Section: New Results

Stochastic Analysis

Participants : Nicolas Champagnat, Julien Claisse, Denis Talay.

• N. Champagnat studied in collaboration with P.-E. Jabin (Univ. of Maryland) strong existence and pathwise uniqueness for stochastic differential equations driven by a Brownian motion and with rough coefficients [34] . The method is an extension of the one of [50] , which studies well-posedness for deterministic dynamical system. Strong existence and pathwise uniqueness can be proved for example if the drift vector is $L^1\left(W^{1,1}\right)$ and the diffusion matrix is uniformly elliptic and $L^q\left(W^{1,p}\right)$ with $2/q+d/p=1$. This improves the previous conditions of [53] .

• J. Claisse and D. Talay studied in collaboration with X. Tan (Univ. of Paris Dauphine) a conditioning argument which is often used to prove the dynamic programming principle [36] . Their study of the literature revealed that previous proofs of this argument are incorrect or incomplete. They provided a rigorous and detailed proof by setting up martingale controlled problems in a original way.