  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.
