Contents of lectures:

1. Noufel Frikha (talk): Well-posedness of McKean-Vlasov SDEs, related PDE on the Wasserstein space and some new quantitative estimates for propagation of chaos
   
   • Summary: I will present some recent results on the well-posedness in the weak and strong sense of some non-linear stochastic differential equations (in the sense of McKean-Vlasov) driven by Brownian and/or jump processes which go beyond those derived from the standard Cauchy-Lipschitz theory (see e.g. the monograph of Sznitman). Then, in the Brownian setting, I will show how the underlying noise regularizes the equation and allows to prove that the transition density of the dynamics exists and is smooth, especially in the measure direction, under the uniform ellipticity assumption. Such smoothing effects then in turn allow to establish the existence and uniqueness for the Cauchy problem associated to a Kolmogorov PDE stated on the Wasserstein space (the space of probability measures with finite second order moment) with irregular terminal condition and source term. This PDE stated on an infinite dimensional space plays a key role in deriving new quantitative estimates of propagation of chaos for the mean-field approximation by systems of interacting particles. This presentation is based on several recent works in collaboration with: P.-E. Chaudru de Raynal (Université Savoie Mont Blanc), V. Konakov (HSE Moscow), L. Li (UNSW Sydney) and S. Menozzi (Université d'Evry Val d'Essone).
2. Reinhard Höpfner (course): Point process models and local asymptotics in statistics
   • Likelihood ratio processes for point processes (purely probabilistic)
   • Main results of the statistical theory of local asymptotic normality (LAN) in the sense of LeCam (purely statistical)
   • Detailed treatment of one example where probability and statistics together allow to specify asymptotically optimal estimators for parameters in a multivariate point process model
3. Arnaud Gloter (course): Malliavin calculus for pure jump processes and statistical applications
   • Malliavin calculus and integration by parts setting
   • Malliavin calculus on the Poisson space and application to S.D.E driven by jump processes
   • Statistical applications for S.D.E driven by stable like processes: approximation of the score function and LAMN properties
4. Arturo Kohatsu-Higa (course): Interaction between simulation and theory
   • Simulation of Poisson random measures
   • Exact simulation methods for jump driven SDE's
   • Simulation methods for infinite activity Lévy processes and their associated functionals
   • Summary: I will mostly concentrate on simulation methodology with associated theoretical results such as convergence results and their rate. I also plan to discuss the interactions between theory and applications. Some interesting and feasible open problems will be discussed, too.
5. Alex Kulik (course): Lectures on the parametrix method
• Basic constructions: diffusion processes and parabolic PDEs
• Sensitivities and approximations
• Non-local PDEs: new effects and methods
• Summary: I will provide a self-contained introduction to the parametrix method, which is a powerful analytical tool for construction and investigation of heat kernels of diffusion and Lévy-type processes.
6. Andrea Pascucci (talk): A class of Kolmogorov SPDEs and applications to stochastic filtering
7. René Schilling (talk): Detecting Independence with Distance Multivariance based on Lévy Metrics
   • Summary: We introduce two new measures for the dependence of n>1 random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted L^2-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Székely, Rizzo and Bakirov) from pairs of random variables to n-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of n random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.
     This is joint work with B. Böttcher and M. Keller-Ressel.