Regularity of McKean-Vlasov stochastic differential equations and applications

Abstract

In this thesis, we study time-inhomogeneous and McKean-Vlasov type stochastic differential equations (SDEs), along with related partial di erential equations (PDEs). We are particularly interested in regularity estimates and their applications to numerical methods. In the rst part of the thesis, we build on the work of Kusuoka & Stroock to develop sharp estimates on the derivatives of solutions to time-inhomogeneous parabolic PDEs. The basis of these estimates is an integration by parts formula for derivatives of the solution under the UFG condition, which is weaker than the uniform Hormander condition. This integration by parts formula is obtained using Malliavin Calculus. The formula allows us to extend the notion of classical solution to a framework where differentiability does not necessarily hold in all directions. As an application, we extend the error analysis for the cubature on Wiener space method to time-inhomogeneous stochastic di erential equations. We then present two cubature on Wiener space algorithms for the numerical solution of McKean-Vlasov SDEs with smooth scalar interaction. The analysis involves the regularity estimates proved previously and takes place under a uniform strong Hormander condition. Finally, we develop integration by parts formulas on Wiener space for solutions of SDEs with general McKean-Vlasov interaction and uniformly elliptic coe cients. These formulas hold both for derivatives with respect to a real variable and derivatives with respect to a measure in the sense of Lions. This allows us to develop estimates on the density of solutions of the McKean-Vlasov SDEs. We also prove the existence of a classical solution to a related PDE with irregular terminal condition.