McKean-Vlasov equations : a probabilistic and pathwise approach

This thesis divides neatly into four collections of results. In the first (Part II), we provide existence and uniqueness results along with several properties for a class of McKean-Vlasov Equations having random coefficients and drifts of superlinear growth. We show a Freidlin-Wentzell-type Large Deviations Principles (LDP) in the Hölder topology for the solution process of McKean-Vlasov Equations using techniques that directly address the presence of the law in the coefficients. Our methods avoid using decoupling tricks or particle system approximations. Secondly (Part III), we close an unexpected gap in the literature concerning the Malliavin and Parametric Differentiability of Stochastic Differential Equations with drifts of super linear growth and with random coefficients. We establish Stochastic Gâteaux Differentiability and Ray Absolute Continuity and take limits in probability rather than mean square or almost surely, bypassing the potentially non-integrable error terms from the unbounded drift. As an application, we recover representations linking both derivatives as well as a Bismut- Elworthy-Li formula. Thirdly (Part IV), we prove a representation for the support of McKean-Vlasov Equations. To do so, we construct functional quantizations for the law of Brownian motion as a measure over the (non-reflexive) Banach space of Hölder continuous paths. By solving optimal Karhunen Loève expansions and exploiting the compact embedding of Gaussian measures, we obtain a sequence of deterministic finite supported measures that converge to the law of a Brownian motion with explicit rate. We show the approximation sequence is near optimal with very favourable integrability properties and prove these approximations remain true when the paths are enhanced to rough paths. These results are of independent interest. The functional quantization results then yield a novel way to build deterministic, finite supported measures that approximate the law of the McKean-Vlasov Equation driven by the Brownian motion which crucially avoid the use of random empirical distributions. These are then used to solve an approximate skeleton process that characterises the support of the McKean-Vlasov Equation. We give explicit rates of convergence for the deterministic finite supported measures in rough path Hölder metrics and determine the size of the particle system required to accurately estimate the law of McKean-Vlasov equations with respect to the Hölder norm. Finally (Part V), we study the Small Ball Probabilities of Gaussian rough paths. While many works on rough paths study the Large Deviations Principles for stochastic processes driven by Gaussian rough paths, it is a noticeable gap in the literature that Small Ball Probabilities have not been extended to the rough path framework. LDPs provide macroscopic information about a measure for establishing Integrability type properties. Small Ball Probabilities provide microscopic information and are used to establish a locally accurate approximation for a random variable. Given the compactness of a Reproducing Kernel Hilbert space ball, its Metric Entropy provides invaluable information on how to approximate the law of a Gaussian rough path. As an application, we are able to find upper and lower bounds for the rate of convergence of an empirical rough Gaussian measure to its true law in pathspace.