Dominique Lépingle

Sur l'intgrabilit uniforme des martingales exponentielles

SommaireNous donnons des conditions suffisantes pour que la martingale localeℰ (M) définie par C. Doléans-Dade soit une martingale uniformément intégrable. Le lien est établi avec la martingale locale exponentielle α(N, z, Μ) de Kunita-Watanabe, ce qui permet de généraliser un théorème de Novikov.

Euler scheme for reflected stochastic differential equations

Using some exponential variables in the time discretization of some reflected stochastic differential equations yields the same rate of convergence as in the usual Euler-Maruyama scheme.

Un schéma multipas d'approximation de l'équation de Langevin

In the approximation of solutions of some second-order stochastic differential equations, a multistep method converges faster than the Cauchy-Euler scheme for usual stochastic differential equations.

No Multiple Collisions for Mutually Repelling Brownian Particles

Brownian particles in electrostatic interaction may pairwise collide when the interaction parameter is small. But multiple collisions are never possible.

Approximating and Simulating Multivalued Stochastic Differential Equations

We propose a two-step simulation scheme for the solution of a singular stochastic differential equation with exploding drift. First we estimate the strong order of the Yosida approximation. Then we use a semi-implicit Euler scheme to discretize the approximate solution. Numerical experiments are displayed for the paths of Brownian particles with strong repulsive interaction. We also present two simple simulation schemes for Bessel processes with arbitrary dimension.

Brownian Motion, Reflection Groups and Tanaka Formula

In the setting of finite reflection groups, we prove that the projection of a Brownian motion onto a closed Weyl chamber is another Brownian motion normally reflected on the walls of the chamber. Our proof is probabilistic and the decomposition we obtain may be seen as a multidimensional extension of Tanakas formula for linear Brownian motion. The paper is closed with a description of the boundary process through the local times at zero of the distances from the initial process to the facets.

Approximating systems of differential equations with random inputs or boundary conditions

It is not very easy to get reasonable simulation schemes with high order of convergence for reflected stochastic differential equations, because of the intricate behavior of the diffusion near the boundary. But a simple Milstein-type scheme can be performed if diffusion and reflection act into separate directions; this is the case in many practical situations, some of which are presented.