David Ruiz Baños: On the regularity of densities of Itô-type processes via stochastic control.

The study of the existence and regularity of densities of random variables has been an active area of research during the last decades for its applications and for its own interest. There are many known results for the case of Itô-type processes or random variables given as solutions of stochastic differential equations (SDEs). Nevertheless, one usually needs to assume that the processes involved are well-behaved. A common tool used in such problems is the so-called Malliavin calculus, which actually, was motivated to prove that the finite-dimensional laws of a solution of an SDE with smooth coefficients (plus Hörmander's condition) are absolutely continuous with smooth densities.

Nevertheless, it turns out to be quite difficult to approach the problem in the irregular setting, e.g. when the coefficients are non-Lipschitz. For this reason, in this work we present a different approach based on a stochastic control argument. We prove that solutions of SDEs at a given time with bounded, time-inhmogeneous and path-dependent coefficients admit a Hölder-continuous density in dimension one. As a side-effect we are able to improve the regularity of the fundamental solution of Fokker-Planck's equation in dimension one. This is a joint work with Dr. Paul Krühner (Technische Universität Wien).