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Ezzedine Mliki

Publications and source records attributed to Ezzedine Mliki.

8 recordsLinked to original sources
arXiv2026

Exact maximum likelihood inference for drifted multi-sub-fractional Brownian motion at discrete observation

Sub-fractional Brownian motion is self-similar and long-range dependent but has no stationary increments, so the increment covariance is not Toeplitz and no spectral density is available. We show that a complete finite-sample likelihood theory survives nonetheless. The model is a linear trend observed at $N$ equidistant times through a superposition of $m$ independent sub-fractional Brownian motions with known Hurst indices and a common scale. Nondegeneracy follows from realising the process as the even part of a two-sided fractional Brownian motion, and the maximum likelihood estimators of the trend and of the scale are explicit. The statistics on which inference rests are pivotal, their laws depending on the sample size alone, so intervals and tests of exact level are available at every $N\ge2$, together with complete sufficiency, minimum variance unbiasedness and attainment of the Cramér--Rao bound. An explicit variance bound gives strong consistency and asymptotic normality, and simulations confirm the exact coverage and the predicted effect of a misspecified Hurst vector.

math.ST↗
arXiv2026

Exact finite-sample inference for multi-mixed fractional Brownian motion with drift

In this paper we study a linear drift perturbed by a superposition of $m$ independent fractional Brownian motions with known Hurst parameters and a common scale, observed at $N$ equidistant times. Inference for such models is usually asymptotic; we show that here it is exact. We derive the maximum likelihood estimators of the drift $θ$ and of the scale $α^{2}$ in closed form and obtain their exact finite-sample joint law: $\widehatθ$ is Gaussian, $N\widehatα^{\,2}/α^{2}$ is chi-square with $N-1$ degrees of freedom, and the two are independent. As this law is free of every model parameter, we deduce Student and chi-square confidence intervals and tests of exact level for every $N\ge2$, whatever the Hurst vector. We also prove that the estimators are uniformly minimum variance unbiased with $\widehatθ$ attaining the Cramér--Rao bound at every $N$, that both are strongly consistent and asymptotically normal, and that the drift estimators form, in law, a Brownian motion run along their own variance scale. A sharp non-asymptotic bound shows that the accuracy of the drift is governed by the length of the observation window and not by the mesh, and a Monte Carlo study confirms exact coverage, even at small sample sizes, and quantifies what is lost when the Hurst vector is misspecified.

math.ST↗
arXiv2026

The Fujita exponent across an interface

We consider the semilinear parabolic equation \[ \partial_t u = Δu + 2\mathfrak{q}\,δ_{\mathbb{S}}\,\nabla u + |u|^{p-1}u \qquad \text{in } (0,\infty)\times\mathbb{R}^N, \] where $|\mathfrak{q}|\le 1$, $p>1$, and $\mathbb{S}$ is a fixed interface hyperplane. Working in Lebesgue spaces, we first establish local well-posedness of mild solutions. This is achieved by combining Gaussian bounds for the associated fundamental solution with a contraction mapping argument adapted to the lack of spatial homogeneity induced by the interface term. We then prove a sharp Fujita-type dichotomy for nonnegative solutions. Specifically, we show that every nontrivial solution blows up in finite time when $1 1+\frac{2}{N}$ global solutions exist for sufficiently small initial data. The blow-up analysis relies on a suitably adapted test-function method that accounts for the presence of the interface. It is noteworthy that the critical exponent coincides with the classical Fujita exponent for the heat equation, indicating that the Fujita phenomenon remains stable under the presence of discontinuous diffusion effects and interface transmission conditions. To the best of our knowledge, this is the first result of this type for operators involving a singular drift supported on a hypersurface.

math.AP↗
arXiv2025

Long-Time Asymptotics for Subordinated Fractional Diffusion Equations

We study the long-time behavior of solutions to a class of evolution equations arising from random-time changes driven by subordinators. Our focus is on fractional diffusion equations involving mixed local and nonlocal operators. By combining techniques from probability theory, asymptotic analysis, and partial differential equations (PDEs), we characterize the dynamics of the subordinated solutions. This approach extends classical fractional dynamics and establishes a deeper connection between stochastic processes and deterministic PDEs.

math.AP↗
arXiv2023

On the fractional mixed fractional Brownian motion Time Changed by Inverse alpha Stable Subordinator

A time-changed fractional mixed fractional Brownian motion by inverse alpha stable subordinator with index alpha in (0, 1) is an iterated process L constructed as the superposition of fractional mixed fractional Brownian motion N(a, b) and an independent inverse α-stable subordinator Talpha. In this paper we prove that the process LT alpha(a, b) is of long range dependence property under a smooth condition on the Hirsh index H1 and H2. We deduce that the fractional mixed fractional Brownian motion has long range dependence for every H1 < H2.

math.PR↗
arXiv2021

Mixed Generalized Fractional Brownian Motion

To extend several known centered Gaussian processes, we introduce a new centered mixed self-similar Gaussian process called the mixed generalized fractional Brownian motion, which could serve as a good model for a larger class of natural phenomena. This process generalizes both the well known mixed fractional Brownian motion introduced by Cheridito [10] and the generalized fractional Brownian motion introduced by Zili [31]. We study its main stochastic properties, its non-Markovian and non-stationarity characteristics and the conditions under which it is not a semimartingale. We prove the long range dependence properties of this process.

math.PR↗
arXiv2021

On the long range dependence of time-changed mixed fractional Brownian motion model

A time-changed mixed fractional Brownian motion is an iterated process constructed as the superposition of mixed fractional Brownian motion and other process. In this paper we consider mixed fractional Brownian motion of parameters a, b and H\in(0, 1) time-changed by two processes, gamma and tempered stable subordinators. We present their main properties paying main attention to the long range dependence. We deduce that the fractional Brownian motion time-changed by gamma and tempered stable subordinators has long range dependence property for all H\in(0, 1).

math.PR↗