Department of Mathematics - Hong Kong-Singapore joint Seminar Series in Financial Mathematics/Engineering - A coupling approach to the turnpike phenomenon in stochastic control and McKean-Vlasov control

Coupling methods provide a powerful toolbox for the quantitative analysis of the long-time behaviour of Markov processes. In particular, coupling by reflection allows to establish sharp exponential convergence results in Wasserstein distance for the Fokker-Planck equation without having to rely on pointwise assumptions on the confinement potential. The purpose of this talk is to illustrate the construction of a variant of coupling by reflection that applies to optimally controlled diffusion processes, including controlled McKean-Vlasov processes. Such construction opens the door for a precise study of the long-time behavior of optimizers: in particular it provides with uniform in time gradient (and Hessian) estimates for the solution of Hamilton-Jacobi-Bellman equations that enable to prove various kind of exponential turnpike properties for the optimal processes and controls. This talk is partially based on joint work with Katharina Eichinger, Alain Durmus, and Alekos Cecchin.