Antoine Lejay

On the constructions of the skew Brownian motion

This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion.

An Introduction to Rough Paths

This article aims to be an introduction to the theory of rough paths, in which integrals of differential forms against irregular paths and differential equations controlled by irregular paths are defined. This theory makes use of an extension of the notion of iterated integrals of the paths, whose algebraic properties appear to be fundamental. This theory is well-suited for stochastic processes.

A SCHEME FOR SIMULATING ONE-DIMENSIONAL DIFFUSION PROCESSES WITH DISCONTINUOUS COEFFICIENTS

The aim of this article is to provide a scheme for simulating diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of the behavior of their trajectories when they reach the points of discontinuity. This description is supplied with the local comparison of the trajectories of the diffusion processes with those of a Skew Brownian Motion.

Homogenization of divergence-form operators with lower order terms in random media

Abstract. The probabilistic machinery (Central Limit Theorem, Feynman-Kac formula and Girsanov Theorem) is used to study the homogenization property for PDE with second-order partial differential operator in divergence-form whose coefficients are stationary, ergodic random fields. Furthermore, we use the theory of Dirichlet forms, so that the only conditions required on the coefficients are non-degeneracy and boundedness.

Simulating diffusion processes in discontinuous media

In this article, we propose new Monte Carlo techniques for moving a diffusive particle in a discontinuous media. In this framework, we characterize the stochastic process that governs the positions of the particle. The key tool is the reduction of the process to a Skew Brownian motion (SBM). In a zone where the coefficients are locally constant on each side of the discontinuity, the new position of the particle after a constant time step is sampled from the exact distribution of the SBM process at the considered time. To do so, we propose two different but equivalent algorithms: a two-steps simulation with a stop at the discontinuity and a one-step direct simulation of the SBM dynamic. Some benchmark tests illustrate their effectiveness.

Computing the principal eigenvalue of the Laplace operator by a stochastic method

We describe a Monte Carlo method for the numerical computation of the principal eigenvalue of the Laplace operator in a bounded domain with Dirichlet conditions. It is based on the estimation of the speed of absorption of the Brownian motion by the boundary of the domain. Various tools of statistical estimation and different simulation schemes are developed to optimize the method. Numerical examples are studied to check the accuracy and the robustness of our approach.

BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization

Backward stochastic differential equations (BSDE) also gives the weak solution of a semi-linear system of parabolic PDEs with a second-order divergence-form partial differential operator and possibly discontinuous coefficients. This is proved here by approximation. After that, a homogenization result for such a system of semi-linear PDEs is proved using the weak convergence of the solution of the corresponding BSDEs in the S-topology.

A Monte Carlo method without grid for a fractured porous domain model

The double porosity model allows one to compute the pressure at a macroscopic scale in a fractured porous media, but requires the computation of some exchange coefficient characterizing the passage of the fluid from and to the porous media (the matrix) and the fractures. This coefficient may be numerically computed by some Monte Carlo method, by evaluating the time a Brownian particle spends in the matrix and the fissures. Although we simulate some stochastic processes, the approach presented here does not use approximation by random walks, and then does not require any discretization. We are interested only in the particles in the matrix. A first approximation of the exchange coefficient may then be computed. In a forthcoming paper, we will present the simulation of the particles in the fissures.

New Monte Carlo schemes for simulating diffusions in discontinuous media

We introduce new Monte Carlo simulation schemes for diffusions in a discontinuous media divided in subdomains with piecewise constant diffusivity. These schemes are higher order extensions of the usual schemes and take into account the two dimensional aspects of the diffusion at the interface between subdomains. This is achieved using either stochastic process techniques or an approach based on finite differences. Numerical tests on elliptic, parabolic and eigenvalue problems involving an operator in divergence form show the efficiency of these new schemes.

Séminaire de Probabilités XLIII

This is a new volume of the Seminaire de Probabilites which is now in its 43rd year. Following the tradition, this volume contains about 20 original research and survey articles on topics related to stochastic analysis. It contains an advanced course of J. Picard on the representation formulae for fractional Brownian motion. The regular chapters cover a wide range of themes, such as stochastic calculus and stochastic differential equations, stochastic differential geometry, filtrations, analysis on Wiener space, random matrices and free probability, as well as mathematical finance. Some of the contributions were presented at the Journees de Probabilites held in Poitiers in June 2009.