scholarly journals Berry–Esséen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes

2019 ◽  
Vol 20 (04) ◽  
pp. 2050023 ◽  
Author(s):  
Yong Chen ◽  
Nenghui Kuang ◽  
Ying Li
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Continuous Time ◽  
Index Formula ◽  
Hurst Index ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Time Observation ◽  
Statistical Estimations ◽  
Drift Parameter

For an Ornstein–Uhlenbeck process driven by fractional Brownian motion with Hurst index [Formula: see text], we show the Berry–Esséen bound of the least squares estimator of the drift parameter based on the continuous-time observation. We use an approach based on Malliavin calculus given by Kim and Park [Optimal Berry–Esséen bound for statistical estimations and its application to SPDE, J. Multivariate Anal. 155 (2017) 284–304].

scholarly journals A Special Study of the Mixed Weighted Fractional Brownian Motion

2021 ◽  
Vol 5 (4) ◽  
pp. 192
Author(s):  
Anas D. Khalaf ◽  
Anwar Zeb ◽  
Tareq Saeed ◽  
Mahmoud Abouagwa ◽  
Salih Djilali ◽  
...  
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Estimation Problem ◽  
Special Study ◽  
Parameter Estimation Problem ◽  
Uhlenbeck Process ◽  
Sample Paths ◽  
Ornstein Uhlenbeck Process ◽  
Drift Parameter

In this work, we present the analysis of a mixed weighted fractional Brownian motion, defined by ηt:=Bt+ξt, where B is a Brownian motion and ξ is an independent weighted fractional Brownian motion. We also consider the parameter estimation problem for the drift parameter θ>0 in the mixed weighted fractional Ornstein–Uhlenbeck model of the form X0=0;Xt=θXtdt+dηt. Moreover, a simulation is given of sample paths of the mixed weighted fractional Ornstein–Uhlenbeck process.

scholarly journals Least squares estimation for non-ergodic weighted fractional Ornstein-Uhlenbeck process of general parameters

2021 ◽  
Vol 6 (11) ◽  
pp. 12780-12794
Author(s):  
Abdulaziz Alsenafi ◽  
◽  
Mishari Al-Foraih ◽  
Khalifa Es-Sebaiy
Keyword(s):  
Least Squares ◽  
Continuous Time ◽  
Strong Consistency ◽  
Least Square ◽  
Uhlenbeck Process ◽  
Type Method ◽  
Ornstein Uhlenbeck Process ◽  
Discrete Time Observations ◽  
Theoretical Results ◽  
Drift Parameter

<abstract><p>Let $ B^{a, b}: = \{B_t^{a, b}, t\geq0\} $ be a weighted fractional Brownian motion of parameters $ a &gt; -1 $, $ |b| &lt; 1 $, $ |b| &lt; a+1 $. We consider a least square-type method to estimate the drift parameter $ \theta &gt; 0 $ of the weighted fractional Ornstein-Uhlenbeck process $ X: = \{X_t, t\geq0\} $ defined by $ X_0 = 0; \ dX_t = \theta X_tdt+dB_t^{a, b} $. In this work, we provide least squares-type estimators for $ \theta $ based continuous-time and discrete-time observations of $ X $. The strong consistency and the asymptotic behavior in distribution of the estimators are studied for all $ (a, b) $ such that $ a &gt; -1 $, $ |b| &lt; 1 $, $ |b| &lt; a+1 $. Here we extend the results of <sup>[<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>]</sup> (resp. <sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup>), where the strong consistency and the asymptotic distribution of the estimators are proved for $ -\frac12 &lt; a &lt; 0 $, $ -a &lt; b &lt; a+1 $ (resp. $ -1 &lt; a &lt; 0 $, $ -a &lt; b &lt; a+1 $). Simulations are performed to illustrate the theoretical results.</p></abstract>

scholarly journals Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 716 ◽  
Author(s):  
Pavel Kříž ◽  
Leszek Szała
Keyword(s):  
Brownian Motion ◽  
Least Squares ◽  
Fractional Brownian Motion ◽  
Numerical Experiments ◽  
Covariance Structure ◽  
Mesh Size ◽  
Uhlenbeck Process ◽  
Least Squares Estimators ◽  
Discrete Time Observations ◽  
Drift Parameter

We introduce three new estimators of the drift parameter of a fractional Ornstein–Uhlenbeck process. These estimators are based on modifications of the least-squares procedure utilizing the explicit formula for the process and covariance structure of a fractional Brownian motion. We demonstrate their advantageous properties in the setting of discrete-time observations with fixed mesh size, where they outperform the existing estimators. Numerical experiments by Monte Carlo simulations are conducted to confirm and illustrate theoretical findings. New estimation techniques can improve calibration of models in the form of linear stochastic differential equations driven by a fractional Brownian motion, which are used in diverse fields such as biology, neuroscience, finance and many others.

scholarly journals On conditional Ornstein–Uhlenbeck processes

1984 ◽  
Vol 16 (04) ◽  
pp. 920-922
Author(s):  
P. Salminen
Keyword(s):  
Brownian Motion ◽  
Three Dimensional ◽  
Bessel Process ◽  
Uhlenbeck Process ◽  
Brownian Motions ◽  
Ornstein Uhlenbeck Process ◽  
The Law ◽  
Similar Description

It is well known that the law of a Brownian motion started from x &gt; 0 and conditioned never to hit 0 is identical with the law of a three-dimensional Bessel process started from x. Here we show that a similar description is valid for all linear Ornstein–Uhlenbeck Brownian motions. Further, using the same techniques, it is seen that we may construct a non-stationary Ornstein–Uhlenbeck process from a stationary one.

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