Α-stable Processes Research Articles

From Markovian to non-Markovian persistence exponents

We establish an exact formula relating the survival probability for certain Lévy flights (viz. asymmetric α-stable processes where ) with the survival probability for the order statistics of the running maxima of two independent Brownian particles. This formula allows us to show that the persistence exponent δ in the latter non-Markovian case is simply related to the persistence exponent θ in the former Markovian case via: . Thus, our formula reveals a link between two recently explored families of anomalous exponents: one exhibiting continuous deviations from Sparre-Andersen universality in a Markovian context, and one describing the slow kinetics of the non-Markovian process corresponding to the difference between two independent Brownian maxima.

Numerical analysis for neutral SPDEs driven by α-stable processes

In this note, we discuss strong convergence of exponential integrator scheme based on spatial and time discretization for a class of neutral stochastic partial differential equations driven by α-stable processes.

Harnack Inequalities for SDEs Driven by Cylindrical α-Stable Processes

By using the coupling argument, we establish the Harnack and log-Harnack inequalites for stochastic differential equations with non-Lipschitz drifts and driven by additive anisotropic subordinated Brownian motions (in particular, cylindrical α-stable processes). Moreover, the gradient estimate is also derived when the drift is Lipschitz continuous.

On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index α is in (0,2), equal to 2, and in (2,∞), respectively. The partial sum weakly converges to a functional of α-stable process when α<2 and converges to a functional of Brownian motion when α≥2. When the process is of short-memory and α<4, the autocovariances converge to functionals of α/2-stable processes; and if α≥4, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on α and β (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2-stable processes; (ii) Rosenblatt processes (indexed by β, 1/2<β<3/4); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index α and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1] with either (i) the J1 or the M1 topology (Skorokhod, 1956); or (ii) the weaker form S topology (Jakubowski, 1997). Some statistical applications are also discussed.

Trace Estimates for Relativistic Stable Processes

In this paper, we study the asymptotic behavior, as the time t goes to zero, of the trace of the semigroup of a killed relativistic α-stable process in bounded C 1,1 open sets and bounded Lipschitz open sets. More precisely, we establish the asymptotic expansion in terms of t of the trace with an error bound of order t 2/α t −d/α for C 1,1 open sets and of order t 1/α t −d/α for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders t −d/α and t (2−d)/α for C 1,1 open sets, and, when α ∈ (0, 1), between the orders t −d/α and t (1−d)/α for Lipschitz open sets.

Serial Amplify-and-Forward Relay Transmission Systems in Nakagami- &lt;inline-formula&gt; &lt;tex-math notation="TeX"&gt;$m$&lt;/tex-math&gt;&lt;/inline-formula&gt; Fading Channels With a Poisson Interference Field

In this paper, the end-to-end performance of a wireless relay transmission system that employs amplify-and-forward (AF) relays and operates in an interference-limited Nakagami- m fading environment is studied. The wireless links from one relay node to another experience Nakagami- m fading, and the number of interferers per hop is Poisson distributed. The aggregate interference at each relay node is modeled as a shot-noise process whose distribution follows an α-stable process. For the considered system, analytical expressions for the moments of the end-to-end signal-to-interference ratio (SIR), the end-to-end outage probability (OP), the average bit-error probability (ABEP), and the average channel capacity are obtained. General asymptotic expressions for the end-to-end ABEP are also derived. The results provide useful insights regarding the factors affecting the performance of the considered system. Monte Carlo simulation results are further provided to demonstrate the validity of the proposed mathematical analysis.

Unavoidable collections of balls for censored stable processes

We study avoidability of collections of balls in bounded C1,1 opens sets for censored α-stable processes, α∈(1,2). The results are analogous to the ones obtained for Brownian motion in Gardiner and Ghergu (2010) [11]. On the way we derive a Wiener–Aikawa-type criterion for minimal thinness with respect to the censored stable processes.

Ergodicity for time-changed symmetric stable processes

In this paper we study ergodicity and related semigroup property for a class of symmetric Markov jump processes associated with time-changed symmetric α-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric α-stable processes, yield sharp criteria for their various ergodic properties and corresponding functional inequalities.

Stable process with singular drift

Suppose that d≥2 and α∈(1,2). Let μ=(μ1,…,μd) be such that each μi is a signed measure on Rd belonging to the Kato class Kd,α−1. In this paper, we consider the stochastic differential equation dXt=dSt+dAt, where St is a symmetric α-stable process on Rd and for each j=1,…,d, the jth component Atj of At is a continuous additive functional of finite variation with respect to X whose Revuz measure is μj. The unique solution for the above stochastic differential equation is called an α-stable process with drift μ. We prove the existence and uniqueness, in the weak sense, of such an α-stable process with drift μ and establish sharp two-sided heat kernel estimates for such a process.

Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory

We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals. We then turn to applications on spectral theory and provide for (measure perturbation of) general regular Dirichlet forms an Allegretto–Piepenbrink type theorem, which is based on a ground state transform, and a Shnol' type theorem. Our setting includes Laplacian on manifolds, on graphs and α-stable processes.

Empirical likelihood approach toward discriminant analysis for dynamics of stable processes

Discriminant analysis for time series models has been studied by many authors in these few decades, but many of them deal with second order stationary processes. In this paper, we introduce an empirical likelihood statistic based on a Whittle likelihood as a classification statistic, and consider problems of classifying an α-stable linear process into one of two categories described by pivotal quantities θ1 and θ2 of time series models. It is shown that misclassification probabilities by the empirical likelihood criterion converge to 0 asymptotically without assuming that the true model is known. We also evaluate misclassification probabilities when θ2 is contiguous to θ1, and carry out simulation studies to make a comparison between goodness of the empirical likelihood classification statistic and that of an existing method. We observed that the empirical likelihood ratio discriminant statistic performs better than the existing method in some cases even if a family of score functions does not contain the true model. Since the stable processes do not have the finite second moment, this extension is not straightforward, and contains a lot of innovative aspects.

Stability in distribution of neutral stochastic partial differential delay equations driven by α-stable process

We consider a class of neutral stochastic partial differential equations driven by an α-stable process. We prove the existence and uniqueness of the mild solution to the equation by the Banach fixed-point theorem under some suitable assumptions. Sufficient conditions for the stability in the distribution of the mild solution are derived.MSC:39A11, 37H10.

Ergodicity of Stochastic Dissipative Equations Driven by α-Stable Process

It is shown that the transition semigroup (P t ) t≥0 corresponding to stochastic dissipative equations driven by α-stable is strong Feller and irreducible for α ∈ (1, 2). This result ensures the ergodicity for the equation.

Exponential Ergodicity of Stochastic Burgers Equations Driven by α-Stable Processes

In this work, we prove the strong Feller property and the exponential ergodicity of stochastic Burgers equations driven by α/2-subordinated cylindrical Brownian motions with α∈(1,2). To prove the results, we truncate the nonlinearity and use the derivative formula for SDEs driven by α-stable noises established in (Zhang in Stoch. Process. Appl. 123(4):1213–1228, 2013).

Dirichlet Heat Kernel Estimates for Stable Processes with Singular Drift In Unbounded C 1,1 Open Sets

Suppose d ≥ 2 and α ∈ (1, 2). Let D be a (not necessarily bounded) C1,1 open set in ℝd and μ = (μ1, . . . , μd) where each μj is a signed measure on ℝd belonging to a certain Kato class of the rotationally symmetric α-stable process X. Let Xμ be an α-stable process with drift μ in ℝd and let Xμ,D be the subprocess of Xμ in D. In this paper, we derive sharp two-sided estimates for the transition density of Xμ,D.

Stable CLTs and Rates for Power Variation of α-Stable Lévy Processes

In a central limit type result it has been shown that the pth power variations of an α-stable Levy process along sequences of equidistant partitions of a given time interval have \(\frac{\alpha}{p}\)-stable limits. In this paper we give precise orders of convergence for the distances of the approximate power variations computed for partitions with mesh of order \(\frac{1}{n}\) and the limiting law, measured in terms of the Kolmogorov-Smirnov metric. In case 2α < p the convergence rate is seen to be of order \(\frac{1}{n}\), in case α < p < 2α the order is \(n^{1-\frac{p}{\alpha}}.\)

Two-term trace estimates for relativistic stable processes

We prove trace estimates for the relativistic α-stable process extending the result of Bañuelos and Kulczycki (2008) in the stable case.

Lowest Eigenvalue Bounds for Markov Processes with Obstacles

We derive lower and upper bound for the first eigenvalue of an infinitesimal generator for symmetric cadlag Markov process via an obstacle enlargement method. Several applications, including α-stable processes on a torus in ℝ d , are presented.

Spectral estimates for high-frequency sampled continuous-time autoregressive moving average processes

In this article, we consider a continuous-time autoregressive moving average (CARMA) process driven by either a symmetric α-stable Levy process with α ∈ (0,2) or a symmetric Levy process with finite second moments. In the asymptotic framework of high-frequency data within a long time interval, we establish a consistent estimate for the normalized power transfer function by applying a smoothing filter to the periodogram of the CARMA process. We use this result to propose an estimator for the parameters of the CARMA process and exemplify the estimation procedure by a simulation study.

Fokker-Planck Type Equations Associated with Subordinated Processes Controlled by Tempered α-stable Processes

In this paper we introduce two models of stochastic processes driven by Brownian motion and fractional Brownian motion subordinated with tempered α-stable waiting times. By using a new integro-differential operator we obtain the generalized Fokker-Planck type equations associated with these subordinated stochastic processes.