Reflected Ornstein Uhlenbeck Process Research Articles

Moderate Deviations for Drift Parameter Estimations in Reflected Ornstein–Uhlenbeck Process

In this paper, we study the asymptotic properties of drift parameter estimations in reflected Ornstein–Uhlenbeck process, and establish their moderate deviations in both cases with one-sided barrier and two-sided barriers. The main methods consist of regenerative process techniques and the strong Markov property, as well as moderate deviations for martingales.

Estimation of all parameters in the reflected Ornstein–Uhlenbeck process from discrete observations

Assuming that a reflected Ornstein–Uhlenbeck process is observed at discrete time instants, we propose generalized moment estimators to estimate all the drift and diffusion parameters via the celebrated ergodic theorem. With the sampling time step h>0 arbitrarily fixed, we prove the strong consistency and asymptotic normality of our estimators as the sampling size n tends to infinity. This provides a complete solution to an open problem left in Hu et al. (2015).

Convergence rate to equilibrium in Wasserstein distance for reflected jump–diffusions

Convergence rate to the stationary distribution for continuous-time Markov processes can be studied using Lyapunov functions. Recent work by the author provided explicit rates of convergence in special case of a reflected jump–diffusion on a half-line. These results are proved for total variation distance and its generalizations: measure distances defined by test functions regardless of their continuity. Here we prove similar results for Wasserstein distance, convergence in which is related to convergence for continuous test functions. In some cases, including the reflected Ornstein–Uhlenbeck process, we get faster exponential convergence rates for Wasserstein distance than for total variation distance.

Stationary distribution convergence of the offered waiting processes for $$GI/GI/1+GI$$ queues in heavy traffic

A result of Ward and Glynn (Queueing Syst 50(4):371–400, 2005) asserts that the sequence of scaled offered waiting time processes of the $$GI/GI/1+GI$$ queue converges weakly to a reflected Ornstein–Uhlenbeck process (ROU) in the positive real line, as the traffic intensity approaches one. As a consequence, the stationary distribution of a ROU process, which is a truncated normal, should approximate the scaled stationary distribution of the offered waiting time in a $$GI/GI/1+GI$$ queue; however, no such result has been proved. We prove the aforementioned convergence, and the convergence of the moments, in heavy traffic, thus resolving a question left open in 2005. In comparison with Kingman’s classical result (Kingman in Proc Camb Philos Soc 57:902–904, 1961) showing that an exponential distribution approximates the scaled stationary offered waiting time distribution in a GI / GI / 1 queue in heavy traffic, our result confirms that the addition of customer abandonment has a non-trivial effect on the queue’s stationary behavior.

On a class of reflected AR(1) processes

AbstractIn this paper we study a reflected AR(1) process, i.e. a process (Zn)n obeying the recursion Zn+1= max{aZn+Xn,0}, with (Xn)n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Zn (in terms of transforms) in case Xn can be written as Yn−Bn, with (Bn)n being a sequence of independent random variables which are all Exp(λ) distributed, and (Yn)n i.i.d.; when |a|<1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (Bn)n are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein–Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.

Asymptotic behaviour of parametric estimation for nonstationary reflected Ornstein–Uhlenbeck processes

Reflected Ornstein–Uhlenbeck process is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. This work aims at the study of asymptotic behaviour of parametric estimator for nonstationary reflected Ornstein–Uhlenbeck processes, including a limiting theorem, strong consistency, and asymptotic distribution. We focus on the further investigation of asymptotic distribution of parametric estimation for ergodic case.

A general lower bound of parameter estimation for reflected Ornstein–Uhlenbeck processes

AbstractA reflected Ornstein–Uhlenbeck process is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. It is an extended model of the traditional Ornstein–Uhlenbeck process being extensively used in finance as a one-factor short-term interest rate model. Under some mild conditions, this paper is devoted to the study of the analogue of the Cramer–Rao lower bound of a general class of parameter estimation of the unknown parameter in reflected Ornstein–Uhlenbeck processes.

On drift parameter estimation for reflected fractional Ornstein–Uhlenbeck processes

We consider a reflected Ornstein–Uhlenbeck process X driven by a fractional Brownian motion with Hurst parameter . Our goal is to estimate an unknown drift parameter on the basis of continuous observation of the state process. We establish Girsanov theorem for the process X, derive the standard maximum likelihood estimator of the drift parameter , and prove its strong consistency and asymptotic normality. As an improved estimator, we obtain the explicit formulas for the sequential standard maximum likelihood estimator and its mean squared error by assuming the process is observed until a certain information reaches a specified precision level. The estimator is shown to be unbiased, uniformly normally distributed, and efficient in the mean square error sense.

About the infinite dimensional skew and obliquely reflected Ornstein–Uhlenbeck process

Based on an integration by parts formula for closed and convex subsets [Formula: see text] of a separable real Hilbert space [Formula: see text] with respect to a Gaussian measure, we first construct and identify the infinite dimensional analogue of the obliquely reflected Ornstein–Uhlenbeck process (perturbed by a bounded drift [Formula: see text]) by means of a Skorokhod type decomposition. The variable oblique reflection at a reflection point of the boundary [Formula: see text] is uniquely described through a reflection angle and a direction in the tangent space (more precisely through an element of the orthogonal complement of the normal vector) at the reflection point. In case of normal reflection at the boundary of a regular convex set and under some monotonicity condition on [Formula: see text], we prove the existence and uniqueness of a strong solution to the corresponding SDE. Subsequently, we consider an increasing sequence [Formula: see text] of closed and convex subsets of [Formula: see text] and the skew reflection problem at the boundaries of this sequence. We present concrete examples and obtain as a special case the infinite dimensional analogue of the [Formula: see text]-skew reflected Ornstein–Uhlenbeck process.

Modeling the Exchange Rates in a Target Zone by Reflected Ornstein-Uhlenbeck Process

In this paper, we model the exchange rate in a target zone by a so-called reflected Ornstein-Uhlenbeck process. A simulation-based maximum likelihood estimation strategy of the parameters involved in the model is proposed and studied. The model fits data on exchange rates in the European Monetary System well.

On the reflected Ornstein–Uhlenbeck process with catastrophes

The reflected Ornstein–Uhlenbeck process in the presence of catastrophes that occur at exponential rate is considered. Explicit expressions for the transition probability density and for related moments are determined. The first visit time to zero state is also investigated. A systematic computational analysis is performed to elucidate the role played by the parameters.

Sequential maximum likelihood estimation for reflected Ornstein–Uhlenbeck processes

The paper studies the properties of a sequential maximum likelihood estimator of the drift parameter in a one dimensional reflected Ornstein–Uhlenbeck process. We observe the process until the observed Fisher information reaches a specified precision level. We derive the explicit formulas for the sequential estimator and its mean squared error. The estimator is shown to be unbiased and uniformly normally distributed. A simulation study is conducted to assess the performance of the estimator compared with the ordinary maximum likelihood estimator.

A note on transition density for the reflected Ornstein–Uhlenbeck process

This note focuses on the Ornstein–Uhlenbeck process reflected at its long-run level (or long-run mean). The analytical closed-form of the transition density is obtained by virtue of the Skorokhod equation and the time-change for martingales. Our result is consistent with that presented by Linetsky (2005). Finally, an open problem concerning the general cases (reflected at an arbitrary level) is proposed.

The Stationary Distributions of Two Classes of Reflected Ornstein–Uhlenbeck Processes

In this paper we consider two classes of reflected Ornstein–Uhlenbeck (OU) processes: the reflected OU process with jumps and the Markov-modulated reflected OU process. We prove that their stationary distributions exist. Furthermore, for the jump reflected OU process, the Laplace transform (LT) of the stationary distribution is given. As for the Markov-modulated reflected OU process, we derive an equation satisfied by the LT of the stationary distribution.

The Stationary Distributions of Two Classes of Reflected Ornstein–Uhlenbeck Processes

In this paper we consider two classes of reflected Ornstein–Uhlenbeck (OU) processes: the reflected OU process with jumps and the Markov-modulated reflected OU process. We prove that their stationary distributions exist. Furthermore, for the jump reflected OU process, the Laplace transform (LT) of the stationary distribution is given. As for the Markov-modulated reflected OU process, we derive an equation satisfied by the LT of the stationary distribution.

Stationary distribution of reflected O–U process with two-sided barriers

In this paper, we develop a one-sided reflected Ornstein–Uhlenbeck process proposed by Ward and Glynn [Ward, A., Glynn, P., 2003. A diffusion approximation for Markovian queue with reneging. Queueing Systems 43, 103–128] and explore the reflected process with two-sided barriers. Consequently the stationary distribution of the process is derived by Markovian infinitesimal generator argument.

A Diffusion Approximation for a Markovian Queue with Reneging

Consider a single-server queue with a Poisson arrival process and exponential processing times in which each customer independently reneges after an exponentially distributed amount of time. We establish that this system can be approximated by either a reflected Ornstein–Uhlenbeck process or a reflected affine diffusion when the arrival rate exceeds or is close to the processing rate and the reneging rate is close to 0. We further compare the quality of the steady-state distribution approximations suggested by each diffusion.

Properties of the Reflected Ornstein–Uhlenbeck Process

Consider an Ornstein–Uhlenbeck process with reflection at the origin. Such a process arises as an approximating process both for queueing systems with reneging or state-dependent balking and for multi-server loss models. Consequently, it becomes important to understand its basic properties. In this paper, we show that both the steady-state and transient behavior of the reflected Ornstein–Uhlenbeck process is reasonably tractable. Specifically, we (1) provide an approximation for its transient moments, (2) compute a perturbation expansion for its transition density, (3) give an approximation for the distribution of level crossing times, and (4) establish the growth rate of the maximum process.