Section: New Results

Numerical Probability

Parametrix method for reflected SDEs

With A. Kohatsu-Higa and M. Hayashi, Aurelien Alfonsi is investigating how to apply the parametrix method recently proposed by V. Bally and A. Kohatsu-Higa for reflected SDEs. This method allows them to obtain an unbiased estimator for expectations of general functions of the process.

Regularity of probability laws using an interpolation method

Participants : Vlad Bally, Lucia Caramellino.

This work was motivated by previous papers of Nicolas Fournier, J. Printemps, E. Clément, A. Debussche and V. Bally on the regularity of the law of the solutions of some equations with coefficients with little regularity - for example diffusion processes with Hölder coefficients (but also many other examples including jump type equations, Bolzmann equation or Stochastic PDE's). Since we do not have sufficient regularity the usual approach by Malliavin calculus fails in this framework. Then one may use an alternative idea which roughly speaking is the following: We approximate the law of the random variable $X$ (the solution of the equation at hand) by a sequence $X\left(n\right)$ of random variables which are smooth and consequently we are able to establish integration by parts formulas for $X\left(n\right)$ and we are able to obtain the absolutely continuity of the law of $X\left(n\right)$ and to establish estimates for the density of the law of $X\left(n\right)$ and for its derivatives. Notice that the derivatives of the densities of $X\left(n\right)$ generally blow up - so we can not derive directly results concerning the density of the law of $X$. But, if the speed of convergence of $X\left(n\right)$ to $X$ is stronger than the blow up, then we may obtain results concerning the density of the law of $X$. It turns out that this approach fits in the framework of interpolation spaces and that the criterion of regularity for the law of $X$ amounts to the characterization of an interpolation space between a space of distributions and a space of smooth functions. Although the theory of interpolation spaces is very well developed and one already know to characterize the interpolation spaces for Sobolev spaces of positive and negative indices, we have not found in the (huge) literature a result which covers the problem we are concerned with. So, although our result may be viewed as an interpolation result, it is a new one. As an application we discussed the regularity of the law of a Wiener functional under a Hörmander type non degeneracy condition. These papers will appear in Annals of Probability.

Regularity of the solution of jump type equations

Continuing the above work we study, in collabration with Lucia Caramellino, the regularity of the solution of jump type equations. This subject has been extensively treated in the literarure using different hypothesis and different variants of Malliavin calculus adapted to equations with jumps. The case of Poisson Point measures with absolutely continuous intensity measure is already well understood with the paper of Bichteler, Garereux and Jacod in the 80's. But the case of discrete intensity measures is more subtle. In this case J. Picard has succeded to obtain regularity results using a variant of Malliavin Calculus based on finite differences. We work also in this framework but we do not use directly some variant of Malliavin calculus but we use an interpolation argument . These are still working papers.

An invariance principle for stochastic series (U- Statistics)

In collaboration with L. Caramellino we work on invariance principles for stochastic series of polynomial type. In the case of polynomials of degree one we must have the classical Central Limit Theorem (for random variables which are not identically distributed). For polynomials of higher order we are in the framework of the so called U statistics which have been introduced by Hoffdings in the years 1948 and which play an important role in modern statistics. Our contribution in this topic concerns convergence in total variation distance for this type of objects. We use abstract Malliavin calculus and more generally, the methods mentioned in the above paragraph.