This paper introduces a novel stochastic partial differential equation (SPDE) driven by space-only fractional Lévy noise ${\dot{L}_{x}^{d}}$ with $d\in (0,1/2)$, governed by a second-order elliptic operator with piecewise constant coefficients. We establish a new framework for analyzing diffusion in heterogeneous media, proving the existence, uniqueness, and equivalence of mild and weak solutions for the resulting non-Gaussian random field. A key original contribution is the explicit characterization of the covariance kernel in terms of transmission-reflection paths induced by the interface. We provide sharp estimates for the sample path regularity, proving that the solution is almost surely Hölder continuous of order $\gamma \lt d$ in space and $\beta \lt \frac{2d+1}{4}$ in time. Our results demonstrate that the fractional order of the noise robustly dictates path regularity, which remains preserved despite the singular transmission conditions and the jump structure of the driving noise.
In this paper, we introduce a family of semi-parametric estimators for the positive extreme value index γ, parameterized in two tuning parameters. The asymptotic normality of the introduced estimators is proved. It is shown that the partial case of newly introduced estimators (a subfamily with one tuning parameter) has quite good asymptotic properties and dominates several previously introduced estimators. Small Monte-Carlo simulations are included. Also, the performance of this parameterized subfamily of estimators is illustrated for pair exchange ratio data sets.
Let B-H be a fractional Brownian motion with Hurst index 1/2 < H < 1 In this paper, we consider the time fractional functional differential equation of the form {(c) D(t)(gamma )x(t) = f(t, x(t)) + G(t, x(t))d/dt B-H (t), t is an element of (0,T], x(0)(t) = eta(t), t is an element of [-r, 0], where 3/2 - H < gamma < 1 (c) D-t(gamma), denotes the Caputo derivative, and x(t) is an element of C-r = C([-r, 0], R) with X-t (u) = x(t + u), u is an element of [-r, 0]. We prove the global existence and uniqueness of the solution of the equation and study its viability. As an application, we also discuss the existence of positive solutions.
The problem of estimating the drift parameter is considered for an Ornstein–Uhlenbeck-type process driven by a tempered fractional Brownian motion (tfBm) or tempered fractional Brownian motion of the second kind (tfBmII). Unlike most existing studies, which assume continuous-time observations, a more realistic setting of discrete-time data is in focus. The strong consistency of a discretized least squares estimator is established under an asymptotic regime where the observation interval tends to zero while the total time horizon increases. A key step in the analysis involves deriving almost sure upper bounds for the increments of both tfBm and tfBmII.
The first-return time is the time that it takes a random walker to go back to the initial position for the first time. In this paper, the first-return time is studied when random walkers perform fractional kinetics, specifically fractional diffusion, that is modelled within the framework of the continuous-time random walk on homogeneous space in the uncoupled formulation with Mittag-Leffler distributed waiting-times. Both the Markovian and non-Markovian settings are considered, as well as any kind of symmetric jump-size distributions, namely with finite or infinite variance. It is shown that the first-return time density is indeed independent of the jump-size distribution when it is symmetric, and therefore it is affected only by the waiting-time distribution that embodies the memory of the process. The analysis is performed in two cases: first jump then wait and first wait then jump, and several exact results are provided, including the relation between results in the Markovian and non-Markovian settings and the difference between the two cases.
