Ratio Limit Theorems for Random Walks and Lévy Processes

In this paper, we study quasi-symmetric random walks and Lévy processes, a property first introduced by C.J. Stone, discuss the -invariant Radon measures for random walks and Lévy processes, and formulate some nice ratio limit theorems which are closely related to -invariant Radon measures. Mathematics Subject Classifications (2000) 60G51, 60G50.Research supported in part by NSFC 10271109.     

On Subordinated Multivariate Gaussian Lévy Processes

We discuss criteria for the selfdecomposability of multivariate Lévy processes. We consider in detail Thorin subordinated multivariate Gaussian Lévy processes. Partially on the basis of the author’s recent results (MII preprint No. 2004-33, 2004), in this paper, we consider the properties of the Pólya subordinated multivariate Gaussian Lévy processes. We define, as a special class, the multivariate generalized z-processes. The one-dimensional case was investigated in (Grigelionis, B.: Liet. Mat. Rink. 41(3), 303–309, 2001).     

Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity

We prove some limiting results for a Lévy process X _t as t0 or t, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are mostly concerned with ideas related to relative stability and attraction to the normal distribution on the one hand and divergence to large values of the Lévy process on the other. The aim is to find analytical conditions for these kinds of behaviour which are in terms of the characteristics of the process, rather than its distribution. Some surprising results occur, especially for the case t0; for example, we may have X _t /t ^P + (t0) (weak divergence to +), whereas X _t /t a.s. (t0) is impossible (both are possible when t), and the former can occur when the negative Lévy spectral component dominates the positive, in a certain sense. Almost sure stability of X _t , i.e., X _t tending to a nonzero constant a.s. as t or as t0, after normalisation by a non-stochastic measurable function, reduces to the same type of convergence but with normalisation by t, thus is equivalent to strong law behaviour. Boundary crossing problems which are amenable to the methods we develop arise in areas such as sequential analysis and option pricing problems in finance.     

Random Fractals Determined by Lévy Processes

Several indices, such as the Blumenthal–Getoor indices, have been defined to help describe various sample path properties for Lévy processes. These indices can be used to obtain bounds on the Hausdorff dimension of the range, graph, and zero set for a special subclass of Lévy processes. However, there has yet to be found an index that precisely determines the dimension of the graph for a general Lévy process. While surveying many of these results with a focus on general Lévy processes, some of the results are generalized or improved. The culmination of this synthesis is a new index that specifies the dimension of the graph of a general multidimensional Lévy process.     

On the Existence and Explicit Estimates for the Coupling Property of Lévy Processes with Drift

In this article, we first establish new criteria for the coupling property of Lévy processes with drift. The criteria are sharp for Lévy processes and Ornstein-Uhlenbeck processes with jumps, and also strengthen the recent result of Lin and Wang (Sci China Math 55:1735–1748, Theorem 1.1, 2012). Then, using the time-change technique, we derive explicit estimates for the coupling property of subordinated Brownian motions with drift. These estimates are optimal for a large class of subordinated Brownian motions.     

Strong and Weak Disorder for Lévy Directed Polymers in Random Environment

Abstract

We consider Lévy directed polymers in the Poisson random environment. We give conditions for strong or weak disorder in terms of the Lévy exponent of symmetric Lévy process.     

Continuous-State Branching Processes in Lévy Random Environments

A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a Lévy process with no jump less than \(-1\). We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another Lévy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey’s condition. In that case, a characterization of the extinction probability is given using a random differential equation with blowup terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcritical CBRE-process with immigration.     

Branching Processes in a Lévy Random Environment

In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813–857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1–13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.     

Kato Classes for Lévy Processes

We prove that the definitions of the Kato class through the semigroup and through the resolvent of the Lévy process in \(\mathbb {R}^{d}\) coincide if and only if 0 is not regular for {0}. If 0 is regular for {0} then we describe both classes in detail. We also give an analytic reformulation of these results by means of the characteristic (Lévy-Khintchine) exponent of the process. The result applies to the time-dependent (non-autonomous) Kato class. As one of the consequences we obtain a simultaneous time-space smallness condition equivalent to the Kato class condition given by the semigroup.     

Perpetual Integrals for Lévy Processes

Given a Lévy process \(\xi \), we find necessary and sufficient conditions for almost sure finiteness of the perpetual integral \(\int _0^\infty f(\xi _s)\hbox {d}s\), where \(f\) is a positive locally integrable function. If \(\mu =\mathbb {E}[\xi _1]\in (0,\infty )\) and \(\xi \) has local times we prove the 0–1 law

$$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )\in \{0,1\} \end{aligned}$$

with the exact characterization

$$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )=0\qquad \Longleftrightarrow \qquad \int ^\infty f(x)\,\hbox {d}x=\infty . \end{aligned}$$

The proof uses spatially stationary Lévy processes, local time calculations, Jeulin’s lemma and the Hewitt–Savage 0–1 law.

     

A Law of Large Numbers for the Power Variation of Fractional Lévy Processes

We prove a law of large numbers for the power variation of an integrated fractional process in a pure jump model. This yields consistency of an estimator for the integrated volatility where we are no longer restricted to a Gaussian model.     

Maximal Inequalities for Fractional Lévy and Related Processes

In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.     

Hyperfinite Lévy Processes

A hyperfinite Lévy process is an infinitesimal random walk (in the sense of nonstandard analysis) which with probability one is finite for all finite times. We develop the basic theory for hyperfinite Lévy processes and find a characterization in terms of transition probabilities. The standard part of a hyperfinite Lévy process is a (standard) Lévy process, and we show that given a generating triplet (γ, C, μ) for standard Lévy processes, we can construct hyperfinite Lévy processes whose standard parts correspond to this triplet. Hence all Lévy laws can be obtained from hyperfinite Lévy processes. The paper ends with a brief look at Malliavin calculus for hyperfinite Lévy processes including a version of the Clark-Haussmann-Ocone formula.     

A Smooth Approach to Malliavin Calculus for Lévy Processes

An approach to Malliavin calculus for Lévy processes, discrete in time and smooth in chance, is presented. Each Lévy triple can be satisfied by a Lévy process living on a fixed sample space Ω, which is, in a certain sense, a finite dimensional Euclidean space. The probability measures on Ω characterize the Lévy processes. We compare these measures with the associated Lévy measures, and present several examples. Using chaos expansions for Lévy functionals, even for those having no moments, we can represent all these functionals by polynomials in several variables. There exists an effective method to compute the kernels of the chaos decomposition. Finally, we point out several applications, which are postponed to a succession of papers. Dedicated to Helmut Schwichtenberg.     

Harnack Inequalities for some Lévy Processes

In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝ ^d . We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Lévy process in ℝ ^d with a Lévy density given by $c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}$c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}, where 0 ≤ j(r) ≤ cr ^{− d − α} , ∀ r > 1, for some constant c. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion with subordinator with Laplace exponent ϕ(λ) = λ ^α/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2).     

Precise Asymptotic Approximations for Kernels Corresponding to Lévy Processes

By using basic complex analysis techniques, we obtain precise asymptotic approximations for kernels corresponding to symmetric α-stable processes and their fractional derivatives. We use the deep connection between the decay of kernels and singularities of the Mellin transforms. The key point of the method is to transform the multi-dimensional integral to the contour integral representation. We then express the integrand as a combination of gamma functions so that we can easily find all poles of the integrand. We obtain various asymtotics of the kernels by using Cauchys Residue Theorem with shifting contour integration. As a byproduct, exact coefficients are also obtained. We apply this method to general Lévy processes whose characteristic functions are radial and satisfy some regularity and size conditions. Our approach is based on the Fourier analytic point of view.     

Strong Laws of Large Numbers for Random Walks in Random Sceneries

In this paper, we study strong laws of large numbers for random walks in random sceneries. Some mild sufficient conditions for the validity of strong laws of large numbers are obtained.     

On Anticipative Girsanov Transformations for Lévy Processes

The smooth approach to Malliavin calculus for Lévy processes in (Osswald in J. Theor. Probab., 2008) is used to study time-anticipative Girsanov transformations for a large class of Lévy processes by means of the substitution rule in finite-dimensional analysis. Dedicated to Wolfram Pohlers on the occasion of his 65th birthday.     

Hitting Times of Points and Intervals for Symmetric Lévy Processes

For one-dimensional symmetric Lévy processes, which hit every point with positive probability, we give sharp bounds for the tail function P ^x (T _B >t), where T _B is the first hitting time of B which is either a single point or an interval. The estimates are obtained under some weak type scaling assumptions on the characteristic exponent of the process. We apply these results to prove sharp two-sided estimates of the transition density of the process killed after hitting B.