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Rachid Belfadli

Publications and source records attributed to Rachid Belfadli.

11 recordsLinked to original sources
arXiv2026

On the Besov-Orlicz path regularity of some Gaussian processes

In this paper, we rely on the additive decomposition in law satisfied by a class of stochastic processes, combined with the well-known regulariy properties of fractional Brownian motion, to establish Besov-Orlicz regularity of their sample paths. This provides a unified and direct proof for a broad class of processes, including bifractional Brownian motion with parameters $H\in (0, 1]$, $ K\in (0, 2)$ such that $HK \in (0, 1)$, subfractional Brownian motion with Hurst parameter $H\in (0, 1)$, and certain class of self-similar processes. %associated with the stochastic heat equation.

math.PR↗
arXiv2026

Regularization of Hyperbolic Stochastic Partial Differential Equations By Two Fractional Brownian Sheets

In this paper, we establish existence and uniqueness of strong solutions for a stochastic differential equation driven by an additive noise given by the sum of two correlated fractional Brownian sheets with different Hurst parameters. Our analysis relies on techniques from two-parameter fractional calculus and a tailored version of Girsanov's theorem. The main challenge arises from the correlation between the two noises and the technical requirements for applying Girsanov's theorem in this setting. We show that, despite these difficulties, the additive noise regularizes the equation, allowing well-posedness under weak assumptions on the drift.

math.PR↗
arXiv2025

On Malliavin differentiability and absolute continuity of one-dimensional doubly perturbed diffusion processes

In this paper, we establish Malliavin differentiability and absolute continuity for $α, β$-doubly perturbed diffusion process with parameters $α<1$ and $β<1$ such that $|ρ| < 1$, where $ ρ: = \frac{αβ}{(1-α)(1-β)}$. Furthermore, under some regularity conditions on the coefficients, we prove that the solution $X_t$ has a smooth density for all $t\in(0, t_0)$ for some finite number $t_0>0$. Our results recover earlier works by Yue and Zhang (2015) and Xue, Yue and Zhang (2016), and the proofs are based on the techniques of the Malliavin calculus.

math.PR↗
arXiv2020

Statistical analysis of the non-ergodic fractional Ornstein-Uhlenbeck process with periodic mean

Consider a periodic, mean-reverting Ornstein-Uhlenbeck process $X=\{X_t,t\geq0\}$ of the form $d X_{t}=\left(L(t)+αX_{t}\right) d t+ dB^H_{t}, \quad t \geq 0$, where $L(t)=\sum_{i=1}^{p}μ_iϕ_i (t)$ is a periodic parametric function, and $\{B^H_t,t\geq0\}$ is a fractional Brownian motion of Hurst parameter $\frac12\leq H<1$. In the "ergodic" case $α<0$, the parametric estimation of $(μ_1,\ldots,μ_p,α)$ based on continuous-time observation of $X$ has been considered in Dehling et al. \cite{DFK}, and in Dehling et al. \cite{DFW} for $H=\frac12$, and $\frac12 0$, and for all $\frac12\leq H<1$. We analyze the strong consistency and the asymptotic distribution for the estimator of $(μ_1,\ldots,μ_p,α)$ when the whole trajectory of $X$ is observed.

math.PR↗
arXiv2020

Berry-Esséen bound for drift estimation of fractional Ornstein Uhlenbeck process of second kind

In the present paper we consider the Ornstein-Uhlenbeck process of the second kind defined as solution to the equation $dX_{t} = -αX_{t}dt+dY_{t}^{(1)}, \ \ X_{0}=0$, where $Y_{t}^{(1)}:=\int_{0}^{t}e^{-s}dB^H_{a_{s}}$ with $a_{t}=He^{\frac{t}{H}}$, and $B^H$ is a fractional Brownian motion with Hurst parameter $H\in(\frac12,1)$, whereas $α>0$ is unknown parameter to be estimated. We obtain the upper bound $O(1/\sqrt{T})$ in Kolmogorov distance for normal approximation of the least squares estimator of the drift parameter $α$ on the basis of the continuous observation $\{X_t,t\in[0,T]\}$, as $T\rightarrow\infty$. Our method is based on the work of \cite{kp-JVA}, which is proved using a combination of Malliavin calculus and Stein's method for normal approximation.

math.PR↗
arXiv2019

On Skorokhod Problem with Two RCLL Reflecting Completely Separated Barriers

In this paper we deal with Skorokhod problem for right continuous left limited (rcll) barriers. We prove existence and uniqueness of the solution when the barriers are only supposed to be rcll and completely separated. Then, we apply our results to prove existence and uniqueness of the solution of a reflected stochastic differential equation (SDE).

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↗
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↗