Andrew Stuart (California Institute of Technology, USA)
Jean-Francois Jabir (Moskau)
Abstract: This seminar aims to give a broad and straightforward presentation on fundamental aspects related to the theory and application of continuous-time stochastic processes and the more specific class of McKean-Vlasov dynamics. The first part of the talk will present/recall some essentials of stochastic analysis, stochastic modeling, some notion of numerical probability, and some elemental examples and applications in Physics and Finance. We will in particular review the notions of diffusion processes, Brownian motion, martingales, Itô's calculus, Stochastic Differential Equations and their link with partial differential equations. These notions will serve as prerequisite for the second part of the talk which will be dedicated to the general subject of McKean-Vlasov dynamics. These stochastic models were originaly introduced in the last century for the probabilistic interpretation of nonlinear equations arising from statistical physics and emerge typically from the asymptotic of some interacting particle systems. Nowadays McKean-Vlasov dynamics are more largely studied and applied in the fields of game theory (Mean-Field Games), optimal control problem and the modeling of interacting multiagent systems. After a short presentation on the mathematical characteristics of such models, we will review some past and current trends related to these models in the framework of fluid mechanics, Finance, stochastic interacting particle/agents models, and Economy.