Levy 2010 – Talks

The 6th International Conference on

Levy Processes: Theory and Applications

26.07.2010-30.07.2010

Dresden, Germany

Contents

Cylindrical Levy processes in Banach Spaces 6David Applebaum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Heat and Weyl asymptotics for stable and relativistic stable processes 7Rodrigo Banuelos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Estimates of the Green function for fractional Laplacian perturbed by gradient 8Krzysztof Bogdan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Recurrence and Transience of stable-like Processes 9Bjorn Bottcher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Brownian motion in Poissonian potential with renormalized energy 10Xia Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Modeling and simulation with operator scaling 11Serge Cohen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1

A non-anticipative calculus for functionals of semimartingales 12RamaCont . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Invariance principles for local times of Levy processes and random walks 13Ron Doney . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Hausdorff and packing measures of Levy trees 14Thomas Duquesne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Unlimited Liabilities, reserve capital requirements and the taxpayer put 15Ernst Eberlein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Modeling Network Traffic by a Cluster Poisson Input Process with Heavy and Light-Tailed File Sizes 16Vicky Fasen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

The numerical approximation of an SPDE driven by a Levy noise 17Erika Hausenblas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Metric Measure Spaces Associated with Continuous Negative Definite Functions and the Behaviour of Transition Functions of Jump Processes 18Niels Jacob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Some questions in high-frequency statistics for semimartingales 19Jean Jacod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

On a Heath-Jarrow-Morton approach for stock options 20Jan Kallsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Generalized fractional Levy processes with fractional Brownian motion limit 21Claudia Klueppelberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2

Meromorphic Levy processes (and a new Wiener-Hopf simulation technique) 22Andreas E. Kyprianou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Strong solutions for stochastic differential equations with jumps 23Zenghu Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

More about limits of nested subclasses of classes of infinitely divisible distributions 24Makoto Maejima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes and loop soups 25Michael Marcus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Space-time duality for fractional diffusion 26Mark M Meerschaert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Symmetrization of Levy processes and applications 27Pedro Mendez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Stable limits for dependent stationary sequences 28Thomas Mikosch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Intertwining relation between fractional operators 29Pierre Patie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

On the exit times of Levy driven SDEs 30Ilya Pavlyukevich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Levy-Ornstein-Uhlenbeck processes in Hilbert spaces 31Szymon Peszat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3

Quasi-stationary distributions and Yaglom type limits for self-similar Markov processes 32Rivero Victor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Finite variation of Levy driven moving average-like processes 33Jan Rosinski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Geometric characteristics of the excursion sets over highl levels of non-Gaussian infinitely divisible random fields 34Gennady Samorodnitsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Some applications of duality for Levy Processes in a half-line 35Mladen Savov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Wavelet Solution of Kolmogoroff Equations for Feller-Levy Processes 36Christoph Schwab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Some properties of positive stable densities 37Thomas Simon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Heat kernel estimates for Dirichlet fractional Laplacian Perturbed by Gradient Operators in C1,1 Open Sets 38Renming Song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Multivariate SupOU Processes 39Robert Stelzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Asymptotics of Transition Densities of Jump Processes 40Pawel Sztonyk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Technique for computing the PDFs and CDFs of non-negative infinitely divisible random variables 41Murad Taqqu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4

Local Malliavin Calculus for Levy Processes and Applications 42Frederic Utzet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Fractional Levy processes: capturing stylized facts of empirical data 43Jeannette H.C. Woerner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Packing Dimension Results for the Images of Levy Processes 44Yimin Xiao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Regularity of solutions to spdes with Levy noise 45Jerzy Zabczyk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5

Cylindrical Levy processes in Banach Spaces

David Applebaum

d.applebaum[-at-]sheffield.ac.uk

Let (µt, t ≥ 0) be a weakly continuous convolution semigroup of probabilitymeasures on a compact group G with neutral element e. Let (Tt, t ≥ 0) bethe associated Feller-Markov semigroup of linear operators on the L2-space ofnormalised Haar measure. The operator Tt is trace-class for t > 0 if and onlyif µt has an L2-density kt. If µt is central then the co-ordinate functions ofirreducible representations are a complete orthogonal set of eigenfunctions forTt and if kt is also continuous we have

kt(e) = tr(Tt) =∑

π

d2πλπ(t),

where dπ is the dimension of the irreducible representation π and λπ(t) is thecorresponding eigenvalue of Tt. We compute the small-time asymptotics of kt(e)on SU(2) when µt has a Cauchy distribution.

6

Heat and Weyl asymptotics for stable andrelativistic stable processes

Rodrigo Banuelos

banuelos[-at-]math.purdue.edu

After recalling some of the well known results on the heat and Weyl asymp-totics for the classical Laplacian (which is intimately connected with Brownianmotion), we will address similar questions for the fractional and relativisticLaplacian. The latter are pseudo differential operators associated with Levyprocesses and this will be the point of view in this talk.

7

Estimates of the Green function for fractionalLaplacian perturbed by gradient

Krzysztof Bogdan

bogdan[-at-]pwr.wroc.pl

For bounded smooth domains the Green function of the fractional Laplacianof differential order bigger than one and the Green function of its gradientperturbation are comparable if the drift function is in a Kato class. This is ajoint work with Tomasz Jakubowski.

References:

1. K. Bogdan, T. Jakubowski. Estimates of heat kernel of fractional Lapla-cian perturbed by gradient operators. Comm. Math. Phys., 271(1):179–198, 2007.

2. T. Jakubowski, K. Szczypkowski. Time-dependent gradient perturbationsof fractional Laplacian. Journal of Evolution Equations, 2009.http://dx.doi.org/10.1007/s00028-009-0051-5.

8

Recurrence and Transience of stable-likeProcesses

Bjorn Bottcher

bjoern.boettcher[-at-]tu-dresden.de

We have developed an overshoot approach to prove recurrence and transienceof one-dimensional Markov processes which have jumps and oscillate between+∞ and −∞.

In this talk we will concentrate on stable-like processes to illustrate thebasic ideas of the approach. In particular we show that a stable-like processwith generator −(−∆)α(x)/2 such that α(x) = α for x < −R and α(x) = β forx > R for some R > 0 and α, β ∈ (0, 2) is transient if and only if α + β < 2,otherwise it is recurrent.

As a special case this yields also a new proof for the recurrence, point recur-rence and transience of symmetric α-stable processes.

References:

1. An Overshoot Approach to Recurrence and Transience of Markov Pro-cesses, 2010, arXiv:1007.2055v1.

9

Brownian motion in Poissonian potential withrenormalized energy

Xia Chen

xchen[-at-]math.utk.edu

The model of Brownian motion in Poissonian potential describes a typicaltrajectory of a Brownian particle surviving from being attracted by the obstaclesrandomly located in the space (think about the stars in the universe). In theexisting literature, the random potential is defined as the convolution betweena Poissonian field and a bounded and locally supported function.

According to the Newton’s law of universal attraction and some other relatedlaws in physics, the most natural way of constructing the random potential isto define it as the Riesz potential of the Poissonian field. On the other hand,the Riesz potential of the Poissonian field blows up.

In this talk, this problem will be fixed by the way of renormalization. Inaddition, some asymptotic patterns of our models will be established and moreproblems will be asked. Part of the talk is based on some collaborative workswith Kulic and Rosen.

10

Modeling and simulation with operator scaling

Serge Cohen

serge.cohen[-at-]math.univ-toulouse.fr

Self-similar processes are useful in modeling diverse phenomena that ex-hibit scaling properties. Operator scaling allows a different scale factor in eachcoordinate. This paper develops practical methods for modeling and simulat-ing stochastic processes with operator scaling. A fast and accurate simulationmethod for operator stable Levy processes is developed, based on a series repre-sentation, along with a Gaussian approximation of the small jumps. The methodreproduces sample paths at an arbitrarily fine scale. Several examples are givento illustrate practical applications. Since operator scaling can be complicatedby symmetries, we provide a classification of operator stable Levy processes intwo dimensions according to their exponents and symmetry groups. When theexponent takes a Jordan form that is not a scalar multiple of the identity, allsymmetries must be orthogonal. This allows us to identify a unique “best” ex-ponent for operator stable Levy processes in two dimensions.

(joint work with Mark M. Meerschaert and Jan Rosinski)

http://www.math.utk.edu/∼rosinski/manuscripts.html

11

A non-anticipative calculus for functionals ofsemimartingales

RamaCont

Rama.Cont[-at-]gmail.com

We develop a non-anticipative calculus for path-dependent functionals of asemimartingale, using a notion of pathwise functional derivative proposed by B.Dupire. A key result is a change of variable formula which extends the Ito for-mula to functionals, without requiring Frechet or Malliavin differentiability. Asan application, we obtain an explicit martingale representation formula, which,in contrast to the Clark-Ocone formula, is based on non-anticipative quantitieswhich may be computed pathwise. Functionals defining local martingales arecharacterized as solutions of a functional differential equation for which we provea uniqueness result.

References:

1. R Cont & D Fournie (2010) A functional extension of the Ito formula,Comptes Rendus Mathematiques de l’Academie des Sciences Ser. I, Vol-ume 348, Issues 1-2, January 2010, Pages 57-61.

2. R Cont & D Fournie (2010) Change of variable formulas for non-anticipativefunctional on path space, 2009. To appear in: Journal of Functional Anal-ysis, Vol 259 (2010). http://arxiv.org/abs/1004.1380

3. R Cont & D Fournie (2009) Functional Ito calculus and stochastic integralrepresentation of martingales, http://arxiv.org/abs/1002.2446

http://arxiv.org/abs/1002.2446

12

Invariance principles for local times of Levyprocesses and random walks

Ron Doney

rad[-at-]ma.man.ac.uk

We prove that when a sequence of Levy processes X(n) or a normed sequenceof random walks S(n) converges a.s. on the Skorokhod space toward a Levyprocess X, the sequence L(n) of local times at the supremum of X(n) convergesuniformly on compact sets in probability toward the local time at the supremumof X. A consequence of this result is that the sequence of (quadrivariate) ladderprocesses (both ascending and descending) converges jointly in law towards theladder processes of X. As an application, we show that in general, the sequenceS(n) conditioned to stay positive converges weakly, jointly with its local timeat the future minimum, towards the corresponding functional for the limitingprocess X. From this we deduce an invariance principle for the meander whichextends known results for the case of attraction to a stable law.This is joint work with Loic Chaumont and will appear shortly in the Annalsof Probability.

13

Hausdorff and packing measures of Levy trees

Thomas Duquesne

thomas.duquesne at upmc.fr

In this paper we discuss Hausdorff and packing measures of random contin-uous trees called Levy trees . Levy trees have been introduced by Le Gall andLe Jan in 1998. Aldouss continuum random tree corresponds to the Browniancase. We provide results for the whole stable trees and for their level sets thatare the sets of points situated at a given distance from the root. We first showthat there is no exact packing measure for levels sets. We also prove that non-Brownian stable trees and their level sets have no exact Hausdorff measure withregularly varying gauge function. We also prove that Levy trees have an exactpacking measure.

14

Unlimited Liabilities, reserve capitalrequirements and the taxpayer put

Ernst Eberlein

eberlein[-at-]stochastik.uni-freiburg.de

When firms access unbounded liability exposures and are granted limitedliability, then an all equity firm holds a call option, whereby it receives a freeoption to put losses back to the taxpayer. We call this option the taxpayer put.We value this option and determine the level for reserve capital. The challengeis to model the risky asset and liability processes appropriately and to extractthe corresponding parameters from market data.

Assets excluding cash plus short term investments and liabilities less debtare modeled as exponentials of two Levy processes. The two Levy processesare modeled as linear mixtures of four independent Levy factors. Two of thesefactors drive assets and liabilities with positive correlations while two of theminduce negative correlations. Equity is then a call option on the spread ofassets over liabilities. We employ recently developed methods by Hurd andZhou (2009) to value these spread options using a two dimensional Fourierinversion. The reserve capital required by the taxpayer is determined by makingthe aggregate risk acceptable to the general external economy at stress level 0.75for the distortion minmaxvar. The compound spread option model is calibratedto equity option data at market close on year end to identify the joint law ofrisky assets and liabilities. This is joint work with Dilip Madan.

References:

1. E. Eberlein, D. Madan: Unlimited Liabilities, Reserve Capital Require-ments and the Taxpayer Put. Preprint Freiburg Institute for AdvancedStudies (FRIAS), 2010.

15

Modeling Network Traffic by a Cluster PoissonInput Process with Heavy and Light-Tailed FileSizes

Vicky Fasen

fasen[-at-]ma.tum.de

We consider a cluster Poisson model with heavy-tailed interarrival times andcluster sizes as a generalization of an infinite source Poisson model, where thefile sizes have a regularly varying tail distribution function or a finite second mo-ment. One result is that this model reflects long range dependence of teletrafficdata. We show that depending on the heaviness of the file sizes, the interarrivaltimes and the cluster sizes we have to distinguish different growths rates forthe time scale of the cumulative traffic. The mean corrected cumulative inputprocess converges to a fractional Brownian motion in the fast growth case. How-ever, in the intermediate and the slow growth case we can have convergence toa stable Levy motion or a fractional Brownian motion as well depending on theheaviness of the underlying distributions. These results are contrary to the ideathat cumulative broadband network traffic converge in the slow growth case toa stable process. Furthermore, we derive the asymptotic behavior of the clusterPoisson point process which models the arrival times of data packets and theindividual input process itself.

16

The numerical approximation of an SPDEdriven by a Levy noise

Erika Hausenblas

erika.hausenblas[-at-]sbg.ac.at

In this talk we present our result on the Numerical Approximation of Stochas-tic Partial differential Equations driven by Levy noise. Here, the main emphasiswill be on the simulation of a Levy noise. The project is a joint work AndreasProhl and Thomas Dunst from Tuebingen.

17

Metric Measure Spaces Associated withContinuous Negative Definite Functions and theBehaviour of Transition Functions of JumpProcesses

Niels Jacob

N.jacob[-at-]swansea.ac.uk

Using the fact that in many cases the characteristic exponent of a Levyprocess give rise to a metric we give a geometric description of a large class oftransition functions of Levy processes. The diagonal behaviour is controlled byvolume growth of balls measured by the intrinsic metric coming from the char-acteristic exponent whereas the off-diagonal decay is controlled by a furtherintrinsic metric and is then essentially as in the Gaussian case. The fact thatthe Gaussians are essentially fixed points of the Fourier transform leads to theuntypical situation that both metrics coincide. The result naturally suggestshow to handle more general processes, i.e. jump processes having a suitablesymbol: Riemannian geometry should be replaced by a geometry induced bymetrics (being x-dependent) being induced by continuous negative definite func-tions.

(This is joint work with Viktoria Knopova, Sandra Landwehr and ReneSchilling.)

18

Some questions in high-frequency statistics forsemimartingales

Jean Jacod

jean.jacod[-at-]upmc.fr

In the context of high frequency data, many questions have been solved inthe recent years: for example the estimation of the volatility is by now quitewell understood under very general hypotheses, as are some questions about thepresence and some specific features of the jumps when they are present.

In this talk we will give a survey of some recent advances on this topic,with emphasis on questions which are still open, especially in the case wherethe underlying process is a ”purely discontinuous” semimartingale. This in factamounts to the existence of appropriate limit theorems for discretized processes,when the discretization mesh goes to 0.

19

On a Heath-Jarrow-Morton approach for stockoptions

Jan Kallsen

kallsen[-at-]math.uni-kiel.de

In the Heath-Jarrow-Morton (HJM) approach in interest rate theory thewhole forward rate curve rather than the short rate is considered as state variablefor a stochastic model. Absence of arbitrage then leads to consistency and driftrestrictions, in particular the HJM drift condition.

Several attempts have been made to transfer this idea to options on a stock,cf. e.g. by Schonbucher (1999), Schweizer & Wissel (2008), Carmona & Nad-tochiy (2009), Jacod & Protter (2006). Here, the underlying stock plays the roleof the short rate. The implied volatility surface or a reparametrisation serves asstate variable and hence as counterpart of the forward rate curve in the classicalframework of HJM.

Our approach to this problem resembles Carmona & Nadtochiy (2009) inthat we try to preserve main features of the HJM setup. However, it is based ona different parametrisation or codebook, which allows to simplify both theoryand application.

(The talk is based on joint work with Paul Kruhner.)

20

Generalized fractional Levy processes withfractional Brownian motion limit

Claudia Klueppelberg

Cklu[-at-]ma.tum.de

Fractional Levy processes generalize fractional Brownian motion in a naturalway. We go a step further and extend the usual fractional kernel functions tothe more general class of regularly varying functions with the correspondingfractional integration parameter. This requires an extension of the Riemann-Liouville fractional integrals, which leads to a new fractional analysis. We invokethen this analysis to define stochastic integrals with respect to a generalizedfractional Levy process and investigate some of their properties, in particulartheir second order structure. For the fractional compound Poisson process (alsoknown as a special multiplicative Poisson shot noise process) with finite variancea functional central limit theorem to fractional Brownian motion is well-known.We extend this result to a general Levy framework; i.e. we prove a functionalcentral limit theorem for stochastic integrals of a generalized fractional Levyprocess. As a specific example we present our result for an Ornstein-Uhlenbeckprocess driven by a time scaled generalised fractional Levy process. This is jointwork with Muneya Matsui (Keio University).

21

Meromorphic Levy processes (and a newWiener-Hopf simulation technique)

Andreas E. Kyprianou

a.kyprianou[-at-]bath.ac.uk

We introduce a new and robust family of Levy processes (Meromorphic Levyprocesses) which permit semi-analytical tractability of certain barrier crossingproblems which appear throughout a number of different scenarios in appliedprobability. Moreover, our new class has pertinence to a new Monte Carlosimulation method, that we also outline in the talk, which makes heavy use ofthe explicit representation of the Wiener-Hopf factorisation that is available forthe Meromorphic class. This talk is based on joint work with Alexey Kuznetsov,Juan Carlos Pardo and Kees van Schaik.

References:

1. Kuznetsov, A., Kyprianou, A.E. and Pardo, J.C. (2010) MeromorphicLevy processes and their fluctuation identities. Preprint.

2. Kuznetsov, A., Kyprianou, A.E., Pardo, J.C. and van Schaik, K. (2009) AWiener-Hopf Monte Carlo simulation technique for Levy process. Preprint.

22

Strong solutions for stochastic differentialequations with jumps

Zenghu Li

lizh[-at-]bnu.edu.cn

General stochastic equations of non-negative processes with jumps are stud-ied. We provide criteria for the strong existence and uniqueness of solutionsunder non-Lipschitz conditions of Yamada-Watanabe type. The results are ap-plied to stochastic equations driven by spectrally positive Levy processes. (Thisis based on a joint work with Leonid Mytnik.)

http://math.bnu.edu.cn/∼lizh/research/papers.htm

23

More about limits of nested subclasses of classesof infinitely divisible distributions

Makoto Maejima

maejima[-at-]math.keio.ac.jp

Let I(Rd) be the class of infinitely divisible distributions on Rd and {X

(µ)t , t ≥

0} a Levy process on Rd with L

(

X(µ)1

)

= µ ∈ I(Rd), where L(X) denotes the

law of a random variable X. We define a mapping from a subclass D of I(Rd),which is the domain of the mapping, into I(Rd) by

Φf (µ) = L

(∫

∞

0

f(t)dX(µ)t

)

, µ ∈ D.

Many known subclasses of I(Rd) have been characterized as the ranges of map-pings with suitably chosen f ’s. There are many interesting problems related tothese mappings. In this talk, I pick up one problem that is to find the limitsof ranges of these mappings by their iterations. Even if the original subclassesare different, many different mappings give us one limit of nested subclassesgiven by the iterations of the mappings. It is the closure of the class of stabledistributions on R

d, as shown in Maejima and Sato (2009). Then a next naturalproblem is whether the closure of the class of stable distributions is only oneclass appearing as the limit given by such a procedure. We can construct severalmappings which produce different classes as their limits.

References:

1. M. Maejima and K. Sato, The limits of nested subclasses of several classesof infinitely divisible distributions are identical with the closure of the classof stable distributions. Probab. Theory Relat. Fields 145 (2009), 119-142.

2. M. Maejima and Y. Ueda, Compositions of mappings of infinitely divis-ible distributions with applications to finding the limits of some nestedsubclasses, Elect. Comm. in Probab. 15 (2010), 227-239.

3. M. Maejima and Y. Ueda, Nested subclasses of the class of α-selfdecomposabledistributions, preprint (http://arxiv.org/abs/1006.1047), 2010.

24

A sufficient condition for the continuity ofpermanental processes with applications to localtimes of Markov processes and loop soups

Michael Marcus

mbmarcus[-at-]optonline.net

We provide a sufficient condition for the continuity of real valued perma-nental processes. When applied to the subclass of permanental processes whichconsists of squares of Gaussian processes, we obtain the sufficient condition forcontinuity which is also known to be necessary. Using an isomorphism theoremof Eisenbaum and Kaspi which relates Markov local times and permanental pro-cesses we obtain a general sufficient condition for the joint continuity of the localtimes. We show that for certain Markov processes the associated permanentalprocess is equal in distribution to the loop soup local time.

25

Space-time duality for fractional diffusion

Mark M Meerschaert

mcubed[-at-]stt.msu.edu

Zolotarev proved a duality result that relates stable densities with differentindices. In this paper, we show how Zolotarev duality leads to some interestingresults on fractional diffusion. Fractional diffusion equations employ fractionalderivatives in place of the usual integer order derivatives. They govern scal-ing limits of random walk models, with power law jumps leading to fractionalderivatives in space, and power law waiting times between the jumps leading tofractional derivatives in time. The limit process is a stable Levy motion thatmodels the jumps, subordinated to an inverse stable process that models thewaiting times. Using duality, we relate the density of a spectrally negative sta-ble process with index 1.

(Co-authors on this work are:Boris Baeumer, Department of Mathematics and Statistics, University of Otago,PO. Box 56, Dunedin, New ZealandErkan Nane, 221 Parker Hall, Department of Mathematics and Statistics, AuburnUniversity, Auburn, AL 36849)

26

Symmetrization of Levy processes andapplications

Pedro Mendez

pedro.mendez[-at-]ucr.ac.cr

It is shown that many of the classical generalized isoperimetric inequalitiesfor the Laplacian when viewed in terms of Brownian motion extend to a wideclass of Levy processes. The results are derived from the multiple integralinequalities of Brascamp, Lieb and Luttinger but the probabilistic structure ofthe processes plays a crucial role in the proofs.

(This is joint work with R. Banuelos and was publish in Journal of FunctionalAnalysis Volume 258, Issue 12, 15 June 2010, Pages 4026-4051.)

http://blogs.emate.ucr.ac.cr/pmendez/files/2008/09/

symmetrization-levy.pdf

27

Stable limits for dependent stationary sequences

Thomas Mikosch

mikosch[-at-]math.ku.dk

The talk is based on the paper [1]. The aim is to provide conditions whichensure that the affinely transformed partial sums of a strictly stationary processconverge in distribution to an infinite variance stable distribution. Conditionsfor this convergence to hold are known in the literature; see e.g. [2,3,4]. However,most of these results are qualitative in the sense that the parameters of the limitdistribution are expressed in terms of some limiting point process; see [2]. Here,we will be able to determine the parameters of the limiting stable distributionin terms of some large deviation characteristics of the partial sums. We willapply our results to some standard time series models, including the GARCHprocess, and solutions to stochastic recurrence equations.

References::

1. Bartkiewicz, K., Jakubowski, A. Mikosch, T. and Wintenberger,

O. (2010) Stable limits for sums of dependent infinite variance randomvariables. Probab. Theory Rel. Fields to appear.

2. Davis, R.A. and Hsing, T. (1995) Point process and partial sum con-vergence for weakly dependent random variables with infinite variance.Ann. Probab. 23, 879–917.

3. Jakubowski, A. (1993) Minimal conditions in p-stable limit theorems.Stoch. Proc. Appl. 44, 291–327.

4. Jakubowski, A. (1997) Minimal conditions in p-stable limit theorems -II. Stoch. Proc. Appl. 68, 1–20.

28

Intertwining relation between fractionaloperators

Pierre Patie

ppatie[-at-]ulb.ac.be

29

On the exit times of Levy driven SDEs

Ilya Pavlyukevich

ilya.pavlyukevich[-at-]uni-jena.de

Let X0 be a one–dimensional dynamical system described by the differentialequation X0

t = −U ′(X0t ) with a unique stable point at the origin. We perturb

the system by a Levy process εL of a small amplitude ε, to obtain the stochasticdifferential equation dXε

t = −U ′(Xεt )dt + εdLt, and study the first exit times

of Xε from an interval [−a, b], a, b > 0, in the limit ε → 0. In particular,we consider perturbations by symmetric Levy processes whose jump measurehas power, sub– or super–exponential tails, and compare the results with thewell–known Gaussian case.

(Based on works with P. Imkeller and T. Wetzel, Humboldt–UniversityBerlin.)

30

Levy-Ornstein-Uhlenbeck processes in Hilbertspaces

Szymon Peszat

napeszat[-at-]cyf-kr.edu.pl

It is shown that the transition semigroup of a a Markov family defined bya Ornstein–Uhlenbeck equation driven by a Levy process is a second quantizedoperator on Poisson Fock space of the semigroup generated by the linear partof the equation.

31

Quasi-stationary distributions and Yaglom typelimits for self-similar Markov processes

Rivero Victor

rivero[-at-]cimat.mx

We consider positive self-similar Markov processes (pssMp), that is, posi-tive valued strong Markov processes having the self-similarity property, and forwhich 0 is a cemetery state. This class of processes and real valued Levy pro-cesses are in a one to one relation, viz. a pssMp, say X, is the exponential ofsome real valued Levy process, say ξ, time changed. In this work we study theexistence of quasi-stationary like measures and obtain Yaglom type limit resultsfor pssMp that hit zero in a finite time a.s. We establish that the existence ofquasi-stationary measures is closely related to the existence of a random variableR independent of ξ such that either

(∫

∞

0

eξsds

)

R ∼ Exponential(1),

or(

∫

∞

0

eξsds

)

R ∼ Pareto(γ), for some γ > 0.

The existence of a random variable R such that the former factorization holdswas established by Bertoin and Yor (2001) in the case where −ξ is a subordina-tor. We prove that this is in fact the only case where the former factorizationhold and provide NASC for the latter factorization to hold. Besides, we provethat a Yaglom limit type result hold for X if and only if the random variable∫

∞

0eξsds is in the maximum domain of attraction of a Gumbel or Frechet dis-

tribution, and provide sufficient conditions, in terms of the underlying Levyprocess, for this to hold.

32

Finite variation of Levy driven movingaverage-like processes

Jan Rosinski

rosinski[-at-]math.utk.edu

Processes of finite variation can be used as integrators for pathwise stochasticintegration. We investigate conditions under which Levy driven mixed movingaverages and the related generalized fractional processes have finite variation.This is closely related to the problem when such processes are semimartingales,previously investigated for moving averages by Basse, A. and J. Pedersen (2009).The talk is based on a joint work with Andreas Basse-O’Connor.

33

Geometric characteristics of the excursion setsover highl levels of non-Gaussian infinitelydivisible random fields

Gennady Samorodnitsky

gs18[-at-]cornell.edu

We consider smooth infinitely disivible random fields of the type (X(t), t ∈[−1, 1]d), with regularily varying Levy measure. For such random fields we areinterested in the geometric characteristics of the excursion sets

Au = {t ∈ [−1, 1]d : X(t) > u}

over high level u.For a large class of random fields we compute the asymmtotic (as u ? 8),

conditional on Au being non-empty) joint distribution of the of the numbersof critical points over the level u of all types. This allows us, for example, toobtain the asymptotic conditional distribution of the Euler characteristic of theexcursion set.

In a significant departure from the Gaussian situation, the excursion set overa high level for smooth random fields we are considering, can have complicatedgeometry. In the Gaussian case the excursion set, unless it is empty, is nearlycertain to be ”a ball-like” and have its Euler characteristic equal to one. Incontrast, the Euler characteristic of the excursion sets in our model can have ahighly non-degenerate conditional distribution.

34

Some applications of duality for Levy Processesin a half-line

Mladen Savov

mladensavov[-at-]hotmail.com

The central topic of this talk is an analytic duality relation for real-valuedLevy processes killed upon exiting a half-line. By Nagasawas theorem, thisyields a time-reversal identity involving the Levy process conditioned to staypositive. As examples of applications, we construct a version of the Levy processindexed by the entire real line and started from -infinity which enjoys a naturalspatial-stationarity property, and point out that the latter leads to a naturalLamperti-type representation for self-similar Markov processes in (0;∞) startedfrom the entrance point 0+.

35

Wavelet Solution of Kolmogoroff Equations forFeller-Levy Processes

Christoph Schwab

schwab[-at-]math.ethz.ch

We review the design and analysis of multiresolution (wavelet) methods forthe numerical solution of the Kolmogorov equations arising, among others, inno-arbitrage valuations of derivative contracts when Feller-Levy (FL) or additiveprocesses are used to model the dynamics of the markets.

In particular, the Dirichlet and free boundary problems connected to barrierand American style contracts are specified and solution algorithms based onwavelet representations of the FL processesDirichlet forms are presented.

We address in particular Feller-Levy processes with generators that give riseto Sobolev spaces of variable differentiation order (corresponding to a state-dependent jump intensity).

A copula construction for the systematic construction of parametric mul-tivariate FL-processes from univariate ones is presented. The domains of thegenerators of the resulting multivariate FL-processes are shown to correspondto Sobolev spaces nonconstant order of differentiation.

New multiresolution norm equivalences in such Sobolev spaces allow forwavelet compression of the matrix representations of the Dirichlet forms. Weaddres implementational aspects, in particular the regularization of the FL-processesDirichlet form and the singularity-free, fast numerical evaluation ofmoments of the Dirichlet form with respect to piecewise linear, continuousbiorthogonal wavelet bases.

Numerical experiments illustrate multilevel preconditioning of the momentmatrices for several exotic contracts as well as for Feller-Levy processes withvariable order jump intensities.

Model sensitivity of Levy models embedded into Feller classes is studiednumerically for several types of plain vanilla, barrier and exotic contracts.

36

Some properties of positive stable densities

Thomas Simon

simon[-at-]math.univ-lille1.fr

I will discuss several unimodality and monotonicity properties for positivestable distributions and their power transforms. Part of this research has beendone together with N. Demni (Bizerte).

37

Heat kernel estimates for Dirichlet fractionalLaplacian Perturbed by Gradient Operators inC

1,1 Open Sets

Renming Song

rsong[-at-]illinois.edu

Suppose that α ∈ (1, 2). Consider the fractional Laplacian perturbed bygradient operator −(−∆)α/2 + b(x) · ∇ on an bounded C1,1 open subset in Rd

with zero exterior condition. We will assume b = (b1, · · · , bd) is such that eachcomponent bi, i = 1, . . . , d belongings to an appropriate Kato class. In this talk,I will present sharp two-sided estimates for the heat kernel of such operators.

References:

1. Z.-Q. Chen, P. Kim and R. Song, Sharp density function estimates forFeynman-Kac semigroups of stable processes in open sets, preprint, 2010.

2. Z.-Q. Chen, P. Kim and R. Song, Heat kernel estimates for Dirichlet frac-tional Laplacian Perturbed by Gradient Operators in C1,1 Open Sets,preprint, 2010.

38

Multivariate SupOU Processes

Robert Stelzer

rstelzer[-at-]ma.tum.de

Multivariate supOU processes are defined using a Levy basis on the realnumbers times the set of square matrices with all eigenvalues having strictlynegative real parts. We discuss the existence, the finiteness of moments, thesecond order structure and important path properties, noting that the peculiar-ities of the underlying matrices cause new phenomena and features compared tothe known univariate case. In particular, we give precise conditions for the va-lidity of an analogue to the stochastic differential equation satisfied by Ornstein-Uhlenbeck type processes, which has been conjectured in the univariate case byBarndorff-Nielsen [2001, Superposition of Ornstein-Uhlenbeck type processes,Theory Probab. Appl. 45, 175-194], but not yet been proven. Our results alsoimply conditions when supOU processes are compatible with semimartingaleintegration theory.

Furthermore, we discuss the possible occurrence of long memory in multi-variate supOU processes and applications of them.

This talk is based on joint work with Ole Eiler Barndorff-Nielsen AarhusUniversity.

http://www.e-publications.org/ims/submission/index.php/AAP/

user/submissionFile/5327?confirm=4e9c2fa7

39

Asymptotics of Transition Densities of JumpProcesses

Pawel Sztonyk

Pawel.Sztonyk[-at-]pwr.wroc.pl

We investigate a class of jump–type Levy processes and the correspondingconvolution semigroups of measures and give estimates of their transition densi-ties. The primary examples of considered objects are the stable Levy processesbut our present study applies to the more general layered and tempered stableprocesses with marginal tails heavier than Gaussian but lighter than stable pro-cesses. We extend also the results for a class of Markov processes which are notnecessary transition invariant with jump intensities dominated by that of therotation invariant stable Levy proces

40

Technique for computing the PDFs and CDFs ofnon-negative infinitely divisible randomvariables

Murad Taqqu

murad[-at-]math.bu.edu

We present a method for computing the PDF and CDF of non-negativein finitely divisible random variables. This method uses the Levy-Khintchinerepresentation of the Laplace transform of the distribution. We apply the Post-Widder method for Laplace transform inversion combined with a sequence con-vergence accelerator to obtain accurate results. We demonstrate this techniqueon several examples including the stable distribution, mixtures thereof, and in-tegrals with respect to non-negative Levy processes. This is joint work withMark S. Veillette.

41

Local Malliavin Calculus for Levy Processes andApplications

Frederic Utzet

utzet[-at-]mat.uab.cat

In this talk we will cover the distance between, and put under the same um-brella, two excellent papers. The first one is the pioneering and original paperby Carlen and Pardoux [1], where a Malliavin calculus for Poisson process wasstudied. The second one is Nourdin and Simon [2] where the absolute continuityof the solution of a stochastic differential equation driven by a Levy process isstudied; in that paper there no explicit Malliavin calculus at all, but the readercan realize of deep relationships. Then we develop a Malliavin calculus for Levyprocesses based on a family of true derivative operators that extends Carlen andPardoux results, and we obtain sufficient conditions for the absolute continu-ity of functionals of the Levy process. Also, as an application, we analyze theabsolutely continuity of the law of the solution of some stochastic differentialequations.

(Joint work with Jorge Leon, CINVESTAV (Mexico); Josep Lluıs Sole, Univer-sitat Autonoma de Barcelona (Spain) and Josep Vives, Universitat de Barcelona(Spain))

References:

1. E. A. Carlen and E. Pardoux (1990) Differential calculus and integrationby parts on a Poisson space. In Stochastics, Algebra and Analysis inClassical and Quantum Dynamics, S. Albeverio et al. (eds), pp. 63-73.Kluwer.

2. I. Nourdin and T. Simon (2006) On the absolute continuity of Levy pro-cesses with drift. The Annals of Probability 34(3), pp. 1035-1051.

42

Fractional Levy processes: capturing stylizedfacts of empirical data

Jeannette H.C. Woerner

jwoerner[-at-]mathematik.uni-dortmund.de

Recently there have been proposed different methods for analyzing the em-pirical characteristics of financial data, e.g. behaviour of realized volatility, acertain correlation structure and paths regularity. Possible models capturingthese features might be models based on fractional Levy processes which com-bine the desirable properties of Levy processes with the correlation structure offractional Brownian motion. We consider different types of data, such as tick-by-tick stock data, daily index data and electricity data and show that theirspecific characteristic features, such as semiheavy tails, short- or long-rangedependence and spikes may be reproduced by this class of processes.

43

Packing Dimension Results for the Images ofLevy Processes

Yimin Xiao

xiao[-at-]stt.msu.edu

Let X = {X(t), t ≥ 0} be a Levy process in Rd with characteristic exponentΨ. This talk is concerned with fractal dimensions of the image X(E), whereE ⊂ R+ is a Borel set. Taylor (1986) proved that the packing dimension ofX([0 , 1]) is given by the index

γ′ = sup

{

α ≥ 0 : lim infr→0+

∫ 1

0

P {|X(t)| ≤ r}

rαdt = 0

}

.

We provide an alternative formulation of γ′ in terms of the characteristic ex-ponent Ψ. Secondly we extend the concept of packing dimension profiles, dueto Falconer and Howroyd (1997) and Howroyd (2001), and use our extensionto determine the packing dimension of X(E), where E ⊂ R+ is an arbitraryBorel set. This talk is based on joint works with D. Khoshnevisan and R. L.Schilling.

44

Regularity of solutions to spdes with Levy noise

Jerzy Zabczyk

zabczyk[-at-]impan.gov.pl

The talk is devoted to time and spatial regularity of Ornstein-Uhlenbeckprocesses with Levy perturbations. Extensions to solutions of non-linear equa-tions will be discussed as well. The following topics will be covered: CylindricalLevy processes; Examples of time irregular solutions; Stochastic continuity andstrong Feller properties of solutions; Spatial regularity for solutions of equationswith subordinated Wiener process. The presentation is based on joint researchwith Z. Brzezniak and E. Priola.

References:

1. Z. Brzezniak and J. Zabczyk, Regularity of Ornstein-Uhlenbeck processesdriven by a Levy white noise, Potential Analysis (2010), to appear

2. E. Priola and J. Zabczyk, Structural properties of semilinear spdes drivenby cylindrical stable processes, PTRF (2010), to appear

45