit theorems for reflected Ornstein–Uhlenbeck processes

Abstract

This paper studies one‐dimensional Ornstein–Uhlenbeck (OU) processes, with the distinguishing feature that they are reflected on a single boundary (put at level 0) or two boundaries (put at levels 0 and d > 0). In the literature, they are referred to as reflected OU (ROU) and doubly reflected OU (DROU), respectively. For both cases, we explicitly determine the decay rates of the (transient) probability to reach a given extreme level. The methodology relies on sample‐path large deviations, so that we also identify the associated most likely paths. For DROU, we also consider the ‘idleness process’ Lt and the ‘loss process’ Ut, which are the minimal non‐decreasing processes, which make the OU process remain ≥ 0 and ≤ d, respectively. We derive central limit theorems (CLTs) for Ut and Lt, using techniques from stochastic integration and the martingale CLT.