Quantitative propagation of chaos of McKean-Vlasov equations via the master equation

Abstract

McKean-Vlasov stochastic differential equations (MVSDEs) are ubiquitous in kinetic theory and in controlled games with a large number of players. They have been intensively studied since McKean, as they pave a way to probabilistic representations for many important nonlinear/ nonlocal PDEs. Classically, their simulation involves using standard particle systems, which replace the evolving law in MVSDEs by the evolving empirical measure of the particles. However, this type of simulation is costly in terms of computational complexity, due to the interaction between the particles. Apart from classical techniques in stochastic analysis, the approach in this thesis relies heavily on the calculus on Wasserstein space, presented by P. Lions in his course at Collège de France. An important object in our study, is a PDE written on the product space of the space of time horizon and the Wasserstein space, which is a generalisation of the classical Feynman- Kac PDE. This PDE, namely the master equation, provides a new insight into the study of mean-field limits of particles and consequently allows us to solve many problems on MVSDEs that are very difficult/impossible to solve by classical techniques. The layout of the thesis is as follows. We start by a recap on classical results of MVSDEs (Chapter 2), followed by a full exposition of Wasserstein calculus on the results that we need (Chapter 3). Chapters 4 and 5 propose approximating systems to MVSDEs (as alternatives to the classical particle system) via Romberg extrapolation and Antithetic Multi-level Monte-Carlo estimation respectively, which are less costly in terms of computational complexity. Finally, in Chapter 6, we explore the converse: given a standard particle system, we hope to find an alternative mean-field limit that gives a better approximation to the standard particle system.