Explicit Strong Solutions of SPDE’s with Applications to Non-Linear Filtering

A differential calculus for random fields is developed and combined with the S-transform to obtain an explicit strong solution of the Cauchy problem

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.     

On an Explicit Skorokhod Embedding for Spectrally Negative Lévy Processes

We present an explicit solution to the Skorokhod embedding problem for spectrally negative Lévy processes. Given a process X and a target measure μ satisfying an explicit admissibility condition we define functions φ _± such that the stopping time T=inf?{t>0:X _t ∈{?φ _?(L _t ),φ ₊(L _t )}} induces X _T ～μ, where (L _t ) is the local time in zero of X. We also treat versions of T which take into account the sign of the excursion straddling time t. We prove that our stopping times are minimal and we describe criteria under which they are integrable. We compare our solution with the one proposed by Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]). In particular, we compute explicitly the quantities introduced in Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]) in our setup. Our method relies on some new explicit calculations relating scale functions and the Itô excursion measure of X. More precisely, we compute the joint law of the maximum and minimum of an excursion away from 0 in terms of the scale function.     

Estimation of Lévy Processes via Stochastic Programming and Kalman Filtering

The estimation of Lévy process has received a lot of attention in recent years. Evidence of this is the extensive amount of literature concerning this problem which can be classified in two categories: the nonparametric approach, and the parametric approach. In this paper, we shall concentrate on the latter, and in particular the parameters will be estimated within a stochastic programming framework. To be more specific, the first derivative of the characteristic function and its empirical version shall be used in objective function. Furthermore, the parameter estimates are recursively estimated by making use of a modified extended Kalman filter (MEKF). Some properties of the parameter estimates are studied. Finally, a number of simulations will be carried out and the results are presented and discussed.     

Stochastic Processes Associated with a Sum of the Lévy Laplacians

Abstract

In this paper, we give a stochastic expression of a semigroup generated by a sum of the Lévy Laplacians acting on a class of S-transforms of white noise distributions in terms of an infinite sequence of independent Brownian motions.     

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.

     

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.     

Continuity Conditions for a Class of Gaussian Chaos Processes Related to Continuous Additive Functionals of Lévy Processes

Let be sequences of real numbers which are symmetric in k. Let be independent sequences of independent normal random variables with mean zero and variance one. For each fixed choice of we consider

Explicit Representation of Strong Solutions of SDEs Driven by Infinite-Dimensional Lévy Processes

We develop a white noise framework for Lévy processes on Hilbert spaces. As the main result of this paper, we then employ these white noise techniques to explicitly represent strong solutions of stochastic differential equations driven by a Hilbert-space-valued Lévy process.     

Transition Density Estimates for a Class of Lévy and?Lévy-Type Processes

We show on- and off-diagonal upper estimates for the transition densities of symmetric Lévy and Lévy-type processes. To get the on-diagonal estimates, we prove a Nash-type inequality for the related Dirichlet form. For the off-diagonal estimates, we assume that the characteristic function of a Lévy(-type) process is analytic, which allows us to apply the complex analysis technique.     

Lévy Approximation of Increment Processes with Markov Switching

We study weak convergence of increment processes with embedded Markov chain switching in a series scheme. The limit process is a Lévy process where the jump part is a compound Poisson process. A result concerning the rate of convergence is also given. This study is motivated by risk theory and its applications.     

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).     

The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes

We study the asymptotic tail behaviour of the first passage time over a moving boundary for asymptotically \(\alpha \)-stable Lévy processes with \(\alpha <1\). Our main result states that if the left tail of the Lévy measure is regularly varying with index \(- \alpha \), and the moving boundary is equal to \(1 - t^{\gamma }\) for some \(\gamma <1/\alpha \), then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary \(1 + t^{\gamma }\) with \(\gamma <1/\alpha \) under the assumption of a regularly varying right tail with index \(-\alpha \).     

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.     

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.     

A Construction of Reflecting Lévy Processes

Doklady Mathematics - For some classes of Lévy processes, the notion of reflection from an interval boundary is introduced. It is shown that trajectories of a reflecting process define random...     

Finite Variation of Fractional Lévy Processes

Various characterizations for fractional Lévy processes to be of finite variation are obtained, one of which is in terms of the characteristic triplet of the driving Lévy process, while others are in terms of differentiability properties of the sample paths. A zero-one law and a formula for the expected total variation are also given.