arXiv subjects

Youssef Ouknine

Publications and source records attributed to Youssef Ouknine.

42 records · Page 3Linked to original sources

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↗

arXiv2010

On Itô's formula for symmetric $α$-stable Lévy process of index $1<α\leq 2 $

We use Young integration (resp, bounded $p,q$-variation theory introduced in \cite{Feng-Zhao}) to establish integration of determinate functions with respect to local time of symmetric $α$-stable Lévy process, for $α\in ]1,2]$, in one parameter case (resp, in two parameter case). We then apply these integrals to write the corresponding generalized Itô formula. Furthermore, some approximations schemes of the area integral w.r.t local time are given.

math.PR↗

arXiv2010

Unicité trajectorielle des équations différentielles stochastiques avec temps local et temps de séjour au bord

English version of the abstract. We study path-wise uniqueness property of a class of stochastic differential equations with local time and sojourn time in the boundary. ----- French version of the abstract. Nous étudions l'unicité trajectorielle des solutions d'une classe d'équations différentielles stochastiques avec temps local et temps de séjour au bord. Nous utilisons le probléme des martingales associé pour montrer qu'il y a unicité en loi, puis nous établissons que le supremum de deux solutions est encore une solution.

math.PR↗

arXiv2010

On one-dimensional stochastic differential equations involving the maximum process

We prove existence and pathwise uniqueness results for four different types of stochastic differential equations (SDEs) perturbed by the past maximum process and/or the local time at zero. Along the first three studies, the coefficients are no longer Lipschitz. The first type is the equation \label{eq1} X_{t}=\int_{0}^{t}σ(s,X_{s})dW_{s}+\int_{0}^{t}b(s,X_{s})ds+α\max_{0\leq s\leq t}X_{s}. The second type is the equation \label{eq2} {l} X_{t} =\ig{0}{t}σ(s,X_{s})dW_{s}+\ig{0}{t}b(s,X_{s})ds+α\max_{0\leq s\leq t}X_{s}\,\,+L_{t}^{0}, X_{t} \geq 0, \forall t\geq 0. The third type is the equation \label{eq3} X_{t}=x+W_{t}+\int_{0}^{t}b(X_{s},\max_{0\leq u\leq s}X_{u})ds. We end the paper by establishing the existence of strong solution and pathwise uniqueness, under Lipschitz condition, for the SDE \label{e2} X_t=ξ+\int_0^t \si(s,X_s)dW_s +\int_0^t b(s,X_s)ds +\al\max_{0\leq s\leq t}X_s +\be \min_{0\leq s \leq t}X_s.

math.PR↗

arXiv2009

Estimation of the drift of fractional Brownian motion

We consider the problem of efficient estimation for the drift of fractional Brownian motion $B^H:=(B^H_t)_{t\in[0,T]}$ with hurst parameter $H$ less than 1/2. We also construct superefficient James-Stein type estimators which dominate, under the usual quadratic risk, the natural maximum likelihood estimator.

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↗