Limit theorems for reflected Ornstein-Uhlenbeck processes

TL;DR

This paper analyzes the decay rates of probabilities and most likely paths for reflected Ornstein-Uhlenbeck processes with one or two boundaries, providing explicit results and CLTs for associated idleness and loss processes.

Contribution

It explicitly determines decay rates and most likely paths for reflected OU processes and derives CLTs for idleness and loss processes, advancing understanding of their probabilistic behavior.

Findings

Decay rates for reaching extreme levels are explicitly calculated.

Most likely paths for the processes are identified via large deviations.

Central limit theorems for idleness and loss processes are established.

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.