H\"older 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. H\"older continuity of the Lebesgue density of that process at any given time is achieved using a different approach than the classical ones in the literature. Namely, the H\"older regularity of the densities is obtained via a control problem by identifying the stochastic differential equation with the worst global H\"older constant. Then we generalise our findings to a larger class of diffusion coefficients. The novelty of this method is that it is not based on a variational calculus and it is suitable for non-Markovian processes.