Layered stable (multivariate) distributions and processes are defined and studied. A layered stable process combines stable trends of two different indices, one of them possibly Gaussian. More precisely, in short time, it is close to a stable process while, in long time, it approximates another stable (possibly Gaussian) process. We also investigate the absolute continuity of a layered stable process with respect to its short time limiting stable process. A series representation of layered stable processes is derived, giving insights into both the structure of the sample paths and of the short and long time behaviors. This series is further used for sample paths simulation.
In this paper we introduce a new class of Lévy processes which we call hypergeometric-stable Lévy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric function plays an essential role. We characterize the Lévy measure of this class and obtain several useful properties such as the Wiener Hopf factorization, the characteristic exponent and some associated exit problems.
We consider the Markov chain approximations for singular stable-like processes. First we obtain properties of some Markov chains. Then we construct the approximating Markov chains and give a necessary condition for weak convergence of these chains to singular stable-like processes.
We investigate a family of discrete-time stationary processes defined by multiple stable integrals and renewal processes with infinite means. The model may exhibit behaviors of short-range or long-range dependence, respectively, depending on the parameters. The main contribution is to establish a phase transition in terms of the tail processes that characterize local clustering of extremes. Moreover, in the short-range dependence regime, the model provides an example where the extremal index is different from the candidate extremal index.
By using the existing sharp estimates of density function for rotationally invariant symmetric $α$-stable Lévy processes and rotationally invariant symmetric truncated $α$-stable Lévy processes, we obtain that Harnack inequalities hold for rotationally invariant symmetric $α$-stable Lévy processes with $α\in(0,2)$ and Ornstein-Uhlenbeck processes driven by rotationally invariant symmetric $α$-stable Lévy process, while logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated $α$-stable Lévy processes.
In this note we prove the well-posedness for stochastic 2D Navier-Stokes equation driven by general Lévy processes (in particular, $α$-stable processes), and obtain the existence of invariant measures.
Conditioning stable Lévy processes on zero probability events recently became a tractable subject since several explicit formulas emerged from a deep analysis using the Lamperti transformations for self-similar Markov processes. In this article we derive new harmonic functions and use them to explain how to condition stable processes to hit continuously a compact interval from the outside.
We consider a new family of $\R^d$-valued Lévy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance \cite{cc}, \cite{ckp}, \cite{kp} and \cite{pp}). This family of processes shares many properties with the tempered stable and the layered stable processes, defined in Rosiński \cite{ro} and Houdré and Kawai \cite{hok} respectively, for instance their short and long time behaviour. Additionally, in the real valued case we find a series representation which is used for sample paths simu…
A new family of stable processes indexed by metric spaces with stationary increments are introduced. They are special cases of a new family of set-indexed stable processes with Chentsov representation. At the heart of the representation, a result on the so-called measure definite kernels is of independent interest. A limit theorem for set-indexed processes is also established.
This paper elucidates the connection between stationary symmetric alpha-stable processes with 0<alpha<2 and nonsingular flows on measure spaces by describing a new and unique decomposition of stationary stable processes into those corresponding to positive flows and those corresponding to null flows. We show that a necessary and sufficient for a stationary stable process to be ergodic is that its positive component vanishes.
We study the exit time $τ=τ_{(0,\infty)}$ for 1-dimensional strictly stable processes and express its Laplace transform at $t^α$ as the Laplace transform of a positive random variable with explicit density. Consequently, $τ$ satisfies some multiplicative convolution relations. For some stable processes, e.g. for the symmetric $\frac23$-stable process, explicit formulas for the Laplace transform and the density of $τ$ are obtained as an application.
We prove that the class of discrete time stationary max-stable process satisfying the Markov property is equal, up to time reversal, to the class of stationary max-autoregressive processes of order $1$. A similar statement is also proved for continuous time processes.
We address the notion of association of sum- and max- stable processes from the perspective of linear and max-linear isometries. We establish the appealing results that these two classes of isometries can be identified on a proper space (the extended positive ratio space). This yields a natural way to associate to any max-stable process a sum-stable process. By using this association, we establish connections between structural and classification results for sum- and max- stable processes.
We develop classification results for max--stable processes, based on their spectral representations. The structure of max--linear isometries and minimal spectral representations play important roles. We propose a general classification strategy for measurable max--stable processes based on the notion of co--spectral functions. In particular, we discuss the spectrally continuous--discrete, the conservative--dissipative, and positive--null decompositions. For stationary max--stable processes, the latter two decompositions arise from connections to non--singular flows and are closely related to …
We investigate long and short memory in $α$-stable moving averages and max-stable processes with $α$-Fréchet marginal distributions. As these processes are heavy-tailed, we rely on the notion of long range dependence suggested by Kulik and Spodarev (2019) based on the covariance of excursions. Sufficient conditions for the long and short range dependence of $α$-stable moving averages are proven in terms of integrability of the corresponding kernel functions. For max-stable processes, the extremal coefficient function is used to state a necessary and sufficient condition for long range dependen…
We study the distribution and various properties of exponential functionals of hypergeometric Levy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applications we present a new proof of some of the results on the density of the supremum of a stable process, which were recently obtained by Kuznetsov "On extrema of stable processes" (2010) and Hubalek and Kuznetsov "A convergent series representation for the density of the supremum of a stable process"…
We study the asymptotics, for small and large values, of the supremum of a product of symmetric stable processes. We show in particular that the persistence exponent remains the same as for only one process, up to some logarithmic terms.
We characterize all possible independent symmetric alpha-stable (SaS) components of an SaS process, 0<alpha<2. In particular, we focus on stationary SaS processes and their independent stationary SaS components. We also develop a parallel characterization theory for max-stable processes.
Several aspects of the laws of first hitting times of points are investigated for one-dimensional symmetric stable Lévy processes. Itô's excursion theory plays a key role in this study.
We prove two limit laws for functionals of one dimensional symmetric 1-stable process using the method of moments, and give a remark on Rosen's paper \cite{Rosen}.
A stochastically continuous process $ξ(t)$, $t\geq0$, is said to be time-stable if the sum of $n$ i.i.d. copies of $ξ$ equals in distribution to the time-scaled stochastic process $ξ(nt)$, $t\geq0$. The paper advances the understanding of time-stable processes by means of their LePage series representations.
We consider a branching stable process with positive jumps, i.e. a continuous-time branching process in which the particles evolve independently as stable L{é}vy processes with positive jumps. Assuming the branching mechanism is critical or subcritical, we compute the asymptotics of the maximum location ever reached by a particle of the process.
We obtain the Lifschitz tail asymptotics for the integrated density of states of the subordinate $α-$stable processes on the Sierpiński gasket $\mathcal G,$ evolving among killing Poissonian obstacles. Simultaneously, we derive the large-time asymptotics for the volume of the $α-$stable sausage on the gasket.
We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly $α$-stable Lévy processes with $1< α\le 2$. This extends a theorem of Kaufman for Brownian motion. Our method is different from that of Kaufman and depends on covering principles for Markov processes.
In this paper we investigate two variants of $α$-stable processes, namely tempered stable subordinators and modified tempered stable process as well as their renormalization. We study the weak convergence in the Skorohod space and prove that they satisfy the uniform tightness condition. Finally, applications to the $α$-dependence of the solutions of SDEs driven by these processes are discussed.
First, we present some results about the Hölder continuity of the sample paths of so called dilatively stable processes which are certain infinitely divisible processes having a more general scaling property than self-similarity. As a corollary, we obtain that the most important (H,delta)-dilatively stable limit processes (e.g., the LISOU and the LISCBI processes, see Igloi [4]) almost surely have a local Hölder exponent H. Next we prove that, under some slight regularity assumptions, any two dilatively stable processes with stationary increments are singular (in the sense that their distribut…
The normalized spectral representation of a max-stable process on a compact set is the unique representation where all spectral functions share the same supremum. Among the class of equivalent spectral representations of a process, the normalized spectral representation plays a distinctive role as a solution of two optimization problems in the context of an efficient simulation of max-stable processes. Our approach has the potential of considerably reducing the simulation time of max-stable processes.
Using a generalization of the skew-product representation of planar Brownian motion and the analogue of Spitzer's celebrated asymptotic Theorem for stable processes due to Bertoin and Werner, for which we provide a new easy proof, we obtain some limit Theorems for the exit time from a cone of stable processes of index $α\in(0,2)$. We also study the case $t\rightarrow0$ and we prove some Laws of the Iterated Logarithm (LIL) for the (well-defined) winding process associated to our planar stable process.
Dilative stability generalizes the property of selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. Inspired by results of Iglói, we will show how dilatively stable processes with independent increments can be represented by integrals with respect to time-changed Lévy processes. Via a Lamperti-type transformation these representations are shown to be closely connected to translatively stable processes of Ornstein-Uhlenbeck-type, where translative stability generalizes the notion of stationarity. The presented results com…
Two kinds of conditionings for one-dimensional stable Lévy processes are discussed via $ h $-transforms of excursion measures: One is to stay positive, and the other is to avoid the origin.
Branching-stable processes have recently appeared as counterparts of stable subordinators, when addition of real variables is replaced by branching mechanism for point processes. Here, we are interested in their domains of attraction and describe explicit conditions for a branching random walk to converge after a proper magnification to a branching-stable process. This contrasts with deep results that have been obtained during the last decade on the asymptotic behavior of branching random walks and which involve either shifting without rescaling, or demagnification.
For a stable process, we give an explicit formula for the potential measure of the process killed outside a bounded interval and the joint law of the overshoot, undershoot and undershoot from the maximum at exit from a bounded interval. We obtain the equivalent quantities for a stable process reflected in its infimum. The results are obtained by exploiting a simple connection with the Lamperti representation and exit problems of stable processes.
The tail correlation function (TCF) is one of the most popular bivariate extremal dependence measures that has entered the literature under various names. We study to what extent the TCF can distinguish between different classes of well-known max-stable processes and identify essentially different processes sharing the same TCF.
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.
For symmetric Lévy processes, if the local times exist, the Tanaka formula has already constructed via the techniques in the potential theory by Salminen and Yor (2007). In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index $α\in (1,2)$ including spectrally positive and negative cases in a framework of Itô's stochastic calculus. Our approach to the existence of local times for such processes is different from Bertoin (1996).
We study avoidability of collections of balls in bounded $C^{1,1}$ opens sets for censored $α$-stable processes, $α\in (1,2)$. The results are analog to the ones obtained for Brownian motion in S. J. Gardiner, M. Ghergu, Champagne subregions of the unit ball with unavoidable bubbles, Ann. Acad. Sci. Fenn. Math. 35 (2010) 321-329. On the way we derive a Wiener-Aikawa-type criterion for minimal thinness with respect to the censored stable processes.
We study the probability of ruin before time $t$ for the family of tempered stable Lévy insurance risk processes, which includes the spectrally positive inverse Gaussian processes. Numerical approximations of the ruin time distribution are derived via the Laplace transform of the asymptotic ruin time distribution, for which we have an explicit expression. These are benchmarked against simulations based on importance sampling using stable processes. Theoretical consequences of the asymptotic formulae are found to indicate some potential drawbacks to the use of the inverse Gaussian process as a …
We determine the Hausdorff dimension of the set of double points for a symmetric operator stable Lévy process in terms of the eigenvalues of its stability exponent.
We prove the existence of an extremal function in the Hardy-Littlewood-Sobolev inequality for the energy associated to an stable operator. To this aim we obtain a concentration-compactness principle for stable processes in $\mathbb{R}^N$.
Dilatively stable processes generalize the class of infinitely divisible self-similar processes. We reformulate and extend the definition of dilative stability introduced by Iglói (2008) using characteristic functions. We also generalize the concept of aggregate similarity introduced by Kaj (2005). It turns out that these two notions are essentially the same for infinitely divisible processes. Examples of dilatively stable generalized fractional Lévy processes are given and we point out that certain limit processes in aggregation models are dilatively stable.
We introduce a class of spatial stochastic processes in the max-domain of attraction of familiar max-stable processes. The new class is based on Cox processes and comprises models with short range dependence. We show that statistical inference is possible within the given framework, at least under some reasonable restrictions.
Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of Lévy processes, which includes subordinated stable and symmetric Lévy processes. We apply this characterisaiton to construct $\varepsilon$-strong simulation ($\varepsilon$SS) algorithms for the convex minorant of stable meanders, the finite dimensional distributions of stable meanders and the convex minorants of weakly stable processes. We prove that the running times of our $\varepsilon$SS algorithms have finite exponential moments. We implement the algorithms in Julia 1.0 (av…
Multivariate max-stable processes are important for both theoretical investigations and various statistical applications motivated by the fact that these are limiting processes, for instance of stationary multivariate regularly varying time series, [1]. In this contribution we explore the relation between homogeneous functionals and multivariate max-stable processes and discuss the connections between multivariate max-stable process and zonoid / max-zonoid equivalence. We illustrate our results considering Brown-Resnick and Smith processes.
We investigate the asymptotic behavior of sample functions of stable processes when $t{\to}\infty$. We compare our results with the iterated logarithm law, results for the first hitting time and most visited sites problems.
