Optimal bounds for the densities of solutions of SDEs with measurable and path dependent drift coefficients

We consider a process given as the solution of a stochastic differential equation with irregular, path dependent and time-inhomogeneous drift coefficient and additive noise. Explicit and optimal bounds for the Lebesgue density of that process at any given time are derived. The bounds and their optimality is shown by identifying the worst case stochastic differential equation. Then we generalise our findings to a larger class of diffusion coefficients.