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Stochastic models in turbulence
Alexandre J. Chorin
Department of Mathematics
University of California
Berkeley CA, 94720-3840
chorin@math.berkeley.edu
http://www.math.berkeley.edu/~chorin/

After a short summary of some stochastic tools used in turbulence over the years, I shall focus on randomized prediction  methods applied to the Navier-Stokes and related equations. The basic idea is to follow in time a relatively small number of "collective variables" (local means of solutions or coefficients in some apprropriate expansion), derive for them stochastic differential or integro-differential equations, and solve the latter by Monte-Carlo methods enhanced by suitable non-linear filters.

As one may expect, conditional expectations play a major role in the analysis; the relations between these prediction methods and stochastic methods in fields such as control theory and data assimilation will be described.
 

Monte Carlo simulations in finance and Malliavin calculus
Pierre-Louis Lions
CNRS and University Paris Dauphine, France

Monte-Carlo approximations for Boltzmann equations without cutoff
Sylvie Méléard
Université Paris 10
MODAL'X, UFR SEGMI
92000 Nanterre, France
sylm@ccr.jussieu.fr

We present a probabilistic interpretation of some spatially homogeneous Boltzmann equations wihout cutoff in terms of nonlinear Markov processes. We show how probabilistic tools allow us to obtain in some cases existence, regularity, and strict positivity of a solution and how this probabilistic approach gives naturally approximating interacting particle systems. We prove a pathwise convergence theorem by non standard coupling techniques and deduce from these theorical results simple trajectorial Monte-Carlo algorithms for the simulation of the solution.
 

Super-resolution in time-reversed signals
George Papanicolaou
Department of Mathematics
Stanford University,  Stanford CA 94305
papanico@math.stanford.edu
http://georgep.stanford.edu

I will present a theory and extensive numerical computations that describe and explain the enhanced reconstruction of a signal by time reversal when random inhomogeneities are present. Enhanced means that the reconstruction is better than the diffraction limit in time reversal in a homogeneous
medium.

Backward stochastic differential equations and applications to PDEs
Etienne Pardoux
University of Provence, France

Backward stochastic differential equations are stochastic equations which generalise both the Black and Scholes model in Mathematical Finance, and the equation for the adjoint state in the theory of stochastic control. They have application in Mathematical Finance and in Stochastic Control. They give probabilistic formulas for the solution of certain semilinear PDEs. The more general class of so--called ``forward--backward SDEs'' provides formulas for the solution of certain quasilinear PDEs.

The talk will survey the current state of the art of the theory, as well as some recent applications to PDEs, essentially to various results in homogenization.
 

Monte Carlo methods in neutron and photon transport equations
Remi Sentis
CEA, France
sentis@bruyeres.cea.fr

We first recall the principle and the main features of a Monte-Carlo method for solving  linear transport equations which describe the  evolution of a neutron density. Then, we deal with a simplified non-linear model for the time evolution of a photon energy density ;i.e. a transport equation with a source term propotional to the fourth power of the material temperature , that  transport equation is coupled with an ordinary differential equation satisfied by  the material temperature. For both models, we emphasize the limits of the Monte Carlo methods : i.e. when the mean free path is too small with respect to a typical simulation length. From a mathematical point of view, this situation corresponds to the case where the approximation of a transport equation by a  diffusion one is valid. We will propose some attempts to overcome these difficulties in such cases.

KINETIC MODELLING AND NUMERICAL METHODS

Organizers:  Benoit Perthame (ENS Paris, France) and Bruno Dubroca (CEA-CESTA, France)

Discrete Boltzmann-BGK Velocity Models for monoatomic and polyatomic rarefied gas flows
P. Charrier,
University Bordeaux I, France
B. Dubroca   and L. Mieussens
CEA-CESTA
BP numero 2
33114 Le Barp Cedex
dubroca@bordeaux.cea.fr

Computation of high altitude flows in the transitional regime between the kinetic and continuum limit are known to be difficult. Indeed, the flow is far from kinetic equilibrium and the Navier-Stokes equations are not enough accurate in boundary layers or across shock fronts.  To solve this problem, we present a numerical method for computing transitional flow as described by the BGK equation of gas kinetic theory:
$$\frac{\partial f}{\partial t} + \vec{v} . \nabla_x f =  {\frac{1}{\tau}} ({\cal E}(f) - f) $$

Using the minimum entropy principle to define a discrete equilibrium function, a discrete velocity model of this equation is proposed. This model, like the continuous one, ensures positivity of solutions, conservation of moments and local dissipation of entropy. The discrete velocity model is then discretized in space and time by an implicit finite volume scheme which is proved to satisfy the previous properties.

Despite the simplicity of the BGK model, several results have been obtained which compare well to DSMC, both for accuracy and CPU time. This method is also able to easily describe flows that are hardly computed by DSMC(such as recirculation zones or near continuum flows). Here, we propose also an extension of this method to polyatomic gas by using a polyatomic BGK model derived from the polyatomic Boltzmann equation.

An introduction to the mesoscopic modelling of polymeric fluids
Patrick Le Tallec
Ecole Polytechnique
91 128 Palaiseau Cedex, France
letallec@polynet.polytechnique.fr

Many materials are governed at a mesoscopic scale by a diffusion process describing the evolution of the average molecular structure of the constitutive materials. In such models, the motion of a molecular structure depends on the macroscopic velocity field, on the intermolecular forces exerted by the neighboring molecular structures, and on a white noise induced by the motion of particles at lower scales. This dynamics controls the distribution of the molecular structures in the configuration space, which governs in turn the macroscopic stress field and thus affects the overall macroscopic behavior of the system under consideration.

The purpose of the talk will be to present different kinetic models and their numerical implementation for approximating such multiscale phenomena, in the spirit of the pioneering work of Doi and Edwards.

On generalized BGK models for Maxwellian molecules
Jacques Schneider
Laboratoire Modelisation Numerique et Couplages, Universite de Toulon et du var,
83162 La Valette Cedex, France
jasch@isitv.univ-tln.fr
 http://www.univ-tln.fr/~mnc

Solving the Boltzmann equation by deterministic numerical methods requires a CPU consuming approximation of the collision integral $Q(f,f)$ (see e.g \cite{Rog}). Besides an attempt to reduce that cost by numerical improvement one may look for simplified models that possess most physical and mathematical properties of $Q(f,f)$. At a minimal level a simplified model must satisfy to mass, momentum and energy conservations and the law of dissipation of entropy. This is the case of the classical BGK model. In \cite{Luc}: in particular the authors show how the computation of the BGK relaxation term reduces to a minimization problem under constraints, allowing both reduction of CPU time and the use of a semi-implicit scheme. However the asymptotic fluid limit reveals the deficiency of that model: ratio between viscous and thermal transfers -so-called Prandtl number- is uncorrect. L. H. Holway (see \cite{And} for a complete study) has proposed a modified BGK model called Ellipsoidal Statistic that adds to the above minimal properties a correct fluid asymptotic, that is a Prandtl number that fits to the physic ($Pr \simeq 2/3$ in realistic case).  But his modification just acts on non-diagonal pressure terms (and heat flux in a certain sense) while higher moments are not taken into account. Even though higher moments have no physical meaning they play an important role in steep phenomena like shock wave or in the Knudsen layer.  This explains why both BGK and ES models fail to describe the proper physic in such region.

Another simplification of the Boltzmann equation consists in considering velocity moments of the density functions. The well-known Grad 13 moments' method \cite{Gra} and the modern "moment closure hierarchies for kinetic theories" by D. Levermore \cite{Lev} propose different ways to reduce the velocity dependant problem to a set of macroscopic equation. But still arises the problem of computing the moments of the collision integral. For Grad's method a diagonalization of the linearized collision operator simplifies that computation and can be extended to arbitrarily high moments of the density function. But eigenvectors of the linearised collision operator are not positive functions so that the approximation of the density function itself becomes negative at some places in the velocity space. This is an important drawback both on theoretical and numerical side. The moment closure hierarchies by D. Levermore does not suffer this drawback and is perfectly fitted to mathematics and physics: non-negativity of the approximate density function, hyperbolicity of the system of macroscopic moments at every level of the hierarchy and dissipation of the entropy. Important feature of those systems are:
- firstly the approximate density function is the solution of the entropy minimization problem under moments constraints, that is a generalization of the idea of Maxwellian distribution.
- secondly solution to that minimization problem exists only in domain of qualified (in the sense of convex analysis) moments constraints and one has to take care of what happens at the boundary of this domain, in particular in the vicinity of the equilibrium state where the so-called realizability problem arises.
- finally, the above generalization of Maxwellian distribution can be applied to construct BGK relaxation terms with pre-defined relaxation rates associated to a sequel of moments of $f$. Restriction to that generalization is that the sequel of relaxation rate must be increasing when considering increasing order of moments, in particular forbiding to recover the correct Prandtl number.

In this talk we propose to take advantages both on the BGK approach and D. Levermore's ideas to design relaxation terms of the form $\nu (G-f)$ that \textbf{fit exactly to the quadratic collision integral for a given number of moments}, that is:
\[   \lan \mu (G-f)m(\vv) \ran= \lan Q(f,f)m(\vv) \ran,\;\; \forall m(\vv) \in E \]
where the space of approximation $E$ is choosen a priori. Notice that including $1,\vv,|\vv|^2$ in $E$ implies the conservation of mass, momentum and energy for our new model. Notice also that if $E$ contains $[\vv \otimes \vv,\vv|\vv|^2]$ then the correct Prandtl number will be automatically recovered provided the collision integral $Q(f,f)$ describes correctly viscous and heat-flux transfers. We start our study by setting our problem in the framework of the linearized Boltzmann equation. Eigenvectors of the linearised operator $\cal{L}$ enables us to construct spaces of approximation $E$ (as in Grad's method). The generalised Maxwellian distribution is then defined as the Levermore solution of a minimisation problem which
constraints are based on the spectral properties of $\cal{L}$. Convexity argument allows us to prove the entropy decay in the linearised case. The non-linear case is treated approximately in the same way except that the constraints of the minimisation problem are now non-linear (but explicit) functions of the moments of $f$. Dissipation of entropy is proved by using results on Fisher information decay for Maxwellian molecules (see e.g \cite{Vil}).  Finally numerical issues are discussed.

\footnotesize{\begin{thebibliography}{10}
\bibitem{Rog} Rogier F. et Schneider J., A Direct Method for Solving the Boltzmann
Equation, proceedings of Colloque Euromech
n0287 Discrete Models in Fluid Dynamics, Transport Theory and Statistical Physics, Vol. 23, numéro 1-3, 1994.
\bibitem{Luc} Mieussens L., Un schéma numérique pour un modèle discret en vitesse de l'équation de Boltzmann,
Intern Report N° 98008 of the university of Bordeaux 1, France, 1998.
\bibitem{And} Andries P., Le Tallec P., Perlat J. P., Perthame B., The Gaussian-BGK Model of Boltzmann Equation
with Small Prandtl Number
\bibitem{Gra} Grad H., On the Kinetic Theory of Rarefied Gases, Comm. Pure and Appl. Math., 2, 331-407, 1949.
\bibitem{Lev} Levermore C. D., Moment Closure Hierarchies for Kinetic Theories, J. Stat. Phys., 83, 1021-1065, 1996.
\bibitem{Vil} Villani C., Fisher information bounds for Boltzmann's collision operator, J. Math. Pures Appl., 77,
821-837, 1998.
\end{thebibliography}
 

NUMERICAL METHODS FOR BOLTZMANN EQUATION

Organizers:  Benoit Perthame (ENS Paris, France) and Bruno Dubroca (CEA-CESTA, France)
 

Kinetic / fluid coupling : application to the SILVA enrichment process
Stephane  Dellacherie
CEA - CEN-Saclay/DCC/DPE/SPCP
91191 Gif sur Yvette, France
stephane.dellacherie@cea.fr

In the field of SILVA process (Atomic Vapor Laser Isotopic Separation), the French Atomic Energy Commission (CEA) has developed a code to simulate the expansion of the vapor of uranium evaporated by electron beam. This code solves with a particle Monte-Carlo method the semi-classical Boltzmann equations $$\partial_t f_i +\overrightarrow v\cdot\nabla_x f_i=\Sigma_{j,k,l}Q_{ij\rightarrow kl}$$  with $$Q_{ij\rightarrow kl}=\int\int\left(f_k(v')f_l(v'_{*}) \frac{g_ig_j}{g_kg_l}-f_i(v)f_j(v_{*})\right)B_{ij\rightarrow kl}dv_{*}d\Omega$$ where $i$, $j$, $k$ and $l$ are the quantum states of the uranium, $g_m$ is the statistical weight of the state $m$ and $B_{ij\rightarrow kl}$ is the collision kernel. Unfortunately, due to the existence of a very dense area - named fluid area, the remaining part being called kinetic area  - when the source flow of uranium is important, the CPU time needed to obtain the stationary solution of the Boltzmann equation is very high. The reason is that the rate of collisions is very high in the fluid area compared to the rate in the kinetic area - which implies that the vapor of uranium is at the local thermodynamical equilibrium in the fluid area - and the code has to simulate a lot of uranium particles to converge to the stationary solution: then, we have to consider the possibility of diminishing the CPU time by applying a special treatment in the fluid area. The idea is to apply a domain decomposition technic by solving the Boltzmann equation in the kinetic area and the fluid limit of the Boltzmann equation which is the classical hyperbolic system of Euler equations in the fluid area: this particular technique is called kinetic / fluid coupling.

In the first part, the Monte-carlo method used to solve the Boltzmann equations in this industrial field is presented. This method named {\it Particle Test Monte-Carlo} (PTMC: see \cite{Roblin}) differs from the classical DSMC method: indeed, it is based on an ergodicity hypothesis of the stationary solution of the Boltzmann equations to compute the distributions. From the CPU time point of view, the PTMC method is more efficient compared to the DSMC method when there are areas with high and small mean free path which is the case in the SILVA process. But, this method can only give stationary solutions of Boltzmann equations and then is not at all adapted to unstationary regimes. In the second part, the coupling technic, which is based on the kinetic schemes (see \cite{Perthame}), is described.The kinetic schemes used to solve the Euler equations are finite volume schemes. They are well adapted because they allow to define the boundary conditions at the kinetic / fluid interface in a very natural way (see \cite{Qiu}). At last, some numerical results are presented.
 

Simulation of kinetic equations  using quasi-random sequences
Christian Lecot
Christian.Lecot@univ-savoie.fr
Laboratoire de Mathematiques, Universite de Savoie 73376 Le Bourget-du-Lac cedex, France

Kinetic equations provide mathematical models for the statistical evolution of particles. A distribution function f(\mbox{\boldmath $x$},\mbox{\boldmath $v$},t)$, which represents the density of particles having position $\mbox{\boldmath $x$}$ and velocity $\mbox{\boldmath $v$}$ at time $t$, is solution of a non-stationary integro-differential equation. One method for approximating kinetic equations is random particle simulation. Particles are sampled from some known initial distribution. The time is discretized and pseudo-random numbers are used to move  the particles in phase space according to the dynamics described in the equation.

In this presentation we study the improvement achieved by using quasi-random sequences in place of pseudo-random numbers. We use $(t,s)$-sequences, which are sequences satisfying strong uniformity properties with respect to their distribution in $I^s := [0,1)^s$. Quasi-random points are not blindly used in place of pseudo-random numbers: at every time step, the number order of the particles is scrambled according to their positions before assigning a new quasi-random point to each particle. The method is to first sort the particles into slabs, according to their first coordinate. The particles of each slab are then sorted into boxes according to their second coordinate, and so on. Convergence is proved for quasi-random simulations using reordering of the particles. Numerical results are presented for model kinetic equations whose solutions can be found analytically. The quasi-random $(t,s)$-sequences of Faure and Niederreiter are compared with pseudo-random sequences.
 

Implicit Monte Carlo Methods for the Boltzmann equation
Lorenzo Pareschi
University of Ferrara
pareschi@dm.unife.it

For the Boltzmann equation, we formulate new Monte Carlo methods, that are robust in the fluid dynamic limit. The methods are based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution (a generalized Wild expansion). A class of unconditionally stable and explicitly implementable numerical schemes is obtained by relaxing the the high order terms in the expansion to the equilibrium Maxwellian distribution. Computational simulations by the new Time Relaxed Monte Carlo (TRMC) methods are presented here for the Variable Hard Sphere model. Comparison to exact solutions and to Direct Simulation Monte Carlo (DSMC) computations show the robustness and the efficiency of the new methods.
 

STOCHASTIC MODELS AND MONTE CARLO ALGORITHMS FOR PARTICLES TRANSPORT

Organizer:  Karl Sabelfeld (WIAS Berlin, Germany)

Forward and Backward Stochastic Lagrangian Models in turbulent transport and mathematical finance related through the well-mixed condition
K. Sabelfeld
Analysis and Stochastics, Mohrenstrasse 39, D -- 10117 Berlin, Germany,
sabelfel@wias-berlin.de
 and I. Shalimova
Institute of Computational Math. and Math. Geophysics, Siberian Branch Russian Acad. Sci., Novosibirsk, Russia
ias@osmf.sscc.ru

In modern Monte Carlo simulation algorithms, one often uses stochastic Lagrangian models to simulate individual Lagrangian trajectories and evaluate different Lagrangian statistical characteristics. We mention, for instance, the dispersion of particles in turbulent flows and the dynamics of the bond and stock prices in financial mathematics.  The Lagrangian stochastic models have a form of Ito type stochastic differential equation, and they are onstructed often in a quite heuristical way. There exists a more rigorous approach, which is based on an accurate description of the Eulerian dynamics. This dynamics however includes random fields, and there arises a nontrivial problem of consistency between the Lagrangian stochastic models and the stochastic dynamic models generated by the Eulerian random fields. The Lagrangian description allows us to analyze directly the motion of material fluid elements. Importance of the Lagrangian trajectories is that the quantities of practical interest are expressed through the n-particle statistical characteristics. In particular, the mean concentration of a passive scalar and its covariance are defined through the one-particle and two-particle statistical characteristics, respectively, and similarly, a hedging portfolio strategy can be expressed as an expectation over solutions to a large system of stochastic differential equations. In stochastic models of turbulent transport an important consistency between the Eulerian and Lagrangian probability density functions is the well-mixed condition due to Thomson. On the other hand, the Girsanov transformation is used in financial mathematics to decrease the variance. In this report we analyse an interrelation between the Girsanov transformation and Thomson's well-mixed condition, and suggest algorithms which use the forward and backward Lagrangian trajectories. It turns out that Thomson's algorithm corresponds to the case of Girsanov transformation with a special choice of the weight-function. On the other hand, the well-mixed condition of the stochastic turbulent models can be used to construct an effective numerical scheme based on the Girsanov transformation. It is interesting to note that the well-mixed condition applied to the stochastic models in mathemetical finance leads to essentially new models which involve nonstandard statistics, some analog of the Eulerian probability density function of the velocity field.
 

Quasi-Monte Carlo Methods for Elliptic Boundary Value Problems
Michael Mascagni, Aneta Karaivanova, Yaohang Li.
Department of Computer Science, Florida State University,
Tallahassee, FL, USA

A Quasi-Monte Carlo method for the solution of elliptic boundary value problems is proposed and studied.  Quasi-Monte Carlo methods employ low-discrepancy point sets in place of pseudorandom numbers in order to improve the slow, $O(N^{-1/2})$, convergence rate of ordinary Monte Carlo methods.  The proposed method uses a local integral representation of the solution whose integral transformation kernel can be thought of as a transition density function in a Markov process for estimating the solution. The stochastic process we use is the well-known random walk on balls. Quasirandom (low discrepancy) sequences, and the acceptance-rejection method are used for calculating the sequence of points in the Markov process.  The points chosen via this Markov chain are used as integration nodes in our integral equation representation of the elliptic boundary value problem.  We derive the accuracy and computational complexity of this method.  In addition we present numerical tests with different low discrepancy sequences such as those of Sobol', Halton, and Faure, and compare these quasirandom results to the results obtained with pseudorandom numbers.  We believe that many Monte Carlo algorithms can achieve a measure of enhanced convergence with the careful application of quasirandom numbers.
 

A New Fluid Permeability Estimation in Periodic Grain Consolidation Models of Overlapping and Nonoverlapping Sphere Models of Porous Media
Chi-Ok Hwang, Michael Mascagni
Department of Computer Science, Florida State University,
Tallahassee, FL, USA

Two new and efficient fluid permeability estimation methods are proposed and applied to periodic grain consolidation models of porous media and nonoverlapping and overlapping sphere models of porous media. The two methods use the average properties of continuous Brownian motion paths  that initiate outside a spherical sample and terminate upon contacting the porous sample.

The first method uses Brownian paths to calculate an effective capacitance for the porous sample, then relates the capacitance, via angle-averaging theorems, to the translational hydrodynamic friction of the sample. Finally, a result of Felderhof is used to relate the translational hydrodynamic
friction to the permeability of the sample.

The second method uses penetration depth, which is the average distance between the radial position where the Brownian motion paths are terminated and the radius of the sample. The permeability is given as the square of the penetration depth. In our calculations, capacitance and penetration depth are obtained simultaneously.

For the sampling of porous media, a new sampling method, the "sharp-boundary" method is proposed and used. We find that both methods are very good in predicting permeability of porous materials.

It is expected that the two new methods also work for other general homogeneous and isotropic porous media.
 

On a class of SPDEs providing probabilistic solutions for nonlinear diffusion equations
Shigeyoshi OGAWA
Laboratory of Applied Math. & Stochastics, Fac. of Technology, Kanazawa Univ., Japan
s_ogawa@t.kanazawa-u.ac.jp
 

The Monte Carlo algorithm for the numerical simulation of physical problems is quite often based on stochastic models of relevant phenomenon. Hence for those who concern with the numerical analysis of the PDEs of mathematical physics, it is of great interest to provide a stochastic model that may suggest us how to construct probabilistic solution, namely the solution whose average solves the specified PDE. In this paper, we are concerned with the Cauchy problem for a class of stochastic partial differential equations (SPDE) and the applications to some problems in mathematical physics. They are the SPDEs of hyperbolic type includining the Gaussian white noise as coefficients of the principal part, which we like to call as the "Brownian particle equations", for they may represent such transportation phenomena carried by virtual Brownian particles.

We will show that this class of SPDEs can serve as an alternative way of constructing the probabilistic solution for the Cauchy problem of some nonlinear diffusion equations. We will also refer to some applications of the idea to numerical analysis.

Here are some key words for the subject: Transport equation, Brownian motion, stochastic calculus, nonlinear diffusions, stochastic conservation law.

STOCHASTIC PARTICLE SYSTEMS AND COAGULATION EQUATIONS

Organizer : Wolfgang Wagner (WIAS Berlin, Germany)

Cluster Coagulation
James Norris
University of Cambridge, UK
J.R.Norris@statslab.cam.ac.uk

We introduce a general class of coagulation models, where clusters of given types may coagulate in more than one way and where the rate at which this happens may depend on the cluster types. In the continuum version of these models there is a generalization of Smoluchowski's coagulation equation.  We introduce a notion of strong solution for this equation and prove the existence of a maximal strong solution, which while it persists is the only solution.  When the total rate of coagulation for particles is bounded above and below by constant multiples of the product of their masses, we show that the maximal strong solution coincides with the maximal mass-conserving solution and does not persist for all time.  Thus, for these models, loss of mass (to infinity) coincides with divergence of the second moment of the mass distribution and takes place in a finite time. When the total rate of coagulation of large particles is proportional to their masses, we establish the existence and uniqueness of solutions for all time.  In a restricted class of `polymer' models, we allow coagulation of weighted shapes in a finite number of ways.  For this class we establish a discrete approximation scheme for the continuum dynamics.

For each continuum coagulation model, there is a corresponding finite-particle-number stochastic model.  We show that, in the polymer case, which includes the case of simple mass coalescence, as the number of particles becomes large, the stochastic model converges weakly to the deterministic continuum model, at an exponential rate.

Different Classes of Coagulation-Fragmentation Problems: Deterministic Versus Stochastic Modeling Approaches
Shay Gueron
University of Haifa, Israel
Dept. of Mathematics
Haifa 31905,  Israel
shay@mathcs2.haifa.ac.il
 

Smoluchowski's Coagulation Equation: Probabilistic Interpretation of Solutions for Constant,  Additive and Multiplicative Kernels
Madalina Deaconu, Etienne Tanre
INRIA Loraine,  France
Madalina.Deaconu@loria.fr
http://altair.iecn.u-nancy.fr/~deaconu/

We study the Smoluchowski's coagulation equation, discrete and continuous version, for the  case of constant, additive and multiplicative kernels. Even though, for the discrete case the results stated in this work are not new, our approach allows the simplification of existing proofs. For the continuous case we obtain new results: a connection between the solutions of the additive and multiplicative cases and renormalisation theorems which show that after a convenient scaling, the solution converges to a  limit which depends on the initial condition only throw its moments of
order 1, 2 and 3.

Some models of Coagulation and gelation
Intae Jeon
Catholic University of Korea,
injeon@www.cuk.ac.kr

We study the finite zero-range process with occupancy-dependent rate function $g(\cdot)$. Under the invariant measure, which can be written explicitly in terms of $g$, particles are distributed over sites and we regard all particles at a fixed site as a cluster. In the density one case, that is, equal numbers of particles and sites, we determine asymptotically the size of the largest cluster, as the number of particles tends to infinity, and determine its dependence on the rate function. We also consider the Smoluchowski coagulation equation with constant source of single particles and its stochastic counterpart. For the well known case of coagulation kernels  $K(i,j)=(ij)^{\alpha}, \ \ 1/2 <\alpha <1$ which exhibit gelation phenomenon, we derive explicitly the stationary solution. Moreover, we get an explicit bound of the explosion time of the tagged particle process associated with the stationary solution.

Stochastic algorithms for studying coagulation dynamics and gelation phenomena
Andreas Eibeck and Wolfgang Wagner
Weierstrass Institute for Applied Analysis and Stochastics
Mohrenstrasse 39
D-10117 Berlin, Germany
eibeck@wias-berlin.de
wagner@wias-berlin.de

The Markus-Lushnikov process provides a stochastic model for Smoluchowski's coagulation equation, and can be used as a direct simulation algorithm. Starting from a transformed equation, we derive an alternative stochastic particle system and prove convergence. Numerical investigations
show that the new process has better approximation properties, i.e. smaller systematic and statistical error, compared with the Markus-Lushnikov process. In particular, it allows us to study numerically the gelation (loss of mass) phenomenon.

Numerical Probabilistic Methods for Polymeric Liquids
Marco Picasso
Ecole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland.
office : MA C2 614, voice : +4121 693 42 47, fax : +4121 693 43 03.
Marco.Picasso@epfl.ch
http://dmawww.epfl.ch/~picasso

A numerical probabilistic model for the flow of a diluted solution of polymers is presented. The fluid is assumed to be a newtonian solvent  containing non interacting polymer chains. According to the kinetic theory of polymeric liquids, the polymer chains can be modelled as dumbbells, that is, two beads connected with a spring. Writing Newton's equations for the dumbbells leads to stochastic differential equations, from which the extra-stress due to the polymer chains can be computed. These equations are then coupled to the incompressible Navier-Stokes equations in order to solve the fluid flow.

A numerical procedure is presented, in which a Monte Carlo method is used to compute the extra-stress.  A finite element method is used to solve Navier-Stokes  equations. A variance reduction algorithm is proposed in order to reduce the noise on the extra-stress. The control variates correspond to dumbbells that yield a closure equation for the extra-stress.

Numerical results are presented on the 4:1 planar contraction flow.

INTERACTING PARTICLE SYSTEMS
 

Interacting Brownian Particles with Strong Repulsion
Dominique Lépingle
MAPMO, Université d'Orléans, BP 6759
45067 Orléans Cedex 2
lepingle@labomath.univ-orleans.fr

This is a joint work with E. Cépa.. The evolution of Brownian particles on the line with electrostatic repulsion has been studied for a long time because of its connection with the stochastic differential system of equations satisfied by the eigenvalues of a symmetric matrix with Brownian entries. When the number of particles tends to infinity, the empirical measure converges to the solution of a  partial differential equation with a Hilbert transform term; a non-linear stochastic differential equation with strong singularity in the drift term is displayed. A similar problem is considered on the circle, and analog results are also obtained in a hyperbolic setting.

A stochastic particle method for a 3D Boltzmann equation without cutoff
Nicolas Fournier,
Université Paris VI, France
fournier@proba.jussieu.fr
Sylvie Méléard
Université Paris 10
MODAL'X, UFR SEGMI
92000 Nanterre, France
sylm@ccr.jussieu.fr

Using the main ideas of Tanaka (1978), the measure solution $\{P_t\}_t$ of  a 3D$spatially homogeneous Boltzmann equation of Maxwellian molecules  without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to $\{P_t\}_t$ of solutions $\{P^l_t\}_t$  of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-M\'el\'eard, (1996) is used, and allows to approximate $\{P^l_t\}_t$ with the empirical measure of an easily simulable interacting particle system.
 

Long time behavior of stochastic particle system on the circle with mean field interaction
Anatoli Manita
Lomonosov Moscow State University,  Russia
manita@mech.math.msu.su

We describe the mean field limit of an interacting system of stochastic particles on the one-dimensional circle. We give examples of interaction kernels when stationary distribution of the limiting non-linear process is not unique. We prove some results about convergence in law of the non-linear  process.
 

BSDEs AND  PDEs

Organizer : Youssef Ouknine (Université Cadi Ayyad, Maroc)

Multidimensional BSDE's with Uniformly Continuous Coefficients
Said Hamadne
Université du Maine
Faculté des Sciences
Laboratoire de Statistique et Processus
Avenue Olivier Messiaen 72085 Le Mans CEDEX 9
hamadene@univ-lemans.fr

In this work we deal with the problem of the existence of a solution for the multidimensional BSDE with uniformly continuous coefficient. An existence result is given.

Some generic properties in backward stochastic differential equations
Khaled Bahlali
Université de Toulon, France
bahlali@univ-tln.fr

We prove that in the sens of Baire category, almost all backward stochastic differential equations with bounded continuous coefficients have unique solutions. That is; the subset of bounded continuous coefficients, for which the corresponding backward stochastic differential equations have unique solution, is of a second category of Baire.
 

BSDE and Homogenization of semilinear PDE
Y. Ouknine
Université Cadi Ayyad
Departement de Mathematiques  et Informatique, Facultedes Sciences et Techniques
B.P. 618, Gueliz, Marrakech, Maroc
ouknine@ucam.ac.ma

We study the homogenization of semilinear PDEs with nonlinear Neumann boundary condition, periodic coefficients and highly oscillating drift and nonlinear term. Our method is entirely probabilistic, and builds upon earlier work of Tanaka and Ouknine & Pardoux.
 

Organizer : Stefano Olla (Université de Cergy-Pontoise, France)

Diffusive behavior of interacting particle systems: bulk-diffusion and self-diffusion
Stefano Olla
Université de Cergy-Pontoise, France
olla@math.u-cergy.fr

An aqueous suspension of interacting particles can be described by two models: a system of interacting Ornstein-Uhlenbeck particles (Langevin equations) and a system of interacting Brownian particles (Smoluchowski equations). In a large (diffusive) space-time scale the two models have the same bulk diffusion coefficient (fluctuations of the density of particles). This can be proven by using hydrodynamic limit methods.  We expect instead that the two systems have different self-diffusion (diffusive behavior of a tagged particle). Then I will relate the tagged particle problem for the interacting  Ornstein-Uhlenbeck particles to other problems always concerning a central limit theorem for non-reversible Markov processes, like the passive tracer in random fields.

Tagged particles of interacting Brownian motions
Hirofumi Osada
Nagoya University, Japan
osada@math.nagoya-u.ac.jp

I will talk about an invariance principle of a tagged particle of interacting Brownian motions with hard core, and also about the positivity of self diffusion matrix. I will prove this for arbitrary activity z of the gran canonical Gibbs measure when the dimension of the space d is greater than or equal to 2. I also consider the skew symmetric interactions.

Weak solution of semi-linear PDE, BSDE and homogenization
Antoine Lejay
Universite de Provence, France
Antoine.Lejay@sophia.inria.fr

The homogenization propety for a system of semi-linear parabolic PDEs is proved using the corresponding BSDE, either in periodic and random media. However, the second-order differential term in the linear part of the operator is a divergence form operator, whose coefficient is not assumed to be regular.  The differences in considering weak solutions and Dirichlet processes instead of classical solutions and Ito diffusion process will be pointed out.

Organizer :  Etienne Pardoux (Universite de Provence, France)

A Monte-Carlo method for discrete-time filtering
Jean Jacod
Universite Paris VI, France

We consider a Monte-Carlo method based on interacting particle systems to approximate the non-linear filter in a discrete-time setting. Some general results are proved, including central limit theorems, and applications to discretely and partially observed continuous-time system are given. We also give some results on the asymptotic behaviour of this method in some special cases.

This talk is based on several joint papers with P. Del Moral and P.  Protter.

Observation Quantisation for Nonlinear Estimators
Nigel J. Newton
Department of Electronic Systems Engineering,
University of Essex,
Wivenhoe Park, Colchester, CO4 3SQ, UK

The use of coarsely quantised observations with nonlinear estimators can substantially reduce the computational complexity of numerical implementations. For example, the nonlinear liklihood functions for quantised observations have finite support, and so can be pre-computed for each possible value of the observation and stored for rapid retrieval in real-time implementations.

The talk will concern the characterisation of the loss of information arising e from the use of observation quantisation, and the numerical benefits it brings. Both estimators with discrete- and continuous-time observations will be considered.  In the former case, the loss of information for estimators with substantially noisy observations sequences can be quantified by means of a functional central-limit theorem, which enables the optimisation of the quantisation parameters, and shows that the loss of information is small, even for coarse quantisation schemes. A similar result applies to the latter case, where it takes the form of as a `stochastic over-sampling theorem'.
 

Regularized particle filters
Francois Le Gland
IRISA / INRIA, Campus de Beaulieu, 35042 RENNES, France
legland@irisa.fr

Particle filters have been proposed recently, in which the optimal filter is approximated by the empirical distribution of a particle system. The particles move independently according to the transition probability kernel of the hidden state, and whenever a new observation is available, resampling occurs in which the particles are selected according to their consistency with the observation (as measured by the likelihood function). The positive effect of the resampling step is to automatically concentrate particles in regions of interest of the state space.

Our contribution has been to propose modifications of the original method, so as to handle efficiently the (difficult) case where the state and / or observation noise are small. The approach is to regularize the empirical distribution associated with the particle system, using a kernel method. In this talk, we will present some implementation details, and we will give error estimates that hold uniformly in time, under some assumption about the transition probability kernel of the hidden state.

This talk is based on joint work with Christian Musso and Nadia Oudjane.

Organizer : Josselin Garnier (CMAP-Ecole Polytechnique, France)

A Wave Automaton for Wave Propagation in Random Media
Michel Pacilli, Patrick Sebbah and Christian Vanneste
Laboratoire de Physique de la Matière Condensée
Université de Nice-Sophia Antipolis
Parc Valrose, 06108 Nice Cedex 2, France
vanneste@unice.fr sebbah@unice.fr

The wave automaton is a numerical method, which uses elementary processes obeying a discrete Huygens' principle to describe wave propagation in random media. It belongs to a family of related models using networks of unitary scattering matrices in different areas of physics: lattice Boltzmann wave model, quantum cellular automata, and Transmission Line Matrix Modeling method...
By properly adjusting the parameters of the model, it can be made equivalent to scalar (Klein-Gordon, Schrödinger), spinor (Dirac) or vector wave equations (Maxwell). Different applications ranging from Anderson localization to wave scattering in random anisotropic media are presented.

Real-time radiative-transfer Monte Carlo results for biomedical applications
E. TINET, J.M. TUALLE, S. AVRILLIER
Laboratoire de Physique des Lasers, UMR7538
99 Av. J.B. Clement
93430 Villetaneuse
FRANCE
tinet@galilee.univ-paris13.fr

The main advantage of Monte Carlo simulations in radiative transfer problems is well known: their ability to deal with any situation regarding the emission-reception geometry, the medium boundaries, and the scattering characteristics of the medium. Any quantity, such as spatial, angular and temporal energy distributions can be calculated. Their main drawback is also well known: calculation time may be prohibitively long. For biomedical diagnosis, using reasonably cheap computers, it is nearly impossible to perform interactively Monte Carlo simulations with a sufficient accuracy. But, using pre-computed results stored in adequate formats, it is possible to obtain results sufficiently fast to be incorporated in some inverse-problem procedure. A Monte Carlo simulation may indeed be divided into two parts: the first part is the "information generator", the simulator itself, which evaluates the particles histories, and the second part, the "information processor", which uses tables created by the first part to evaluate the desired quantities.

Wave propagation in nonlinear and random media
J. Garnier
Centre de Mathematiques Appliquees
Ecole Polytechnique
91128 Palaiseau Cedex
France
garnier@cmapx.polytechnique.fr

The investigation of the competition between randomness and nonlinearity for wave propagation phenomena in the one-dimensional case  is of great interest for applications in nonlinear optics and optical transmission systems. As it is well known, in one-dimensional linear media with random  inhomogeneities strong localization occurs, which means in particular that the transmitted intensity decays exponentially as a function of the size of the medium. On the other hand, in some homogeneous nonlinear media wavepackets called solitons can propagate without change of form or diminution of speed. We consider a randomly perturbed continuum nonlinear Schrödinger equation or a lattice version of this equation with the initial condition given by a soliton.  We study the interplay of nonlinearity, randomness, and discreteness by applying a perturbation theory of the inverse scattering transform.

MCMC ALGORITHMS IN SIGNAL AND IMAGE PROCESSING

Organizer :  Josiane Zerubia (INRIA Sophia Antipolis, France)

An Introduction to Monte Carlo Methods for Bayesian Signal and Image Analysis
William J. Fitzgerald , Christophe Andrieu and Arnaud Doucet
Signal Processing Laboratory, Department of Engineering,
University of Cambridge, CB2 1PZ Cambridge, UK.
wjf, ca226, ad2,@eng.cam.ac.uk

An Introduction to Monte Carlo Methods for Bayesian Signal and Image Analysis In many problems encountered in statistical signal and image processing, it is possible to describe accurately the underlying statistical model using probability distributions. A natural framework that allows one to take into account both the information given by the observations and prior information is the Bayesian framework.  In this talk, we will adopt  a Bayesian approach which considers all unknown parameters to be random variables.

This approach is now widespread in the applied statistics community but not common in many other fields related to signal and image analysis. The Bayesian approach suffered for a long time from severe practical  limitations. That is to say, except for a few simple cases, Bayesian inference cannot be performed analytically as it requires integration and/or maximization of complex multi-dimensional functions.

Most problems encountered in applied research (speech processing, communications, spectral analysis, target tracking etc.) require non-Gaussianity and non-linearity in order to correctly account for the observed data. In such situations it is not possible to develop closed-form estimators based on the standard criteria. One approach to solve this problem is either to make model simplifications or analytic approximations in order to obtain algorithms that can be implemented in closed-form (this is the basis of the extended Kalman filter, for example). However, with the recent availability of high powered computers, numerical simulation based approaches can now be considered and the full complexity of real problems can now be addressed. This talk will develop the theory of Markov Chain Monte Carlo (MCMC) and it will address several application areas taken from signal and image processing.
 

A Bayesian approach for pulse deinterleaving
C. Theys,
lab. I3S, Les Algorithmes, 2000, route des lucioles
BP 121, 06903 Sophia Antipolis cedex - France
Celine.Theys@unice.fr
A. Ferrari, G. Alengrin
UMR 6525 Astrophysique, Universite de Nice Sophia-Antipolis, Parc Valrose
06108 Nice CEDEX 2, France

In passive detection of radars, periodic pulse trains transmitted from a known number of sources are received by a single receiver. The deinterleaving problem is to determine which source contributed to each pulse using a noisy measurement of the pulses time of arrival. The first step consists in choosing a model for the data. From this model,  a posterior distribution for the parameters is obtained. This density is too complicated to evaluate its maximum or its mean analytically, so a Markov Chain Monte Carlo algorithm  is used to draw samples from the distribution. Some simulated examples are presented to illustrate the algorithm's performance.
 

Classification of Digital Modulations using MCMC Methods
Jean-Yves Tourneret, Stéphane Lesage
ENSEEIHT, 2 rue Camichel,
BP 7122, 31071 Toulouse Cedex 7,France
tournere@len7.enseeiht.fr

The classification of digital modulations consists of determining the modulation type of an intercepted signal corrupted by noise and interference. This challenging problem has received much attention for several years. Once a signal is detected, the modulation type and its parameters have to be determined to select the demodulation scheme. Optimal classifiers (minimizing an appropriate cost function such as the error rate), based on the Bayes or Neyman Pearson criteria can be derived when statistical properties regarding signal and noise are available. However, these classifiers suffer from high computational complexity. Moreover, the classification is not robust with respect to model mismatch such as phase and frequency offsets or timing errors. Consequently, suboptimal classifiers based on likelihood approximations or feature extraction have been proposed in the literature. This paper studies a Bayes classifier implementation using MCMC methods. MCMC methods allow to simulate phase and frequency offsets and/or timing errors according to their a posteriori distribution. These simulations are then used to mitigate model mismatch effects.
 

RJMCMC methods in image processing
Xavier Descombes, Sebastien Drot, Mikael Imberty, Radu  Stoica, Josiane Zerubia
Ariana, joint project CNRS/INRIA/UNSA, INRIA, Sophia Antipolis, France
Firstname.Lastname@sophia.inria.fr
http://www-sop.inria.fr/ariana/

We consider Markov Object Processes as priors in a Bayesian framework to solve image processing problems. The objects model the features we want to detect (roads, homogenous areas,...). This alternative to pixelwise models is more robust w.r.t. noise because the likelihood is computed over
the whole object. Moreover, some geometrical constraints can be modelled within this framework. Some previous works have been made in that  direction (Baddeley et al 1992, Rue et al 1998).  We define two kinds of models for both extracting the road networks (Descombes et al 1999) and segmenting an image (Imberty et al 2000). Some interactions between object allows to describe the roads network as a graph composed of segments and an homogenous area as a collection of simple polygons (rectangles, triangles). The likelihood is based on statistical tests assessing the data  homogeneity inside an object. To optimise the models we have to consider some transitions between spaces of different dimensions. Therefore, a RJMCMC algorithm is derived for each model, according to the expected structure of the result.

MCMC METHODS FOR HIDDEN MARKOV MODELS

Organizer : Christian P.  Robert (CREST, France)

On-line Bayesian Parameter Estimation of Hidden Markov modeles
Christophe Andrieu and Arnaud Doucet
Signal Processing Group, Cambridge University Trumpington Street CB2 1PZ Cambridge, UK
Emails: ad2@eng.cam.ac.uk

Many adaptive algorithms have been recently proposed to perform parameter estimation of hidden Markov models (HMM). They are mainly gradient-type methods designed to perform approximate maximum likelihood estimation. We present here an alternative Bayesian approach where the parameters are considered random of known prior distribution. Bayesian inference is then based on the sequence of  evolving posterior distributions. Original sequential Monte Carlo (i.e. particle filtering) methods are developed to solve this estimation problem.
 

Recursive maximum likelihood estimation for general state space models based on particle approximation
O. Cappé, R. Douc, and E. Moulines
ENST Dpt. TSI / LTCI (CNRS URA 820) 46 rue Barrault 75634 Paris cedex 13
Emails: appe, douc, moulines@sig.enst.fr

Discrete-time state space models (hidden Markov models, Markov switching autoregressive models) are used in a variety of applications which range from finance (stochastic volatility models), to signal processing (deconvolution applications) or teletraffic analysis (doubly stochastic Poison  process). Iterative maximum likelihood approaches applicable when the state space of the dynamic variables (non-observable part of the model) is  inite have been known since the 70s. Recent works by Mevel and Le Gland (1997) and others suggest a general approach for recursive maximum likelihood identification based on the  propagation of the one step prediction density and its gradient with respect to the model parameters. In this contribution, extension of this approach to cases where the prediction update cannot be carried out exactly is considered. The proposed solution is based on the particle filtering (or sequential Monte Carlo) approach which is used to recursively approximate the gradient of the one step log-likelihood update. An associated stochastic approximation algorithm is described and its application to the stochastic volatility model is described.
 

Marginal Maximum A Posteriori Estimation using Markov Chain Monte Carlo
Arnaud Doucet and Christian P. Robert
Signal Processing Group, University of Cambridge Trumpington Street CB2 1PZ Cambridge, UK
Statistics Laboratory, CREST, INSEE 92245 Malakoff cedex, France
Emails:  ad2@eng.cam.ac.uk, robert@ensae.fr

Markov chain Monte Carlo (MCMC) methods, while facilitating the solution of many complex problems in Bayesian inference, are not currently well adapted to the problem of marginal maximum a posteriori (MMAP) estimation, especially when the number of parameters is large. We present a simple and novel Markov Chain Monte Carlo (MCMC) strategy, called State-Augmentation for Marginal Estimation (SAME), which leads to MMAP estimates for Bayesian models. We illustrate the simplicity and utility of the approach for missing data interpolation in autoregressive time series and blind deconvolution of impulsive processes.
 

Bayesian Analysis of autoregressive models with unknown order
Anne Philippe
Laboratoire de Probabilites et Statistique, CNRS F.R.E. 2222  UFR de Mathematiques -- Bat M2
Universite des Sciences et Technologie 59655 Villeneuve d'Ascq Cedex, France
Email: Anne.Philippe@univ-lille1.fr

We study from a Bayesian point of view the estimation of order and of parameters for autoregressive models. To evaluate the joint posterior distribution, we propose a MCMC algorithm which allows to move between spaces of different dimensions. For the mixtures with an unknown number of components, Stephens (1998) develops an algorithm based on the construction of a continuous time Markov birth-death process. We follow this methodology and we establish the conditions on the rates at which births and deaths occur to obtain ergodic Markov chain with the joint posterior distribution as its stationary distribution. The performance of these algorithms is illustrated through applications to simulated data and compared with classical methods.

MONTE CARLO METHODS FOR TURBULENCE SIMULATIONS

Organizer : Peter   KRAMER (Courant Institute, New York University, USA)

Review of Some Methods for Simulating Synthetic Turbulence
Peter Kramer
kramerp@courant.nyu.edu
Address through August 2000:  Courant Institute of Mathematical Sciences New York University
251 Mercer Street  New York, NY 10012    USA
Address starting September 2000:  Department of Mathematical Sciences  Amos Eaton Hall
 Rensselaer Polytechnic Institute  110 Eighth Street  Troy, NY 12180  USA

We review four approaches to constructing random velocity fields for numerical simulations: a direct Fourier method, a Fourier method with randomized  choice of wavenumbers, a moving average method, and a wavelet-based method.  We discuss their relative merits in simulating a random  velocity field with self-similar scaling properties mimicing the inertial range of a fully developed turbulent flow.  The results of some numerical  simulations of pair dispersion within the inertial range of a wavelet-based synthetic velocity field are discussed, with implications regarding richardson's law.  This presentation draws largely on the work of Elliot, Horntrop, and Majda.

Lagrangian Monte-Carlo methods in scalar turbulence
Massimo Vergassola
massimo@obs-nice.fr
 CNRS  Observatoire de la Cote d'Azur  BP 4229, 06304 Nice Cedex 4  France

Monte-Carlo methods for the advection-diffusion of passive scalar fields will be discussed.  The method is based on the simulation of tracer  particle  Lagrangian trajectories,  supplemented by a point splitting procedure for coinciding points. Applications will include both the non-perturbative regimes of the Kraichnan  passive scalar model and advection by Navier-Stokes velocity fields.

Probabilistic approach to two-phase flow modelling and simulation :  theoretical and numerical issues
Jean-Pierre Minier
EDF/DER/LNH 6 Quai Watier 78400 Chatou FRANCE
Jean-Pierre.Minier@der.edfgdf.fr

Turbulent two-phase flows, for example particle-laden flows or gas-liquid flows, are currently encountered in numerous industrial processes. Their understanding  and simulation is therefore an important practical issue. Yet, theoretical modelling as well as numerical simulation are still at an early stage and many questions remain to be addressed. The purpose of this article is to present the stochastic approach to two-phase flow modelling. The justification and the interest of this approach is first underlined with respect to given objectives. We then discuss the form of the general equations and the need of a  rigorous and complete (including the two phases) probabilistic framework in  which precise models can be developed and analysed. The particular form of the current state-of-the-art models is not detailed since it is related to physical modelling. The emphasis is mostly put on algorithmic and numerical issues where much work remains to be done. We discuss  constraints on the numerical schemes used to simulate the stochastic differential equations which appear, and the need of suitable Monte Carlo methods to limit computational costs within reasonable limits.

A stochastic particles method with ramdom interaction weights to compute statistical solutions of McKean-Vlasov equations
Olivier Vaillant
OMEGA Project, INRIA Sophia Antipolis, France
Olivier.Vaillant@sophia.inria.fr

The initial condition of some complex physical systems, such as turbulent flows, is often modelled by a random variable. The study of such problems is related to the theory of statistical solutions, that is probability measures on the space of solutions of the problem whose marginal at time 0 is the law of the random initial condition. We present an original and numerically efficient method to compute moments of the statistical solution of a McKean-Vlasov equation. It is a stochastic particles method with random interaction weights, for which we estimate the rate of convergence according to the number of simulated particles and the time discretization step.

MALLIAVIN CALCULUS AND APPLICATIONS

Organizer : Arturo Kohatsu-Higa (Universitat Pompeu Fabra, Barcelona, Spain) and  Fabio Antonelli (Universita di  Chieti,  Pescara,  Italy)
 

Lie algebra and Application of Expectation in Diffusion Model
Shigeo Kusuoka
University of Tokyo, Japan
kusuoka@ms.u-tokyo.ac.jp

We give a method to compute the expectation of functions of diffusion process.  We use time inhomogeneous discrete Markov chain to approximate a diffusion process throu the Lie algebra structure of vector fields. Our method is justified by a certain analytical estimate which is derived by Malliavin calculus.
 

Conditional mixing and asymptotic expansion
Nakahiro Yoshida
University of Tokyo
Graduate School of Mathematical Sciences
3-8-1 Komaba, Meguro-ku
Tokyo 153, Japan
nakahiro@ms.u-tokyo.ac.jp

We consider a stochastic process which satisfies a mixing property under conditioning. Asymptotic expansion for an additive functional is derived. We also discuss its applications to a long-memory time series model and a stochastic differential equation with jumps.

Variance reduction methods for diffusion density simulation
Arturo Kohatsu-Higa
Universitat Pompeu Fabra
Department of Economics
Ramon Trias Fargas 25-27
08005 Barcelona, Spain
kohatsu@upf.es

In this talk we use the integration by parts formula of Malliavin Calculus to construct methods similar to the control variate method with localization in order to obtain variance reduction in the simulation of densities of diffusions. Previous forms of these ideas have been already been given by Bally and Talay and Fournie, Lasry, Lebuchoux, Lions and Touzi. Here we study the asymptotical amount of variance reduction and also carry out some simulations in a particular case that point out that these methods can be used to achieve some reduction of variance.

STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS

Organizer : Arturo Kohatsu-Higa (Universitat Pompeu Fabra, Barcelona, Spain) and Fabio Antonelli (Universita di Chieti,
Pescara, Italy)

Approximation of the invariant probability measure of stochastic Hamiltonian dissipative systems with non globally Lipschitz coefficients
Denis Talay
INRIA Sophia Antipolis, France

We study the convergence rate of the implicit Euler scheme for the approximation of invariant measures of stochastic Hamiltonian dissipative systems with non globally Lipschitz coefficients.

The technical difficulty of the analysis comes from the polynomial growth of the coefficients and the degeneracy of the infinitesimal generators of Hamiltonian dissipative diffusion processes.

We need to prove precise estimates on the exponential decay in time of the solution of some degenerate parabolic partial differential equations with non globally Lipschitz coefficients. These estimates ensure, e.g., that moments of Hamiltonian dissipative systems converge exponentially fast when time goes to infinity.

Some recent results on numerical methods for SDEs
Philip Protter
Purdue University, USA

We will present a collection of recent results concerning methods for the numerical solutions of SDEs, both continuous and with jumps. A highlight is a study of the asymptotic normalized error of the entire procedure of combining the Euler scheme with a Monte Carlo method to estimate the value to the expectation of functionals of the solution.  Such applications arise in Finance Theory and in Electrical Engineering.  We will also present recent results of Liqing Yan on the validity of the Euler method when the coefficients of the SDE are not smooth, and recent results of Xiang Long on a new method for the reduction of variance for SDEs.
Approximation of invariant measures
Gilles  Pages
Université Paris 12,  Creteil, France

We propose a recursive stochastic algorithm with decreasing step to compute the invariant distribution $\nu$ of a Brownian diffusion process. Namely, we approximate $\nu(f)$ for a wide class of possibly unbounded continuous functions $f$. We consider a rather general setting, that includes some cases where the diffusion  may have  several invariant distributions. Our main convergence  result contains  as a corollary the $a.s.$ Central Limit Theorem.  As a second step, we investigate the weak rate of convergence of the algorithm. We show that, in the class of polynomial steps $\ga_n=n^{-\al}$,  it can be at most $n^{\frac{1}{3}}$ when the white noise has third moment zero and $n^{\frac{1}{4}}$ otherwise where $n$ denotes the number of iterations of the algorithm.

REFLECTED  PROCESSES AND PDEs

An approximation method for Reflected Backward Stochastic Differential Equations
Vlad Bally
Laboratoire de Statistique et Processus Département de Mathématiques Université du Maine
Avenue Olivier Messiaen  BP 535  72017 Le Mans Cedex  France
bally@ccr.jussieu.fr

 We give an approximation scheme for reflected BSDE's based on a quantizatin method (From an analitical point of viue reflected BSDE's correspond to variational inequalities). An evaluation of the speed of convergence is obtained. An important appliquation concerns optimal stopping problems and in particular the pricing of american options. We also obtain an algorithme for the approximation of the exercise time.

Rate of Convergence of a Particle Method for the Solution of  a 1D Viscous Conservation Law in a Bounded Interval
M. Bossy
INRIA Sophia Antipolis
2004 route des lucioles
BP 93, 06902 Sophia Antipolis, France
Mireille.Bossy@sophia.inria.fr
B. Jourdain
CERMICS
6 et 8 avenue Blaise Pascal
Cité Descartes - Champs sur Marne
77455 Marne la Vallée Cedex 2
jourdain@cermics.enpc.fr

First, we  give a probabilistic interpretation of the viscous scalar conservation law in a bounded interval by means of a non linear martingale problem. Here, the underlying non linear stochastic process is reflected at the boundary to take into account the Dirichlet conditions. We show a propagation of chaos result and construct a numerical algorithm based on the simulation of the system of reflected interacting particles. We show that the rate of convergence of the error is in $\O(h +1/\sqrt{N})$ when we use the Euler-Lepingle scheme to discretise in time the system  of N particles.

Efficient schemes for the weak approximation of reflecting diffusions
Emmanuel Gobet
Université Paris VI
Tour 56 - 3ème étage
Boite Courrier 188
4, Place Jussieu
75252 PARIS CEDEX 05, France
gobet@ccr.jussieu.fr

We propose new schemes, easy to simulate, to weakly approximate reflecting diffusions in general domains. We prove that the rate of convergence is order 1, improving the order 1/2 obtained with the standard Euler scheme with projection. We discuss both stationnary and non stationnay problems.
 

An elementary variance reduction method for discretized diffusion processes with boundary conditions
Cristina Costantini
Dipartimento di Scienze,
Universita G. D'Annunzio,
viale Pindaro 42, I-65127 Pescara,  Italy
costanti@sci.inch.it

It is shown that an elementary variance reduction method (the method of Antithetic Random Numbers), suitably adapted, can be used in the computation of the expectations of functionals of discretized diffusion processes with boundary conditions. Examples of situations in which the method can be applied are the computation of the price of barrier options under the CEV process and the computation of the expected value of the local time on the boundary of a diffusion reflecting in the positive orthant. Some numerical results for the first example are also given.

STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Organizer : Arnaud Debussche (Universite Paris XI, France)

Theoretical results on the stochastic nonlinear Schrodinger equation
Anne de Bouard
Université Paris XI, France

The nonlinear Schr\"odinger equation is one of the basic models for nonlinear waves. It arises in various areas of physics such as hydrodynamics, nonlinear optics or plasma physics. In some circumstances, randomness has to be taken into account, often to model phenomena with characteristic time scales much smaller than the characteristic time scales of the deterministic phenomena. Hence it is natural that the time-dependence of the random terms appearing then in the equation is a white noise type dependence, while the spacial correlation of these terms may take different forms.

We report here results which were obtained in collaboration with Arnaud Debussche, concerning the well-posedness in a nonlinear Schr\"odinger equation with a random potential, together with some convergence results for a time semi-discretization of this equation. Such an equation has been proposed to model energy transfer in the context of molecular aggregates with thermal fluctuations.
 

Influence  of a noise on the blow-up in a nonlinear Schr\'odinger equation
Arnaud Debussche
Université Paris XI, France

This talk is linked to the preceeding one by A. de Bouard and present results obtained in collaboration with her and L. Di Menza. We study the blow-up in finite time for the solutions of a stochastic nonlinear Schr\"odinger equation.

We first obtain theoritical results in the case of an additive noise which  is assumed to be  smooth in space. We generalize the standard deterministic computation yielding blow-up in finite time and prove that if the expectation of the energy of the initial data is negative, then the expectation of the sup norm of the solution blows up in finite time. Then using a control argument as well as the irreducibility of the transition semigroup, we prove that in fact this result holds true {\it for any initial data} and the blow-up time is arbitrarily small.

We then present numerical simulations. We confirm the theoretical results and show that the above conclusion seems to be still true in the case of a multiplicative smooth noise with a supercritical nonlinearity. However, in the critical case, apparently the noise is not able to create a blow-up.

We therefore conclude that in most cases, if a spatially smooth noise is taken into account, all trajectories blow-up in finite time. This is a drastic change compared to the deterministic case.

A refinment procedure is developped in order to simulate a noise which is delta correlated in space and time and to understand if the influence of such a noise is different. In the additive case, the behaviour does not seem to change. But, with multiplicative noise, it is totally different : {\it the noise prevents the blow-up and  the formation of singularities}.

On the discretization in time of a parabolic stochastic partial differential equation
Jacques Printems
Université Paris XII, Val de Marne, France

Let $H$ be a Hilbert space, we consider the following evolution equation written in the abstract Ito form
\begin{equation}
\label{eq:1}
\dd u + ( A u + f(u) ) \, \dd t = \sigma(u) \, \dd W,
\end{equation}
with the initial condition
\begin{equation}
\label{eq:2}
u(0)= u_0 \in H,
\end{equation}
where $A:D(A) \rightarrow H$ is a self-adjoint, nonnegative, unbounded operator on $H$, such that $D(A) \subset \subset H$, $f$, an application from $H$ to $D(A^{-s})$ for some $s>0$ and $\sigma$, an application from $H$ into ${\cal L}(H,D(A^{-\beta}))$ for some $\beta > 0$. Here, $\{W(t)\}_{t \in [0,T]}$ denotes a cylindrical Wiener process on $H$ (see \cite{dpz}) defined on a given stochastic basis $\proba{\Omega}{F}{P}$. We suppose that here $\beta < (1-\alpha)/2$ where $\alpha$ is such that
$$
{\mathrm Tr}(A^{-\alpha}) < + \infty.
$$
In order to numerically integrate (\ref{eq:1})--(\ref{eq:2}), we study the following numerical scheme
\begin{equation}
\label{eq:3}
u^{n+1} - u^n + \tau (A u^{n+\theta} + f(u^n) )= \sqrt{\tau} \sigma(u^n) \chi^n,
\end{equation}
where $u^{n+\theta} = \theta u^{n+1} + (1-\theta) u^n$ for $\theta > 1/2$ and where $\{\chi^n\}_{n\geq 0}$ is a sequence of i.i.d. normal random variables.

We first generalize, in the abstract framework (\ref{eq:1}), results on the order of convergence of a semi-discretization in time by the implicit Euler scheme (\ref{eq:3}) of stochastic parabolic equations \cite{gyongy-nualart} when all the coefficients ($f$ and $\sigma$) are globally Lipschitz. The case when the nonlinearity is only locally Lipschitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation ($f(y) = y^2/2$). We are not able in this case to compute a pathwise order of approximation. We introduce the weaker notion of {\em order in probability} and generalize to that extent the results of the globally Lipschitz case.

On a Stochastic Parabolic Equation
Laurent Denis
Université du Maine
Departement de Mathematiques
Avenue Messiaen
BP 535, 72 085 LE MANS
ldenis@univ-lemans.fr

The aim here is to study the solutions of the following non linear stochastic partial equation of parabolic type,
\[ du+Ludt+f(t,u)dt+Rg(t,u)dt=h(t,u)dB_t ,\]
$L$ is a non-negative self adjoint operator defined on a Hilbert space. The operator $L^{1/2}$ plays an important role so we consider the space $F=\DD (L^{1/2})$ and $R$ is a bounded linear operator defined on some Hilbert space $K$ with values in the dual $F^{\prime}$ of $F$.\\ $f$ and $g$ are only supposed to be lipschitzian from $\R^+ \times F$ into $H$ (resp. $K$) .\\ A special case is the usual one: $H=L^{2}(\R^d )$, $-L$ is the a second order elliptic operator:
\[ -L=\sum_{i,j} \partial_i a^{i,j}\partial_j ,\]
where coefficients $a$ are only assumed to be measurable and of the form $a=\sigma\sigma^{\ast}$.\\ So we have existence and unicity (in the weak sense) of equation
\[du(t,x)+Lu(t,x)dt+\widetilde{f}(t,u,\sigma\nabla u)dt+\sum_i\partial_i \widetilde{g}_i(t,u,\sigma\nabla u)dt=\widetilde{h}(t,u,\sigma\nabla u)dB_t.\] We recall that $\widetilde{f}$, $\widetilde{g}$ and $\widetilde{h}$ are only assumed to be lipschitz.\\ Finally, we also give a doubly stochastic representation of solutions.

Organizer : Volker Wihstutz (UNC Charlotte, USA)

True randomizers and Shannon entropy
Alex Gordon (Random Number Generator Systems, Inc.)  (talk delivered by Joe Quinn)

The random numbers used in most simulations are produced by pseudorandom number generators, i.e. specialized computer programs with a "random-looking" output. Unlike those, a truly random number generator is based on some physical source of randomness (thermal noise, radioactive decay, etc). The talk will discuss the ways to collect data from such a physical source, to transform them into Bernoulli bits, and to get as many of those as possible.

Theorems for generalized quantum statistics and the testing of randomizers with and without asymptotic assumptions
Joe Quinn
University of North Carolina, Charlotte, USA

All algorithmic RNG?s (PRNG?s) are dynamical systems (possibly truncated) on the space of bit strings of some fixed length, say K on the order of 30 to 48 or more.  There are strong similarities between such RNG?s and certain generalized Fermi-Dirac statistics.  Limit theorems and moment theorems for such statistics with effective estimation of remainders and testing procedures based on these theorems will be discussed.  Testing results for specific PRNG?s will be presented.
 

Numerical schemes for SDE's
Gerard Fleury and Pierre Bernard
Universite Blaise Pascal, Clermont Ferrand, France
bernard@ucfma.univ-bpclermont.fr

In stochastic Mechanics, one has to deal with SDE which coefficients are  only locally Lipschitz, and that are degenerated. In this contribution, we prove the convergence of a class of numerical schemes under weak non-explosin conditions on the solutions of the SDE.This results applies to Euler, Milshstein, Newmark shemes ....
 

Communication structure of discretized diffusion processes and approximation of Lyapunov exponents
Volker Wihstutz
University of North Carolina, Charlotte, USA

Studying the time discretized approximate solution of a stochastic differential equation, simulated with help of a random number generator, we are interested in the (control theoretical) properties like accessibility and their stochastic counter parts which are inherited from the true solution of the original linear or non-linear stochastic differential equation. In view of the Furstenberg-Khasminsky representation for Lyapunov exponents and a formula of Grorud-Talay these properties are crucial for approximating Lyapunov exponents.

Organizer : Andrew Stuart (Warwick University, Coventry, U.K.)

Ergodicity for Numerical Simulations
Jonathan Mattingly
Mathematics Department
Stanford University

The ergodicity of SDEs is studied by use of techniques from the theory of Markov chains on general state spaces. Careful application of these Markov chain results leads to straightforward proofs of ergodicity for a variety of SDEs, in particular for problems with degenerate noise and for problems with locally Lipschitz vector fields. The key points which need to be verified are the existence of a Lyapunov function inducing returns to a compact set, a uniformly reachable point from within that set and some smoothness of the probability densities. Applications include the Langevin equation, the Lorenz equation with degenerate noise and gradient systems. The ergodic theorems proved are quite strong, yielding exponential convergence of expectations for classes of functions restricted only by the condition that they grow no faster than the Lyapunov function.

This is joint work with A.M. Stuart

Ergodicity for Numerical Simulations
Des Higham
Mathematics Department
University of Strathclyde

We study the property of geometric ergodicity for numerical methods applied to stochastic differential equations (SDEs). We are especially concerned with SDEs driven by degenerate noise and with deterministic vector fields that are not globally Lipschtitz. A simple example shows that the Euler-Maruyama method can fail to reproduce ergodicity of an SDE. Using a Markov Chain approach of Mattingly and Stuart we show that certain implicit methods can guarantee to preserve ergodicity on important classes of SDEs, including certain Langevin and dissipative problems. Illustrative numerical experiments will be presented.

This is joint work with J. Mattingly and A.M. Stuart

Convergence Properties of Perturbed Markov Chains
Jeffrey S. Rosenthal
Department of Statistics
University of Toronto

We consider the question of which convergence properties of Markov chains are preserved under small perturbations.  Convergence properties considered include geometric ergodicity and time to stationarity. Perturbations considered include roundoff error from computer simulation. We are motivated partially by interest in Markov chain Monte Carlo algorithms.

This is joint work with G.O. Roberts and P.O. Schwartz.

Pseudo-moderate deviations in the Euler method for real diffusion processes
Emmanuelle Clement
Universite de Marne la Vallee, France

Consider a one-dimensional process $(X_t)$ solution of a stochastic differential equation driven by a Brownian motion. In order to simulate the process $(X_t)$, it is usual in practice to approximate it by an Euler scheme denoted by $(X^n_t)$. We are interested in the study of the asymptotic behaviour of the error process $(X-X^n)$ and we prove that it satisfies a pseudo-moderate deviation principle.

Organizer : Rolando Rebolledo (Universidad Católica de Chile)

Quantum Stochastic Differential Equations
Franco Fagnola
Dipartimento di Matematica, Università di Genova,
Via Dodecaneso 35, I-16146 Genova, Italia
fagnola@dima.unige.it

Several classes of quantum stochastic differential equations (QSDE) have been introduced in the last decade to build mathematical models of quantum evolutions.

In this talk we discuss the theory QSDE of the form
$$ dV(t) = V(t)\left(\sum_{\ell,m\ge 0}L_\ell^m d\Lambda_m^\ell(t)\right), \qquad V(0)=I$$
where
\begin{itemize}
\item[(a)] solutions $(V(t))_{t\ge 0}$ are families of operators acting on a complex Hilbert space of the form $h\otimes\Gamma(L^2({\bf R}_+;{\bf C}))$, $h$ being a Hilbert space and $\Gamma(L^2({\bf R}_+;{\bf C}))$ being the Boson Fock space over $L^2({\bf R}_+;{\bf C})$,
\item[(b)] $L_\ell^m$ are operators on the Hilbert  space $h$,
\item[(c)] $\Lambda_m^\ell$ are the basic noises of Boson Fock quantum stochastic calculus, in particular $\Lambda_0^0 (t)=t$,
\item[(d)] the initial condition $I$ is the identity operator on $h\otimes\Gamma(L^2({\bf R}_+;{\bf C}))$.
\end{itemize}

These equations arise in the study of the irreversible evolution of open systems in quantum mechanics. In this case $h$ is the Hilbert space of a system incteracting with a heat bath described by a Fockspace.

From a probabilistic point of view one can notice that processes
\begin{eqnarray*}
(\Lambda_0^\ell(t) + \Lambda_\ell^0(t))_{t\ge 0},\qquad \ell\ge 1
\end{eqnarray*}
are (non-commuting) operator versions of classical Brownian motions and processes
\begin{eqnarray*}
(\Lambda_\ell^\ell(t) + \Lambda_0^\ell(t) + \Lambda_\ell^0(t) + t)_{t\ge 0},
\qquad \ell\ge 1
\end{eqnarray*}
are (non-commuting) operator versions of classical Poisson processes with intensity $1$. Thus the above QSDE provide also a non-commutative generalization of classical partial SDE.

On the other hand they also generalize the Schr\"odinger equation (take $L_0^0 = iH$ with $H$ self-adjoint and $L_\ell^m=0 $ whenever either $\ell$ or $m$ is not $0$). For this reason they  are also called quantum stochastic Schr\"odinger equations.

In the talk we shall discuss results on the existence and uniqueness of solutions $(V(t))_{t\ge 0}$ given by families of bounded operators and condtions for the $V(t)$ to be isometric or unitaries.
 

Quantum stochastic particles and a model for electronic transport in aperiodic media
Rolando Rebolledo
Pontificia Universidad Catolica de Chile
Facultad de Matematicas
Casilla 306, Santiago 22, Chile
rrebolle@mat.puc.cl

A model of quantum interacting particles is presented which establishes a bridge with classical stochastic interacting particles. Within this framework, three main mathematical problems are considered:
\begin{itemize}
\item The construction of the quantum dynamical semigroup which describes the dynamics,
\item the search for invariant states of this semigroup,
\item the approach to the equilibrium.
\end{itemize}
Moreover, the talk will provide a view on a physical application: a model for electronic transport in aperiodic media.

Stopping quantum processes : the value of the Brownian motion at the Poisson jumping times
Stéphane Attal
Institut Fourier
Université de Grenoble
Stephane.Attal@ujf-grenoble.fr

We will speak about the general problem of stopping times in quantum theory. We will explain the  problems that are attached to their definition  in quantum mechanics. We will finally give a very interesting example and a computation with the Brownian motion and the Poisson process.

Numerical solution of stochastic differential equations and the simulation of a class of quantum dynamical semigroups
Carlos Mora
Pontificia Universidad Catolica de Chile
Facultad de Matematicas
Casilla 306, Santiago 22, Chile
cmora@mat.puc.cl

The talk will provide with new methods of approximating solutions to systems of stochastic differential equations in the weak sense. As a result, numerical schemes with corvengence orders 1 and 2 will be obtained.
In particular, these numerical schemes will be applied to build up simulation procedures for a class of quantum dynamical semigroups arising in the study of  quantum open systems.
 

A view on Quantum Large Deviations from a theory of quantum capacities
Henri Comman
Pontificia Universidad Cat\'olica de Chile
Facultad de Matematicas
Casilla 306, Santiago 22, Chile
hcomman@mat.puc.cl

Quantum Large Deviations are obtained as an application of a Theory of Non commutative capacities. This theory, introduces a new concept of capacity, based on $C^*$ algebras, and includes the definition of their associated weak topologies (vague and narrow)  as well as the characterization of the remarkable class of maxitive capacities which naturally appears in Large Deviations principles.
 
 

Organizer : Bernard Lapeyre (ENPC-CERMICS, France)

Asymptotics of hitting probabilities for diffusions with applications to simulation
Lucia Caramellino
Dipartimento di Matematica,
Università  Roma Tre,
largo San Leonardo Murialdo,1, I-00146 Roma
lucia@mat.uniroma3.it

Let $X$ be a diffusion process, starting at $x$ at time $0$, and consider the associated conditioned diffusion $\hat X^{y,\varepsilon}$ pinned by $X_\varepsilon=y$. Setting $\hat {\rm I\!P}^{x,y}_\varepsilon$ its law on the path space and $\tau$ as the exit time from an open set, possibly time-dependent, we give the asymptotics, as $\varepsilon\to 0$ of the exit probability $\hat {\rm I\!P}^{x,y}_\varepsilon(\tau\leq \varepsilon)$, by studying the probability density function of the law of $\hat X^{y,\varepsilon}$ with respect to the law of a suitable Brownian Bridge. By using these sharp large deviation estimates, we develop simulation techniques which allow to improve usual Monte Carlo procedures in order to evaluate contingent claims whose payoffs % are path-dependent and depend on suitable hitting times on some barrier.  Numerical results and comparisons with existing literature will be provided for single and double barrier options, Parisian barrier options, bonds and other contingent claims that are subject to default risk as well as interest rate risk, in which the default can occur any time before the maturity of the claim.

Numerical Experiments in Risk Management : a simplified approach in life insurance contract
Christophe Berthelot,
Bull, Louveciene, France
Christophe.Berthelot@bull.net
Mireille Bossy
INRIA Sophia Antipolis, France
Mireille.Bossy@sophia.fr
Nathalie Pistre
ENSAE, Malakof, France
pistre@ensae.fr

In addition of the liquidity risk  and credit risk, it is now commonly admitted that Insurance Companies are submitted to the risk due to the embedded options in insurance contracts. Life insurance contracts provide an  early exit option  for the policy holders, which value is connected to the market evolution.

In this work, we tempt to characterise the risk associated to a life insurance contract. We choose the example of a contract with a unique initial deposit which guarantees a fixed minimal rate of return increased by a participation to the financial benefits of the Company.

 With Monte Carlo simulations, we analyse the risk in terms of the contract characteristics, the expected exit of the customer, and the market fundamentals. We compute statistics on the capital return and provide an analysis of the risk with respect to the Company asset portfolio and the customer's behaviour.

Numerical experiments have been done with the LICS V2 simulator for NEC SX computers.

Competitive Monte Carlo methods for the Pricing of Asian Options
Emmanuel Temam
ENPC-CERMICS, France
temam@cermics.enpc.fr

We explain how a carefully chosen scheme can lead to competitive Monte Carlo algorithm for the computation of the price of Asian options. We give evidence of the efficiency of these algorithms with a mathematical study of the rate of convergence and a numerical comparison with some existing methods.
Key Words: Asian option, Monte Carlo methods, Numerical methods, Diffusion process.

Optimal capital structure and endogenous default
L. C. Rogers and Bianca Hilberink
Department of Mathematical Sciences   University of Bath Bath  BA2 7AY GB
lcgr@maths.bath.ac.uk

In a sequence of fascinating papers, Leland and Leland & Toft have investigated various properties of the debt and credit of a firm which keeps a constant profile of debt and chooses its bankruptcy level endogenously, to maximise the value of the equity. One feature of these papers is that the credit spreads tend to zero as the maturity tends to zero, and this is not a feature which is observed in practice. This defect of the modelling appears to be related to the diffusion assumptions made in the papers referred to; in this paper, we take a model for the value of the firm's assets which allows for jumps, and find that the spreads do not go to zero as maturity goes to zero. The modelling is quite delicate, but it just works; analysis takes us a long way, and for the final steps we have to resort to numerical methods.

FINANCIAL MATHEMATICS II

Organizer : Monique Jeanblanc (Universite d'Evry, France)

Numerical approximation of combined stochastic and impulse control problems. Application to portfolio optimisation with  fixed and  proportional transaction costs.
Agnes Sulem
INRIA Rocquencourt
Domaine de Voluceau
BP 105  Rocquencourt
78153 Le Chesnay Cedex, France
Phone: 01 39 63 55 69
Fax : 01 39 63 57 86
Agnes.Sulem-bialobroda@inria.fr
B. Oksendal, J.Ph. Chancelier

We present a numerical method to solve combined stochastic control and impulse control problems. This method is based on the construction of  an iterative sequence of combined stochastic control and  optimal stopping problems. This algorithm is implemented  to solve numerically the  quasi-variational inequality (QVI) associated to a problem of portfolio optimisation with both fixed and  proportional transaction costs. The QVI is obtained as the limit of variational Hamilton-Jacobi-Bellman inequalities. Each variational inequality is approximated by a finite difference scheme and then  solved by a Howard algorithm.

Optimal portfolio management with american capital garantee
Nicole El Karoui
Ecole Polytechnique, France

The aim of the paper is to investigate finite horizon portfolio strategies which maximize a utility criterion when dynamic rebalancing is allowed with no transaction cost and a continuous value constraint is imposed for every intermediary date. The classical automatic strategies such as the All or Nothing strategy, the Cushion method -~also know as Constant Proportional Portfolio Insurance~- as well as the Option Based Portfolio Insurance are studied. In the case of a European guarantee, we extend the optimality of the OBPI method to a general class of utility functions, and we prove the OBPI to be optimal in the American case only for CARA utility functions. We take a special care to fully describe the OBPI method based on American options, and we show the optimal gearing of the portfolio to be an increasing path dependent process, for which we have a quasi closed form solution.
 

A worst case Model Risk Management for Discount Bond Option
Ziyu Zheng
INRIA Sophia Antipolis, France
Zheng.Ziyu@sophia.inria.fr

In this talk we are interested in the management of model risk. Our objective is to propose a new strategy for the trader which, in a sense, guarantees good performances whatever is the unknown model. Our construction corresponds to a `worst case' worry and, in this sense, can be viewed as a continuous-time extension of discrete-time strategies based upon prescriptions issued from VaR analyses at the beginning of each period. The trader chooses trading strategies to decrease the risk and therefore acts as a minimizer; the market systematically acts against the interest of the trader, so that we consider it acts as a maximizer. Thus we consider the model risk control problem as a two players (Trader versus Market) zero-sum stochastic differential game problem. We give a proper mathematical statement for such a game problem. We proved that the value function to this game problem is the unique viscosity solution to a corresponding Hamilton-Jacobi-Bellman-Isaacs Equation and satisfies the Dynamic Programming Principle. We simulate the value function by solving the HJBI equation
numerically.

Exotics Interest rate Derivatives
Guy Kamagne
TREMA, France
guy.kamagne@trema.com

A number of differents models has been used in order to price interest rate derivatives. After background explanations of the requirements that are need in order to fulfill markets conventions, I  will explain how differents models such as Black-Derman-Toy, Hull and White , Health -Jarrow-Morton and Brace-Gatarek-Musiela can be calibrated to match the market price of Caps/Floor/ Swaptions. Those informations can then be used to price Exotics interest rate derivatives.

FINANCIAL MATHEMATICS III

Organizer :  Wolfgang Runggaldier (Università di Padova, Italy)

Modelling and simulating high frequency data for financial models
Omar Zane
Warburg Dillon Read, London
Address starting May  2000 :
Quantitative Risk, Models and Statistics
UBS, 1 Finsbury Avenue
London EC2M 2PG
Omar.Zane@wdr.com

The growing availability of market data makes it important to have models which realistically describe the behaviour of tick-by-tick  datasets. We shall illustrate a general modelling framework for tick data and give specific examples, presenting estimation procedures which perform effectively on simulated samples.
 

Monte Carlo Improvement of Estimates of the Mean Reverting Constant Elasticity of Variance Interest Rate Diffusion
Rolf Poulsen
Department of Statistics and Operations Research
Institute of Mathematics, University of Copenhagen
Unversitetsparken 5
University of Copenhagen
DK-2100 Copenhagen, Denmark
Phone: +45 353 20685
Fax: +45 35 32 06 78
rolf@math.ku.dk
http://www.math.ku.dk/~rolf/
Bent Jesper Christensen
School of Economics and Management
University of Aarhus
350 University Park, DK - 8000 - Aarhus C, Denmark
Phone: (45) 8942 1547
Fax  (45) 8613 5132
bjchristensen@econ.dk

We use simulation based methods and other numerical techniques to construct improved estimation procedures for discretely observed diffusions. The benchmark model used for illustration and comparison is the CKLS short interest rate model dr = - a(r - b) dt + sigma r^gamma dW, where W is a Wiener process. Here, a is the rate of mean reversion, b is the long term interest rate level, sigma is a scaling parameter, and gamma is the constant elasticity of variance parameter. The likelihood function is unknown, and standard method of moments methods lead to biased and inconsistent estimates of the unknown parameters a, b, sigma and gamma. Consistent estimates are obtained through a simulation based correction of the estimating equations (efficient method of moments). These methods are also used to estimate interest rate models with nonlinear drift.
 

Stochastic flows techniques in MC framework to work out the Greeks
Igor Pikovsky
Product Development Group - Relative Value Modelling
CREDIT SUISSE  | Financial Products
One Cabot Square, London, E14 4QJ
Phone  (44) 171 888 3230
Fax   (44) 171 888 2771
Mob (44) 7974 766 394
Igor.Pikovsky@csfb.com

The paper gives theoretical motivation and Monte Carlo scheme recipes for computing sensitivities (deltas, gammas, vegas, etc) for various contingent claims by exploiting methods of stochastic flows. We present the sufficient conditions of Bouleau-Hircsch and Hormander's type for this method to be valid and exhibit examples from industrial practice when the techniques could and could not be applied.
 

American options, critical price and dividends
Damien Lamberton
Equipe d'analyse et de mathematiques appliquees
Universite de Marne-la-Vallee
5, Boulevard Descartes
Cite Descartes, Champs-sur-Marne
77 454 Marne-la-Vall\'ee CEDEX 2 France
dlamb@math.univ-mlv.fr

In this talk, based on a joint work with S. Villeneuve, we discuss the behavior of the critical price near maturity, for an American put on a dividend-paying stock. In the case of high dividend rates, the result is very different from the previously known case of non dividend paying stocks (cf. Barles et al. 1995).
 

FINANCIAL MATHEMATICS IV

Using Monte Carlo Methods to Evaluate Sub-Optimal Exercise Policies for American Options
R.Mallier and G.Alobaidi
Department of Applied Mathematics, University of Western Ontario, London, ON, CANADA N6A 5B7
mallier@julian.uwo.ca

Options are derivative financial instruments which give the holder the right but not the obligation to buy (or sell) the underlying asset. American options are options which can be exercised either on or before a pre-determined expiry date. For such options there is, therefore, the possibility of early exercise, and the issue of whether and when to exercise an American option is one of the best-known problems in mathematical finance, leading to an optimal exercise boundary and an optimal exercise policy, the following of which will maximize the expected return from the option. Ideally, at each instant in time, an investor would be able to calculate the expected return from continuing to hold the option, and if that is less than the return from exercising immediately at that time, he should exercise the option. However, whereas a large institution may well be able to perform those calculations and thereby optimize their return, a retail investor may well be unable to do this, and instead may have his  own set of exercise policies, choosing to exercise the option when certain conditions are met, for example when the value of the option reaches some multiple of the exercise price. The expected return from such sub-optimal strategies will be less than or equal to the expected return when the optimal exercise policy is pursued.

In this study, we use a Monte Carlo scheme to look at several such strategies that a somewhat ill-advised investor might follow, and calculate the expected return from the option using these strategies. A Monte Carlo simulation is well-suited for this particular problem, since the theory of derivative pricing is based on the premise that the price of the underlying asset, the risk-free interest rate, etc, follow random walks which  can be tackled using Monte Carlo methods.

In addition to evaluating several naive strategies, we will also look at how the expected return is affected by the ``frequency of checking'', meaning how often the investor checks the value of the option to see if his exercise criteria have been met. For example,  in real-life, a day-trader is constantly checking his portfolio on-line, whereas someone else might check his holdings daily in the morning paper.

Monte-Carlo computations of American options via Malliavin calculus
Herve Regnier
Caisse des Depots et Consignations
France
herve.regnier@car.caissedesdepots.fr

We introduce a new Monte-Carlo method for the pricing and hedging of American options. As is well known, due to the  intrinsic non linear nature of American options, it's difficult to use Monte Carlo simulations in order to compute the price of an american option. All the computations are made along the simulated trajectories and the heart of this method is the possibility of rewriting conditionnal expectation in a form which is available to Monte-Carlo simulations. This rewriting is based upon the so-called Malliavin calculus and integration by parts.

Monge-Ampere PDE's in Finance and their Direct Numerical Solutions
Srdjan Stojanovic
Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025, USA
http://math.uc.edu/~srdjan/
srdjan@math.uc.edu
 

Monte Carlo, or more precisely, SDE simulation methods for elliptic and parabolic, linear and nonlinear, possibly degenerate, PDE's can be employed for problems for which no direct numerical solutions are known, also for problems in infinite domains for which no natural boundary conditions for the truncated domains are known, etc. On the other hand, for the majority of problems, the SDE simulation methods seem to be computationally less efficient then the direct numerical PDE approach. We illustrate the above views on some examples using Mathematica computer platform. On the other hand, we demonstrate the power of the new direct numerical approach by solving fully nonlinear, possibly degenerate backward parabolic PDE of Monge-Ampere type associated with the stochastic control problem of optimal diversification of portfolio of stocks (and options), under various natural constraints, and with tracking of an index. Arbitrary, possibly discontinuous, utility function, i.e., terminal/boundary condition (causing degeneracy of the equation), is allowed. The performance of the numerical solver (up to the second derivative of the solution) is evaluated on simple but representative problems for which explicit solutions are known. The performance of the solution of the full blown problem is evaluated on real market data through SDE simulations, and also through the empirical experiments, i.e., through the real market trading.

Reducing the Variance for Monte-Carlo and Quasi-Monte-Carlo simulations for the Greeks using Malliavin Calculus
Eric Ben-Hamou
The Financial Markets Group
Room G 303
The London School of Economics
Houghton Street
London WC2A 2AE
benhamou_e@yahoo.com
http://fmg.lse.ac.uk/~benhamou/

Traditionally, Monte Carlo methods and their natural Quasi Monte Carlo extensions have a slow rate of convergence for the numerical computation of the Greeks (sensitivities of the option price to model  parameters) when the payoff function is strongly discontinuous. The reason of this inefficiency lies in the way the Greeks are commonly computed. One estimates the Greeks by simply taking the finite difference of two particular simulation results (forward, central or backward difference scheme). The different classical variance reduction techniques (antithetic variates, control variates, importance sampling, stratified sampling, Latin hypercube sampling and moment matching techniques) or deterministic methods based on low discrepancy sequences (Halton, Sobol, Faure sequences) do not counterweight the finite difference error. Fournie et al (1999) have suggested to use an integration by parts formula to smoothen the payoff function by means of the Malliavin calculus theory. By introducing the weighting function generator defined as the Skorohod integrand of the weight, we show how to derive necessary and sufficient conditions for the weighting function generator. This enables us to characterise the set of solutions for the weight. This extends Fournie et al (1999) results and show that their solutions are particular ones of a bigger set.