May – June – July 2014 : Professor Arturo Kohatsu-Higa has been invited for three months (May, June and July) by the Labex Bézout. He has given a lecture (6 times 3 hours) on”Stochastic Differential Equations with irregular coefficients” in June. This lecture was opened to master students, PhD students and researchers and was followed by roughly 20 participants. Concerning the research activity, Aurélien Alfonsi, Benjamin Jourdain and Arturo Kohatsu-Higa have worked on the uniform in time estimation of the Wasserstein distance between the time-marginals of an elliptic diffusion and its Euler scheme. To generalize in higher dimension the result that they obtained previously in dimension one using the optimality of the explicit inverse transform, they compute the derivative of the Wasserstein distance with respect to the time variable thanks to the theory developed by Ambrosio Gigli and Savare. The abstract properties of the optimal coupling between the time marginals then enable them to estimate this time derivative. This work has been finished during the visit of Arturo, and several continuations of this work have been investigated. In particular, an important question to extend the estimate on marginal laws to pathwise laws would be to get sharp estimates for the diffusion bridges in any dimension. Aurélien Alfonsi and Arturo Kohatsu-Higa have worked on the application of the parametrix method for the exact simulation of SDEs to the case of reflected SDEs. This method has been developed by Arturo and Vlad Bally for general SDEs, and can be adapted in principle to other Markovian processes. However, there are some technical difficulties that are related to reflected SDEs that needs to be settled. This work is still in progress and is in collaboration with Masafumi Hayashi (University of the Ryukyus). Vlad Bally and Arturo investigated the possibility to use the parametrix method in order to obtain more regularity for the density of diffusion processes with Hölder coefficients (in the original work of Arturo and Vlad on the parametrix method one obtains just the continuity of this density). The idea which has been considered is to combine the parametrix method with some interpolation techniques developped in a recent work of Vlad and Lucia Caramellino. For the moment this work is just in progress and some serious difficulties remain to be solved. Arturo KOHATSU-HIGA
