• 10h00 - Lucas IZYDORCZYK (ENSTA Paris et Università Milano Bicocca)

Probabilistic backward McKean numerical methods for PDEs and one application to energy management.

Summary. This thesis concerns McKean Stochastic Differential Equations (SDEs) to represent possibly non-linear Partial Differential Equations (PDEs). Those depend not only on the time and position of a given particle, but also on its probability law. In particular, we treat the unusual case of Fokker-Planck type PDEs with prescribed final data. We discuss existence and uniqueness for those equations and provide a probabilistic representation in the form of McKean type equation, whose unique solution corresponds to the time-reversal dynamics of a diffusion process. We introduce the notion of fully backward representation of a semilinear PDE: that consists in fact in the coupling of a classical Backward SDE with an underlying process evolving backwardly in time. We also discuss an application to the representation of Hamilton-Jacobi-Bellman Equation (HJB) in stochastic control. % namely FBSDE representations in which the forward process evolves backward in time, with application to stochastic control. Based on this, we propose a Monte-Carlo algorithm to solve some control problems which has advantages in terms of computational efficiency and memory when compared to traditional forward-backward approaches. We apply this method in the context of demand side management problems occurring in power systems. Finally, we survey the use of generalized McKean SDEs to represent non-linear and non-conservative extensions of Fokker-Planck type PDEs.