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

A note on the central limit theorem for the idleness process in a one‐sided reflected Ornstein–Uhlenbeck model

“…Let us briefly summarize the result in HUANG et al (). In that paper, the object of study was a doubly reflected (at the lower bound zero and an upper bound d ) OU process Z , satisfying the SDE For a twice continuously differentiable function h on , by Itô's formula, we have On the basis of the key properties of L (e.g., Equation ) and U (which takes care of the reflection at the upper level d ), this reduces to …”

“…Indeed, with this choice of h , we have Boundedness of h ( Z t )( Z is a bounded process) combined with a central limit theorem for the martingale was central in the proof of the central limit theorem for U t ; see HUANG et al () for further details. In the present paper, we deal with the one‐sided reflected process Y and with a twice differential function h one obtains Two facts obstruct a direct application of the method earlier: (i) the process h ( Y t ) is not bounded, and (ii) we cannot immediately apply Lemma to obtain a proper choice of h .…”

“…Proof One easily verifies (like in the proof of the corresponding result in H UANG et al ., ) that the general solution is Then, the initial conditions uniquely determine the values of C 1 , C 2 . Indeed, we obtain C 2 =0 and C 1 =1, and so …”

“…As in HUANG et al (), we use Zhang and Glynn's martingale approach, as developed in ZHANG and GLYNN (), and further elaborated on in GLYNN and WANG () to establish the results. In section 2, we review a previous result from HUANG et al () for doubly reflected processes and explain why one has to modify this approach for the one‐sided reflected case, whereas in section 3, we show which modifications are needed to identify the central limit theorems. We also present results for reflected processes at lower boundaries other than zero and for processes reflected at an upper bound.…”

Continuous-time multi-type Ehrenfest model and related Ornstein-Uhlenbeck diffusion on a star graph

“…> α N +1 are the roots of the polynomial P , due to Lemma 3.1, and where β k is defined in (27). Finally, after straightforward calculations one obtains the expression (25).…”