Asymptotic behaviours for the trajectory fitting estimator in Ornstein–Uhlenbeck process with linear drift

Abstract

We study the asymptotic behaviours for the trajectory fitting estimator in the OrnsteinUhlenbeck process with linear drift. Deviation inequality, moderate deviations, BerryEsseen bound and the law of iterated logarithm (LIL) of this estimator can be obtained. Moreover, as an application of the Berry-Esseen bound, we can get the precise rate in LIL. The main method of this paper is the deviation inequality for multiple Wiener-Itô integrals [P. Major, On a multivariate version of Bernsteins inequality. Electron. J. Prob. 12 (2007), pp. 966-988; P. Major, Tail behavior of multiple integrals and U-statistics. Probab. Surv. 2 (2005), pp. 448-505].

Modified trajectory fitting estimators for multi‐regime threshold Ornstein–Uhlenbeck processes

“…The trajectory fitting estimator (TFE) is proposed first by Kutoyants (1991) as a numerically attractive alternative to the maximum likelihood estimator. The TFEs have been widely studied in stochastic processes, such as traditional OU process (Dietz, 2001; Dietz & Kutoyants, 1997; Jiang & Chen, 2016; Kutoyants, 1991), the OU process driven by ‐stable Lévy motions (Hu & Long, 2007) and the reflected OU process (Zang & Zhang, 2019). Suppose that we have the prior information of the thresholds and diffusion parameters, that is, , , and the diffusion parameter is known.…”