A sharp upper bound for the expected occupation density of It\^o processes with bounded irregular drift and diffusion coefficients

We find explicit and optimal upper bounds for the expected occupation density for an It\^o-process when its drift and diffusion coefficients are unknown under boundedness and ellipticity conditions on the coefficients. This is related to the optimal bound for the expected interval occupation found in Ankirchner and Wendt(2021). In contrast, our bound is for a single point and the resulting formula is less involved. Our findings allow us to find explicit upper bounds for mean path integrals.