Searcharxiv⌕ Search

arXiv subjects

Khalifa Es-Sebaiy

Publications and source records attributed to Khalifa Es-Sebaiy.

31 records · Page 2Linked to original sources
arXiv2016

Least squares estimator for non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian processes

The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein-Uhlenbeck process defined as $dX_t=θX_tdt+dG_t,\ t\geq0$ with an unknown parameter $θ>0$, where $G$ is a Gaussian process. We provide sufficient conditions, based on the properties of $G$, ensuring the strong consistency and the asymptotic distribution of our estimator $\widetildeθ_t$ of $θ$ based on the observation $\{X_s,\ s\in[0,t]\}$ as $t\rightarrow\infty$. Our approach offers an elementary, unifying proof of \cite{BEO}, and it allows to extend the result of \cite{BEO} to the case when $G$ is a fractional Brownian motion with Hurst parameter $H\in(0,1)$. We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.

math.PR↗
arXiv2016

Least squares estimator of fractional Ornstein Uhlenbeck processes with periodic mean

We first study the drift parameter estimation of the fractional Ornstein-Uhlenbeck process (fOU) with periodic mean for every $\frac{1}{2}<H<1$. More precisely, we extend the consistency proved in \cite{DFW} for $\frac{1}{2}<H<\frac{3}{4}$ to the strong consistency for any $\frac{1}{2}<H<1$ on the one hand, and on the other, we also discuss the asymptotic normality given in \cite{DFW}. In the second main part of the paper, we study the strong consistency and the asymptotic normality of the fOU of the second kind with periodic mean for any $\frac{1}{2}<H<1$.

math.PR↗
arXiv2016

Optimal rates for parameter estimation of stationary Gaussian processes

We study rates of convergence in central limit theorems for partial sum of functionals of general stationary and non-stationary Gaussian sequences, using optimal tools from analysis on Wiener space. We apply our result to study drift parameter estimation problems for some stochastic differential equations driven by fractional Brownian motion with fixed-time-step observations.

math.ST↗
arXiv2015

Parameter estimation for SDEs related to stationary Gaussian processes

In this paper, we study central and non-central limit theorems for partial sum of functionals of general stationary Gaussian fields. We apply our result to study drift parameter estimation problems for some stochastic differential equations related to stationary Gaussian processes.

math.PR↗
arXiv2013

Parameter estimation for alpha-fractional bridges

Let alpha,T>0. We study the asymptotic properties of a least squares estimator for the parameter alpha of a fractional bridge defined as dX_t=-alpha*X_t/(T-t)dt+dB_t, with t in [0,T) and where B is a fractional Brownian motion of Hurst index H>1/2. Depending on the value of alpha, we prove that we may have strong consistency or not as t tends to T. When we have consistency, we obtain the rate of this convergence as well. Also, we compare our results to the (known) case where B is replaced by a standard Brownian motion W.

math.PR↗
arXiv2013

Berry-Esséen bounds for the least squares estimator for discretely observed fractional Ornstein-Uhlenbeck processes

Let $θ>0$. We consider a one-dimensional fractional Ornstein-Uhlenbeck process defined as $dX_t= -θ X_t dt+dB_t,\quad t\geq0,$ where $B$ is a fractional Brownian motion of Hurst parameter $H\in(1/2,1)$. We are interested in the problem of estimating the unknown parameter $θ$. For that purpose, we dispose of a discretized trajectory, observed at $n$ equidistant times $t_i=iΔ_{n}, i=0,...,n$, and $T_n=nΔ_{n}$ denotes the length of the `observation window'. We assume that $Δ_{n} \rightarrow 0$ and $T_n\rightarrow \infty$ as $n\rightarrow \infty$. As an estimator of $θ$ we choose the least squares estimator (LSE) $\hatθ_{n}$. The consistency of this estimator is established. Explicit bounds for the Kolmogorov distance, in the case when $H\in(1/2,3/4)$, in the central limit theorem for the LSE $\hatθ_{n}$ are obtained. These results hold without any kind of ergodicity on the process $X$.

math.PR↗
arXiv2012

Almost sure central limit theorems for random ratios and applications to LSE for fractional Ornstein-Uhlenbeck processes

We investigate an almost sure limit theorem (ASCLT) for sequences of random variables having the form of a ratio of two terms such that the numerator satisfies the ASCLT and the denominator is a positive term which converges almost surely to 1. This result leads to the ASCLT for least square estimators for Ornstein-Uhlenbeck process driven by fractional Brownian motion.

math.PR↗
arXiv2012

Berry-Esséen bounds and almost sure CLT for the quadratic variation of the bifractional Brownian motion

Let $B$ be a bifractional Brownian motion with parameters $H\in (0, 1)$ and $K\in(0,1]$. For any $n\geq1$, set $Z_n =\sum_{i=0}^{n-1}\big[n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\E((B_{i+1}-B_{i})^2)\big]$. We use the Malliavin calculus and the so-called Stein's method on Wiener chaos introduced by Nourdin and Peccati \cite{NP09} to derive, in the case when $0<HK\leq3/4$, Berry-Esséen-type bounds for the Kolmogorov distance between the law of the correct renormalization $V_n$ of $Z_n$ and the standard normal law. Finally, we study almost sure central limit theorems for the sequence $V_n$.

math.PR↗
arXiv2011

An extension of bifractional Brownian motion

In this paper we introduce and study a self-similar Gaussian process that is the bifractional Brownian motion $B^{H,K}$ with parameters $H\in (0,1)$ and $K\in(1,2)$ such that $HK\in(0,1)$. A remarkable difference between the case $K\in(0,1)$ and our situation is that this process is a semimartingale when $2HK=1$.

math.PR↗
arXiv2011

Parameter Estimation for Fractional Ornstein-Uhlenbeck Processes: Non-ergodic Case

We consider the parameter estimation problem for the non-ergodic fractional Ornstein-Uhlenbeck process defined as $dX_t=θX_tdt+dB_t,\ t\geq0$, with a parameter $θ>0$, where $B$ is a fractional Brownian motion of Hurst index $H\in(1/2,1)$. We study the consistency and the asymptotic distributions of the least squares estimator $\hatθ_t$ of $θ$ based on the observation $\{X_s,\ s\in[0,t]\}$ as $t\rightarrow\infty$.

math.PR↗
arXiv2009

Approximation of the finite dimensional distributions of multiple fractional integrals

We construct a family $I_{n_{\eps}}(f)_{t}$ of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener-Itô integral $I_{n}^{H}(f1^{\otimes n}_{[0,t]})$ with respect to the fractional Brownian motion. We assume that $H>{1/2}$ and we prove our approximation result for the integrands $f$ in a rather general class.

math.PR↗
arXiv2008

Occupation densities for certain processes related to fractional Brownian motion

In this paper we establish the existence of a square integrable occupation density for two classes of stochastic processes. First we consider a Gaussian process with an absolutely continuous random drift, and secondly we handle the case of a (Skorohod) integral with respect to the fractional Brownian motion with Hurst parameter $H>\frac 12$. The proof of these results uses a general criterion for the existence of a square integrable local time, which is based on the techniques of Malliavin calculus.

math.PR↗