stochastic methods in finance and physics€¦ · nonequilibrium processes for current reservoirs...

Workshop on

Stochastic Methods in Finance and Physics

Book of Abstracts

Archimedes Center for Modelling, Analysis and ComputationHeraklion, Greece · 15–19 July 2013

Contents

1 Sponsors 1

2 Program 3

3 Abstracts 73.1 Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Posters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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1 Sponsors

FP7-REGPOT-2009-1, Grant Agreement 245749Archimedes Center for Modelling, Analysis and Computation

ERC Grant Agreement 258237Rough Path Theory, Differential Equations and Stochastic Analysis

Random Geometry of Large Interacting Systems and Statistical Physics (RGLIS)

Hochschuldialog mit Sudeuropa

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2 Program

Monday

9:00–9:15 Opening Address9:15–10:15 Jean-Dominique Deuschel

Quenched invariance principle for random conductance model I10:15–11:15 Alexandre Gaudilliere

Metastability and quasi-stationnary measures I

11:15–11:45 Coffee Break

11:45–12:45 Stefan GrosskinskyDynamics of condensation in the inclusion process

12:45–13:45 Nikolaos ZygourasContinuum limits for random pinning models and Wienerchaos expansions

13:45–15:15 Lunch and Coffee Break

15:15–16:15 Dimitrios TsagkarogiannisNonequilibrium Processes for Current Reservoirs

16:15–16:45 Rebecca NeukirchMetastability in the zero-range process

16:45–17:15 Paul GassiatPhysical Brownian motion in magnetic field as rough path

Tuesday

9:00–10:00 Hendrik WeberRecent progress in the theory of non-linear stochastic PDEs I

10:00–11:00 Jean-Dominique DeuschelQuenched invariance principle for random conductance model II


11:30–12:30 Claudio LandimMetastability of Markov processes I

12:30–13:30 Alexandre GaudilliereMetastability and quasi-stationnary measures II


15:00–15:30 Paul ChlebounTime scale separation in the low temperature East model

15:30–16:00 Martin SlowikLocal limit theorem for the random conductance model in adegenerate ergodic environment

16:00–17:00 Poster Session

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Wednesday

9:00–10:00 Claudio LandimMetastability of Markov processes II

10:00–11:00 Hendrik WeberRecent progress in the theory of non-linear stochastic PDEs II


11:30–12:30 Ines ArmendarizConvergence to equilibrium of the trajectories inHammersley’s process

12:30–13:30 Elena SartoriProbabilistic models for interacting agents facing binary decisions


15:00– Free Evening and Excursion20:30– Conference Dinner

Thursday

9:00–10:00 Ulrich HorstPrincipal-Agent Games and Equilibria under AsymmetricInformation I

10:00–11:00 Ulrich HorstPrincipal-Agent Games and Equilibria under AsymmetricInformation II


11:30–12:00 Michail AnthropelosAn equilibrium model for commodity spot and forward prices

12:00–12:30 Selim GokaySuperreplication when trading at market indifference prices

12:30–13:00 Christoph MainbergerOptimal supersolutions of convex BSDEs under constraints

13:00–13:30 Asgar JamneshanConditional set theory


15:00–15:30 Martin KlimmekOptimal martingale transport with Monge’s cost function

15:30–16:00 Harald OberhauserRoot’s and Rost’s solution of the Skorohod embedding problem

16:00–17:00 Michael KupperOn the duality of the superhedging price under model uncertainty

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Friday

9:00–10:00 Josef TeichmannTerm structure models in discrete time I

10:00–11:00 Josef TeichmannTerm structure models in discrete time II


11:30–12:30 Zorana GrbacAffine LIBOR models with multiple curves: theory, examplesand calibration

12:30–13:00 Sebastian RiedelRandom Fourier series as rough paths and applications to a classof SPDEs

13:00–13:30 Christos KountzakisCanonical modelling for coherent risk measures in dominatedvariation of tails


15:00–16:00 Christian BayerAsymptotics beats Monte Carlo: The case of correlated localvol baskets

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3 Abstracts

3.1 Talks

Michail Anthropelos (University of Piraeus)

An equilibrium model for commodity spot and forward prices

We consider a market model that consists of financial speculators and producers and con-sumers of a (consumption) commodity. Producers trade the forward contracts to hedge thecommodity price uncertainty, while speculators invest in these contracts to diversify theirportfolios. It is argued that the commodity equilibrium prices are the ones that clear out themarket of spot and forward contracts. Assuming that producers and speculators are utilitymaximizers and that the consumers’ demand and the exogenously priced financial marketare driven by a Levy process, we provide expressions for the equilibrium prices and analyzetheir dependence on the model parameters.

Ines Armendariz (University of Buenos Aires (UBA))

Convergence to equilibrium of the trajectories in Hammersley’s process

The particle trajectories in the Hammersley’s process exclude each other, therefore deter-mining a reflection mechanism. We apply techniques inspired by Mountford and Prabakhar’smethod to obtain a version of Burke’s theorem for continuous queues, in the Hammersley’ssetup, to conclude the convergence (under iteration of the reflection) to the stationary dis-tribution of the tagged particle trajectory. Joint work with Pablo Ferrari and Sergio LopezOrtega.

Christian Bayer (Weierstrass Institute)

Asymptotics beats Monte Carlo: The case of correlated local vol baskets

We consider a basket of options with both positive and negative weights, in the case whereeach asset has a smile, e.g. evolves according to its own local volatility and the drivingBrownian motions are correlated. In the case of positive weights, the model has been con-sidered in a previous work by Avellaneda, Boyer-Olson, Busca and Friz. We derive highlyaccurate analytic formulas for the prices and the implied volatilities of such baskets. Thecomputational time required to calculate these formulas is under two seconds even in thecase of a basket on 100 assets. The combination of accuracy and speed makes these formulaspotentially attractive both for calibration and for pricing. In comparison, simulation basedtechniques are prohibitively slow in achieving a comparable degree of accuracy. Thus thepresent work opens up a new paradigm in which asymptotics may arguably be used forpricing as well as for calibration. This talk is based on joint work with Peter Laurence.

Paul Chleboun (University of Warwick)

Time scale separation in the low temperature East model

We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spinsevolving under a simple Glauber dynamics in the presence of a kinetic constraint whichforbids flips of those spins whose left neighbour is 1. We focus on the glassy effects causedby the kinetic constraint as the equilibrium density of 0’s, given by q, tends to zero. Specif-ically we analyse time scale separation and dynamical heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium. For any mesoscopic length scale

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L = O(q−γ), γ < 1, we show that the characteristic time scales associated with two systemsizes, L and λL, are well separated provided that λ ≥ 2 is large enough. In particular,the evolution of mesoscopic domains, i.e. maximal blocks of the form 111..10, occurs on atime scale which depends sharply on the size of the domain, a clear signature of dynamicalheterogeneity. A key result for this is a very precise computation of the relaxation time ofthe chain as a function of (q, L), well beyond the current knowledge. Finally we show thatno form of time scale separation can occur on the equilibrium scale, contrary to what waspreviously assumed in the physical literature based on numerical simulations.

Refs: P. Chleboun, A. Faggionato, F. Martinelli, Time scale separation and dynamic het-erogeneity in the low temperature East model, arXiv:1212.2399 —, Time scale separationin the low temperature East model: Rigorous results, J. Stat. Mech. (2013) L04001

Jean-Dominique Deuschel (Technical University Berlin)

Quenched invariance principle for random conductance model

We consider a continuous time random walk on the lattice Zd in an environment of symmet-ric random conductances, µx,y. The law of the environment is assumed to be ergodic withrespect to space shifts with P(0 < µx,y < ∞) = 1. In this talk, we show how a quenchedinvariance principle can be established under suitable moment conditions. A key ingredientin the proof is to establish the sub-linearity of the corrector by means of Moser’s iterationscheme. We also get parabolick Harnack inequalities and quenched local limit theorems.This is joint work with Sebastian Andres (U Bonn) and Martin Slowik (TU Berlin).

Paul Gassiat (Technical University Berlin)

Physical Brownian motion in magnetic field as rough path

We consider the indefinite integral of the homogenized OU process, a well-known model forphysical Brownian motion. While uniform convergence, in the homogenization parameter,to Brownian motion holds generically, this is not true for Levy’s stochastic area e.g. inpresence of a magnetic field. The correction term is computed explicitly and agrees withthe expressions previously derived by PDE methods. Secondly, we establish convergence inrough path sense. This justifies viewing the trajectory of a charged Brownian particle withsmall mass in a magnetic field as a ”non-canonical” rough path lift of Brownian motion.Joint work with Peter Friz and Terry Lyons.

Alexandre Gaudilliere (CNRS - Marseille - LATP)

Metastability and quasi-stationnary measures

We will compare restricted ensembles and quasi-stationary distributions to describe themetastability phenomenon. This will be done by elementary spectral techniques to studyconvergence to local equilibrium and by introducing “soft measures” as an interpolationbetween quasi-stationary measures and restricted ensemble to recover in this way the “ex-ponential law”. We will then introduce some potential theoretic tools, and in particular“(κ, λ)-capacities” to compute all the relevant times via variational principal: exit, relax-ation, transition and mixing time.

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Selim Gokay (Technical University Berlin)

Superreplication when trading at market indifference prices

We study superreplication in a discrete-time market setting in a model recently proposedby Bank and Kramkov. In this model, market makers quote prices for the large investor’strades, where these quotes are determined by a certain interaction between the market mak-ers and the large trader. We provide conditions in this framework, when the superreplicatingcost is attained and provide many examples satisfying these conditions. (joint work withPeter Bank)

Zorana Grbac (Technical University Berlin)

Affine LIBOR models with multiple curves: theory, examples and calibration

In this talk we present an extension of the LIBOR market model with stochastic basisspreads, which is developed in the spirit of the affine LIBOR models. This multiple-curvemodel satisfies the main no-arbitrage and market requirements (such as nonnegative LIBOR-OIS spreads) already by construction. Furthermore, we clarify the connection between theaffine LIBOR setup and classical LIBOR market models. The use of multidimensional affineprocesses as driving processes ensures the analytical tractability of the model. We providepricing formulas for caps, swaptions and basis swaptions and discuss their efficient numer-ical implementation. This is joint work with A. Papapantoleon, J. Schoenmakers and D.Skovmand.

Stefan Grosskinsky (University of Warwick)

Dynamics of condensation in the inclusion process

The inclusion process is an interacting particle system where particles on connected sitesattract each other in addition to performing independent random walks. The system hasstationary product measures and exhibits condensation in the limit of strong interactions,where all particles concentrate on a single lattice site. We study the equilibration dynam-ics on finite lattices in the limit of infinitely many particles, which, in addition to jumpsof whole clusters, includes a continuous mass exchange between clusters given by Wright-Fisher diffusions. During equilibration the number of clusters decreases monotonically, andthe stationary dynamics consists of jumps of a single remaining cluster (the condensate),which happens on the same time scale as equilibration, in contrast to other well-studiedmodels such as the zero-range process. This is joint work with Frank Redig and KiamarsVafayi.

Ulrich Horst (Humboldt University Berlin)

Principal-Agent Games and Equilibria under Asymmetric Information

We discuss a mathematical framework within which we study various equilibrium problemsunder asymmetric information. The framework had originally been developed to establishexistence of equilibrium results for repeated games under incomplete information. We re-view the original model in the first session. Suitably extended, it is flexible enough, though,to allow for an analysis of much more general (cooperative or Principal-Agent) games andof optimal risk sharing problems with private information when both parties evaluate theirrisk exposures using convex risk measures.

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Asgar Jamneshan (Humboldt University Berlin)

Conditional set theory

We give an introduction into the theory of conditional sets. Conditional sets are structuresallowing for an action from a Boolean algebra, for example the σ-algebra of a probabilityspace. For conditional sets one can introduce conditional set operations which satisfy theBoolean laws as in classical set theory. Conditional sets together with a proper conceptof conditional functions constitute a place for mathematical discourse. We will show thereal numbers object of this universe for which we prove a Gap theorem. This allows us togive a continuous representation of conditional preferences. We sketch the scope of appli-cations which might be covered by conditional set theory and the mathematics built uponit. This is based on joint work with Samuel Drapeau, Martin Karlizcek and Michael Kupper.

Martin Klimmek (University of Oxford)

Optimal martingale transport with Monge’s cost function

We show how to derive an extremal martingale which has given marginal laws at two fixedtimes and is extremal in the sense that it minimises Monge’s cost function in a martingalesetting. ’Optimal martingale transport problems’ such as this one occur naturally in math-ematical finance. In our setting, the motivating issue is to find a model-independent lowerbound for the price of a forward starting straddle given two marginal laws. Moreover, thereis a dual problem which is to construct the corresponding sub-hedging strategy which workspath-wise for any realisation of the underlying asset price process. Unsurprisingly, there is acorrespondence between the dual (hedging) problem in mathematical finance and the dual(Kantorovich) formulation of Monge’s classical transportation problem.

Christos Kountzakis (University of the Aegean)

Canonical modelling for coherent risk measures in dominated variation of tails

We study the connection among properties of the heavy-tailed distributions, of the radialsubsets of random variables and of the coherent risk measures. A new adjusted expectedshortfall is introduced on radial subsets of heavy-tailed random variables. In the Lundbergand renewal risk models, the solvency capital is calculated in certain subclasses of distri-butions with dominatedly varying tails. Existence and uniqueness of the solution in theoptimization problem associated to the minimization of the risk over a set of financial po-sitions in the same classes of distributions is investigated. The optimization results hold onthe L1+ε-spaces, for any ε > 0, but they collapse on L1, which represents the canonical spacefor the law-invariant coherent risk measures.

Michael Kupper (University of Konstanz)

On the Duality of the Superhedging Price under Model Uncertainty

We give stability and duality results for supersolutions of BSDEs under model uncertaintyand discuss some links to the fundamental theorem of asset pricing. The talk is based onjoint works with Patrick Cheridito and Reinhard Schmidt.

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Claudio Landim (IMPA / CNRS Rouen)

Metastability of Markov processes

We investigate the metastable behavior of three models: zero-range processes which exhibitcondensation, random walks among random traps, and the Kawasaki dynamics for the Isinglattice gas in a large two-dimensional square with periodic boundary conditions.

Christoph Mainberger (Humboldt University Berlin)

Optimal Supersolutions of convex BSDEs under Constraints

We study supersolutions of a backward stochastic differential equation, the control processesof which are constrained to be continuous semimartingales of the form dZ = ∆dt+ΓdW . Thegenerator may depend on the decomposition (∆,Γ) and is assumed to be positive, jointlyconvex and lower semicontinuous, and to satisfy a specific growth condition in ∆ and Γ .We prove existence of supersolutions optimal at finitely many fixed times within the classwhere controls coincide up to these times. Furthermore, we provide a weak formulation ofoptimal supersolutions and discuss a Markovian setting. Given the generator is independentof y, we prove the existence of solutions that are ε-optimal at all times. Finally, we provideduality results within the present framework.

Rebecca Neukirch (University of Bonn)

Metastability in the zero-range process

The zero-range process is an interacting particle system on the lattice Zd, first mentionedby Spitzer 1970. In the talk we will consider a zero-range process on the one dimensionallattice S = {1, ..., L} with periodic boundary conditions and with N particles inside the sys-tem. The dynamic of the particles only depends on the occupation number of the actuallysite, hence the name ”zero-range”. The interesting feature of this model is, that in the highdensity phase it exhibits a condensation phenomenon, namely the zero-range process showscondensation behavior in the sense that a macroscopic fraction of particles is localized on asingle site under the canonical equilibrium measure.Beltran and Landim 2012 studied the metastable behavior of the zero-range process in onedimension on a finite box, S = {1, . . . , L}, in the limit where the number of particles tendsto infinity. In the talk, we will improve their results using the methods of the so-calledpotential theoretic approach to metastability, developed in the paper of Bovier, Eckhoff,Gayrard, and Klein 2002, also in the case that S is infinite. In particular, we show that themodel, considered in the limit, fits perfectly into the (simplest instance) of that approach,and that the definition of metastability given there applies, and that the abstract resultsof that paper provide the usual sharp estimates on mean metastable exit times and theirexponential distribution for the zero-range process. In addition, we show that some of theresults can be extended to the case when L = L(N) ↑ ∞ but L(N)/N ↓ 0.

Harald Oberhauser (Technical University Berlin)

Root’s and Rost’s solution of the Skorohod embedding problem

An intuitive solution of the Skorokhod embedding problem is due to Root and Rost whoshowed that there exists a subset of time-space such that its first hitting time solves theSkorohod embedding problem. We discuss some applications, connections to viscosity the-ory, FBSDEs, etc.

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Sebastian Riedel (Technical University Berlin)

Random Fourier series as rough paths and applications to a class of SPDEs

Fourier series with random coefficients are classical objects which appear in many applica-tions in pure and applied mathematics. A natural question is whether it is possible to lifttheir sample paths to rough paths (in the sense of Lyons) which would lead to a stochasticcalculus for random differential equations driven by such Fourier series. In the first partof our talk, we present sufficient criteria in terms of the Fourier coefficients under whichthis lift exists, provided the series gives rise to a Gaussian process. En passant, we revisita classical result of Jain–Monrad (cf. [Jain and Monrad, Gaussian measures in Bp, Ann.of Prob. 1983]) connecting regularity properties of the sample paths with the covariancefunction, and present a generalization to Gaussian rough paths.

In the second part of the talk, we discuss examples from stochastic PDE theory. Namely,we consider the stochastic heat equation

dΨt = ∂2xxΨt dt+ ξ, x ∈ [0, 2π], t ∈ [0, T ], (1)

ξ being space-time white noise. We discuss several modifications of this equation (differentboundary conditions, fractional Laplace operators, stationary and non-stationary solutions)and show that it can be interpreted as an evolution equation in a rough paths space. As aconsequence, equations of the form (1) with an additional non-linear term can be solved.This idea was first developed by Hairer in [Hairer, Rough stochastic PDEs, CPAM 2011]and later generalized to other equations. We comment on the implication of our results inthis direction.

This is joint work with Peter Friz (TU and WIAS Berlin), Benjamin Gess (HU Berlin)and Archil Gulisashvili (U Ohio).

Elena Sartori (Ca’ Foscari University of Venice)

Probabilistic models for interacting agents facing binary decisions

In this talk I present a class of stochastic models useful to represent the dynamics of asystem of many interacting agents. At each time, actors update their state maximizing theirown individual payoff, which depends on their action, on the state of the system and on arandom noise. I obtain stochastic dynamics for a small number of sufficient statistics that,at the aggregate level, describe the system. In the limit of infinitely many agents, a law oflarge numbers is derived. In particular, I put attention on two different mechanisms for theupdating: sequential and parallel. The former is more similar in spirit to classical statis-tical mechanics, whereas the latter mimics a non-cooperative game played by the agents.Depending on the chosen scheme, different limiting behaviors appear: the attractors canbe stable fixed points or periodic orbits. Moreover, the limiting dynamics may exhibit theexistence of a phase, where the two different types of attractors coexist. The talk is basedon joint works with Paolo Dai Pra and Marco Tolotti.

Martin Slowik (Technical University Berlin)

Local limit theorem for the random conductance model in a degenerate ergodicenvironment

We consider a continuous time random walk on the d-dimensional Euclidean lattice in anenvironment of random conductances, µx,y. The law of the environment is assumed to be

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ergodic with respect to space shifts with P[0 < µx,y < ∞] = 1. In this talk, I will explainhow a quenched local limit theorem can be established under suitable moment conditions.A key ingredient in proof is to establish a parabolic Harnack principle by means of Moser’siteration scheme.

This is joint work with Sebastian Andres (U Bonn) and Jean-Dominique Deuschel (TUBerlin).

Josef Teichmann (ETH Zurich)

Term structure models in discrete time

We present a self-contained and elementary theory for term structure models in discretetime, which can be applied to model the term structure of interest rates, of implied volatil-ity smiles, or of credit spreads.

Dimitrios Tsagkarogiannis (University of Crete)

Nonequilibrium Processes for Current Reservoirs

Stationary non equilibrium states are characterized by the presence of steady currents flow-ing through the system. Currents are produced by external forces and we are interested inforces acting on the boundary trying to establish a given current. We model this processconsidering the simple exclusion process in one space dimension with appropriate bound-ary mechanisms which create particles on the one side (right) and kill particles on theother (left). The system is then ”unbalanced” and in the stationary measure there is anon-zero steady current of particles flowing from right to left. The system is designed tomodel Fick’s law which relates the current to the density gradient. In statistical mechanicsnon-equilibrium is not as well understood as equilibrium, hence the interest from a physicalviewpoint to look at systems which are stationary yet in non-equilibrium: in our case thestationary process is in fact non-reversible and the stationary measure not Gibbsian. In thispresentation we give an overview of the results we obtained in collaboration with Anna DeMasi, Errico Presutti and Maria Eulalia Vares. We prove that the hydrodynamic limit isgiven by the linear heat equation with Dirichlet boundary conditions obtained by solving anon-linear equation which essentially fixes the values of the density at the boundary. Thenwe show that the rescaled limiting density profile of the (unique) invariant measure of theprocess coincides with the unique stationary solution of the hydrodynamic equation. Last,we obtain a spectral gap estimate in the (non equilibrium) stationary process uniformly onthe system size.

Hendrik Weber (University of Warwick)

Recent progress in the theory of non-linear stochastic PDEs

Nonlinear stochastic PDE arise as scaling limits of models in statistical physics near criticalpoints. These natural noise term that arises in these limits is typically quite rough - oftenspace time white noise.

In these lectures I will survey some recent developments when studying these non-linearequations. Among the examples discussed will be stochastic Reaction diffusion equationsand the KPZ equation.

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Nikolaos Zygouras (University of Warwick)

Continuum limits for random pinning models and Wiener chaos expansions

Random pinning phenomena appear when a one dimensional Markov process (e.g. simplerandom walk or more general discrete Bessel type processes) runs in the vicinity of a defectline, where an i.i.d. disorder, independent of the path, lies. Based on the relation betweenthe variance and the mean of the disorder the Markov process undergoes a phase transi-tion between localisation (i.e. a positive fraction of time is spent on the defect line) anddelocalisation (i.e. a zero fraction of time is spent on the defect line).

The quantitative features of the critical line of the phase transition depend non triviallyon the number of intersections of two independent copies of the underlying Markov process.In an attempt to understand this phase transition we study continuum limits of the modelby scaling accordingly the variance and the mean of the disorder. We express the continuumpartition function in the form of a Wiener chaos expansion. Of particular interest is the“marginal” case where for example the underlying process is a simple random walk. In thiscase the continuum object is not expressed in the form of a Wiener chaos, as the formalseries are not L2 convergent.

Joint work with F. Caravenna and R.F. Sun.

3.2 Posters

Tatiana Gonzalez Grandon (Berlin Mathematical School)

Mandelbrot’s Random cutouts to characterize the set of zeros of GWI and CBI

It is considered, as an example of a regenerative set, the closed subset of the nonnega-tive real-line left uncovered by a family of random open intervals formed from a Poissonpoint process (Mandelbrot’s construction of random cutouts). Then, based on Fitzsimmons-Fristedt-Shepp, it can be determined when this set is degenerate, bounded, its Levy measureand its fractal dimensions: the approach relies on Maissoneuve’s correspondence between re-generative sets and the closure of the image of a subordinator. I present this theory tracing aparallelism between discrete and continuous approach to finally construct by the procedureof random cutouts, the set of zeros of Bienayme-Galton Watson processes with immigrationand of continuous-state branching processes with immigration.

Xanthi-Isidora Kartala (Athens University of Economics and Business)

Rational Expectations - Consol Rate Models, Control for Large Investors, andStochastic Viscosity Solutions

We study a class of stochastic saddlepoint systems, represented by fully coupled forwardbackward stochastic differential equations (FBSDEs) with infinite horizon, that gives rise toa continuous time rational expectations / consol rate model with random coefficients. Understandard Lipschitz and monotonicity conditions, and by means of the contraction mappingprinciple, we establish existence, uniqueness and dependence on a parameter of adaptedsolutions. Making further the connection with quasilinear backward stochastic partial dif-ferential equations (BSPDEs), we are led to the notion of stochastic viscosity solutions. Astochastic maximum principle for the optimal control problem of a large investor is alsoprovided as an application to this framework.

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Illia Simonov (University of Leoben)

The Numerical approximation of solution to parabolic SPDEs forced by a Levynoise

Simulation of the Levy process is always complicated. We should always be aware of infiniteactivity of small jumps. There are some examples of simulation replacing small jumps withBrownian motion or even complete truncation, but depending on the process this is a veryrough approximation.

In this work we are interested in a stochastic differential equations of parabolic typedriven by a Levy noise with Lipschitz continuous and bounded coefficients.{

du(t, x) = Au(t, x) dt+ f(u(t, x)) dt+ g dL(t), t ∈ [0, T ],

u(0, x) = x.

We apply the Galerkin method to discretize the space domain. The time-discretizationis done via the exponential Euler scheme.

u1,k+1n,τ = eλ1τu1,kn,τ +

(eλ1τ−1)λ1

f1n(ukn,τ )

+∫ (k+1)τkτ e−λ1((k+1)τ−s)dL1(s)q1φ1

u2,k+1n,τ = eλ2τu2,kn,τ +

(eλ2τ−1)λ2

f2n(ukn,τ )

+∫ (k+1)τkτ e−λ2((k+1)τ−s)dL2(s)q2φ2

. . . = . . .

. . . = . . .

un,k+1n,τ = eλnτun,kn,τ +

(eλnτ−1)λi

fnn (ukn,τ )

+∫ (k+1)τkτ e−λn((k+1)τ−s)dLn(s)qnφn.

Laplace operator is taken as an initial condition. Eigenvalues and eigenvectors are wellknown.

We do the truncation of small jumps with parameter 0 < ε < 1, and do numerical sim-ulation of these jumps on every time step. Then we can simulate uk+1

n,τ .

Marios Stamatakis (University of Crete)

A Hydrodynamic Limit for the Zero Range Process in the presence of conden-sation

A hydrodynamic description of the empirical density µN of a symmetric nearest neighbourZero Range Process for which phase transition may occur with the appearance of a conden-sate, is not known. We prove relative compactness results for the processes induced by twoimportant but not conserved quantities at the microscopic level -the empirical diffusion rateσN and the empirical current WN and that all limit points of the sequence of laws of thetriple (µN , σN ,WN ) are concentrated on trajectories (µ, σ,W ) satisfying the equation

∂tµ = ∆σt = −divWt

in the sense of distributions.

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