English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

\noindent \textbf{Abstract}: We consider the parameter estimation problem forthe Ornstein-Uhlenbeck process $X$ driven by a fractional Ornstein-Uhlenbeckprocess $V$, i.e. the pair of processes defined by the non-Markoviancontinuous-time long-memory dynamics $dX_{t}=-\theta X_{t}dt+dV_{t};\ t\geq 0$,with $dV_{t}=-\rho V_{t}dt+dB_{t}^{H};\ t\geq 0$, where $\theta >0$ and $\rho>0$ are unknown parameters, and $B^{H}$ is a fractional Brownian motion ofHurst index $H\in (\frac{1}{2},1)$. We study the strong consistency as well asthe asymptotic normality of the joint least squares estimator$(\hat{\theta}_{T},\widehat{\rho }% _{T}) $ of the pair $( \theta ,\rho) $,based either on continuous or discrete observations of $\{X_{s};\ s\in \lbrack0,T]\}$ as the horizon $T$ increases to +$\infty $. Both cases qualify formallyas partial-hbobservation questions since $V$ is unobserved. In the latter case,several discretization options are considered. Our proofs of asymptoticnormality based on discrete data, rely on increasingly strict restrictions onthe sampling frequency as one reduces the extent of sources of observation. Thestrategy for proving the asymptotic properties is to study the case ofcontinuous-time observations using the Malliavin calculus, and then to exploitthe fact that each discrete-data estimator can be considered as a perturbationof the continuous one in a mathematically precise way, despite the fact thatthe implementation of the discrete-time estimators is distant from thecontinuous estimator. In this sense, we contend that the continuous-timeestimator cannot be implemented in practice in any na\"ive way, and serves onlyas a mathematical tool in the study of the discrete-time estimators'asymptotics.

Parameter Estimation for a partially observed Ornstein-Uhlenbeck process with long-memory noise