David Ruiz Baños: Optimal bounds and Hölder continuous densities of solutions of SDEs with measurable and path-dependent drift coefficients

We consider a process given as the solution of a one-dimensional stochastic differential equation with irregular, path-dependent and time-inhomogeneous drift coefficient and additive noise. Hölder continuity of any order of the Lebesgue density of that process at any given time is achieved. Explicit and optimal upper and lower bounds for the densities are also obtained. The regularity of the densities is obtained via the inverse Fourier theorem by identifying the stochastic differential equation with the worst characteristic function. The latter is done by employing a method already introduced by the authors in a previous work to find explicit optimal bounds for the densities. Then we generalise our findings to a larger class of diffusion coefficients. In the Markovian setting the densities are directly connected to the fundamental solution of the Fokker-Planck equation. For this reason, this entitles us to improve the regularity of the fundamental solution in dimension one. This is based on a joint work with Dr. Paul Krühner (TU Wien).