Limit theorems for reflected Ornstein-Uhlenbeck processes

Abstract

This paper studies one-dimensional Ornstein-Uhlenbeck 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' $L_t$ and the `loss process' $U_t$, which are the minimal nondecreasing processes which make the OU process remain $\geqslant 0$ and $\leqslant d$, respectively. We derive central limit theorems for $U_t$ and $L_t$, using techniques from stochastic integration and the martingale central limit theorem.