SINGULAR MCKEAN-VLASOV PROBLEMS FROM MATHEMATICAL PHYSICS AND FINANCE

Abstract

The purpose of this thesis is to develop techniques for analyzing the limiting McKean-Vlasov dynamics of interacting particle systems featuring singularities, and arising in physics and mathematical finance. We first investigate the asymptotic stability of unidimensional log gases under non- convex confining potentials by establishing new Entropy-Wassertein-Information (HWI) inequalities. Such gases are obtained as the mean-field limit of particles interacting via a repulsive logarithmic potential. Then, we establish the well-posedness of the supercooled Stefan problem with oscilla- tory initial condition. This classical problem from mathematical physics is reformulated using a probabilistic description of the free boundary as a cumulative distribution function of the hitting time of a Brownian motion with a jumping drift. Finally, we study the well-posedness problem of a class of bidimensional stochastic dif- ferential equations (SDE), whose coefficients depend on the joint density of the unknown process. This class of local stochastic volatility models is important for the calibration of volatility surfaces. Additionally, we solve the long-standing problem of joint S&P 500/VIX calibration by using SDEs controlled by neural networks.