On drift parameter estimation for reflected fractional Ornstein–Uhlenbeck processes

Abstract

We consider a reflected Ornstein-Uhlenbeck process X driven by a fractional Brownian motion with Hurst parameter H ∈ (0, 1 2 ) ∪ ( 1 2 , 1). Our goal is to estimate an unknown drift parameter α ∈ (−∞, ∞) 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 α, and prove its strong consistency and asymptotic normality. As an improved estimator, we obtain the explicit formulas for the sequential 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.

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

“…On the other hand, some future work may investigate some other estimators for the other reflected diffusions. For example, Lee et al [26] proposed an SMLE of the unknown drift of the ROU process without jumps; the reflected jump-diffusion or Levy processes have been extensively investigated in the literature; see [1], [2], [4]- [6], [10]- [12], and [36]); other authors are concerned with the problem of statistical parameter estimation for reflected fractional Brownian motion; see [19] and [25]).…”

Asymptotic behavior of parametric estimation for a class of nonlinear diffusion process

“…See, for example, Lee et al [7] proposed a sequential maximum likelihood estimation of the unknown drift of the reflected Ornstein-Uhlenbeck process without jumps; some others are concerned with the problem of statistical parameter estimation for reflected fractional Brownian motion (cf. Hu and Lee [3]; Lee and Song [8]). …”

Piecewise-Tunneled Captive Processes and Corridored Random Particle Systems

“…Similar principles, concerning the evolution of stochastic diffusions in restrictive conditions, are applied to population dynamic models as well (see [50,51]). Furthermore, the reader is guided to [52,53] for other recent developments.…”