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We find explicit upper bounds for the density of marginals of continuousdiffusions where we assume that the diffusion coefficient is constant and thedrift is solely assumed to be progressively measurable and locally bounded. Inone dimension we extend our result to the case that the diffusion coefficientis a locally Lipschitz-continuous function of the state. Our approach is basedon a comparison to a suitable doubly reflected Brownian motion whose density isknown in a series representation.

Explicit Local density bounds for It\^o-processes with irregular drift