  We find explicit upper bounds for the density of marginals of continuous
diffusions where we assume that the diffusion coefficient is constant and the
drift is solely assumed to be progressively measurable and locally bounded. In
one dimension we extend our result to the case that the diffusion coefficient
is a locally Lipschitz-continuous function of the state. Our approach is based
on a comparison to a suitable doubly reflected Brownian motion whose density is
known in a series representation.
