General Draw-Down Times for Refracted Spectrally Negative Lévy Processes

Methodology and Computing in Applied Probability - In this paper, we prove several results involving a general draw-down time from the running maximum for refracted spectrally negative Lévy...     

Occupation Times of Intervals Until Last Passage Times for Spectrally Negative Lévy Processes

In this paper, we derive the Laplace transform of occupation times of intervals until last passage times for spectrally negative Lévy processes. Motivated by [2], the last passage times before an independent exponential variable are investigated. By a dual argument, explicit formulas are obtained and expressed as a modified version of the analytical identities introduced in Loeffen et al. [13]. As an application, a corridor option and an Omega risk model are studied here.     

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.     

A Distributional Equality for Suprema of Spectrally Positive Lévy Processes

Let \(Y\) be a spectrally positive Lévy process with \({\mathbb {E}}Y_1\!<\!0\) and \(C\) an independent subordinator with finite expectation, and let \(X\!=\!Y\!+\!C\). A curious distributional equality proved in Huzak et al. (Ann Appl Probab 14:1278–1397, 2004) states that if \({\mathbb {E}}X_1<0\), then \(\sup _{0\le t <\infty }Y_t\) and the supremum of \(X\) just before the first time its new supremum is reached by a jump of \(C\) have the same distribution. In this paper, we give an alternative proof of an extension of this result and offer an explanation why it is true.     

Maximum Drawdown and Drawdown Duration of Spectrally Negative Lévy Processes Decomposed at Extremes

Journal of Theoretical Probability - Path decomposition is performed to characterize the law of the pre-/post-supremum, post-infimum and the intermediate processes of a spectrally negative...     

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).     

Dividend Problem with Parisian Delay for a Spectrally Negative Lévy Risk Process

In this paper, we consider dividend problem for an insurance company whose risk evolves as a spectrally negative Lévy process (in the absence of dividend payments) when a Parisian delay is applied. An objective function is given by the cumulative discounted dividends received until the moment of ruin, when a so-called barrier strategy is applied. Additionally, we consider two possibilities of a delay. In the first scenario, ruin happens when the surplus process stays below zero longer than a fixed amount of time. In the second case, there is a time lag between the decision of paying dividends and its implementation.     

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.

     

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.     

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.     

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).     

Explicit Solution of a Non-Linear Filtering Problem for Lévy Processes with Application to Finance

In this paper we explicitly solve a non-linear filtering problem with mixed observations, modelled by a Brownian motion and a generalized Cox process, whose jump intensity is given in terms of a Lévy measure. Motivated by empirical observations of R. Cont and P. Tankov we propose a model for financial assets, which captures the phenomenon of time inhomogeneity of the jump size density. We apply the explicit formula to obtain the optimal filter for the corresponding filtering problem.     

On the Double Points of Operator Stable Lévy Processes

We determine the Hausdorff dimension of the set of double points for a symmetric operator stable Lévy process \(X=\left\{ X(t),t\in \mathbb {R}_+\right\} \) in terms of the eigenvalues of its stability exponent.     

On the Boundary Theory of Subordinate Killed Lévy Processes

Potential Analysis - Let Z be a subordinate Brownian motion in $\mathbb {R}^{d}$ , d ≥ 2, via a subordinator with Laplace exponent ?. We kill the process Z upon exiting a bounded open...     

Harnack’s Inequality for Stable Lévy Processes

We prove Harnacks inequality for harmonic functions of a symmetric stable Lévy process on R^d without the assumption that the density function of its Lévy measure is locally bounded from below. Mathematics Subject Classifications (2000) Primary 60J45, 31C05; Secondary 60G51.Research partially supported by KBN (2P03A 041 22) and RTN (HPRN-CT-2001-00273-HARP).     

Self-Standardized Central Limit Theorems for Trimmed Lévy Processes

Journal of Theoretical Probability - We prove under general conditions that a trimmed subordinator satisfies a self-standardized central limit theorem (SSCLT). Our basic tool is a powerful...     

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.