Showing papers by "Antoine Lejay published in 2010"
TL;DR: The theory of rough paths as discussed by the authors allows one to define controlled differential equations driven by a path which is irregular, in which the integrals are interpreted as Young integrals, and the main results regarding this theory are existence, uniqueness, convergence of the Euler scheme, flow property, etc.
65 citations
TL;DR: Using a kinetic approximation of a linear diffusion operator, an algorithm is proposed that allows one to deal with the simulation of a multi-dimensional stochastic process in a media which is locally isotropic except on some surface where the diffusion coefficient presents some discontinuities.
15 citations
TL;DR: In this article, the authors presented an algorithm to simulate a Brownian motion by coupling two numerical schemes: the Euler scheme with the random walk on the hyper-rectangles.
Abstract: In this paper, we present an algorithm to simulate a Brownian motion by coupling two numerical schemes: the Euler scheme with the random walk on the hyper-rectangles. This coupling algorithm has the advantage to be able to compute the exit time and the exit position of a Brownian motion from an irregular bounded domain (with corners at the boundary), and being of order one with respect to the time step of the Euler scheme. The efficiency of the algorithm is studied through some numerical examples by comparing the analytical solution with the Monte Carlo solution of some Poisson problems. The Monte Carlo solution of these PDEs requires simulating Brownian motions of different types (natural, reflected or drifted) over an irregular domain.
7 citations
TL;DR: In this article, a new Monte Carlo method based on importance sampling techniques for the simulation of stochastic differential equations is introduced. But the main idea is not to combine random walk on squares or rectangles methods with importance sampling, but to compute the weights from the density of one-dimensional Brownian motion.
Abstract: The aim of this paper is to introduce a new Monte Carlo method based on importance sampling techniques for the simulation of stochastic differential equations. The main idea is here to combine random walk on squares or rectangles methods with importance sampling techniques. The first interest of this approach is that the weights can be easily computed from the density of the one-dimensional Brownian motion. Compared to the Euler scheme this method allows to obtain a more accurate approximation of diffusions when one has to consider complex boundary conditions. The method provides also an interesting alternative to perform variance reduction techniques and to simulate rare events.
6 citations
TL;DR: In this paper, a new Monte Carlo method based on importance sampling techniques for the simulation of stochastic differential equations is introduced. But the main idea is to combine random walk on squares or rectangles methods with importance sampling technique.
Abstract: The aim of this paper is to introduce a new Monte Carlo method based on importance sampling techniques for the simulation of stochastic differential equations. The main idea is to combine random walk on squares or rectangles methods with importance sampling techniques. The first interest of this approach is that the weights can be easily computed from the density of the one-dimensional Brownian motion. Compared to the Euler scheme this method allows one to obtain a more accurate approximation of diffusions when one has to consider complex boundary conditions. The method provides also an interesting alternative to performing variance reduction techniques and simulating rare events.
1 citations