We give explicit solutions for utility optimization problems in the presence of Knightian uncertainty in continuous time with nondominated priors and finite time horizon in a diffusion model. We assume that the uncertainty set is compact and time dependent on $[0,T]$. We solve the robust optimization problem explicitly both when the investor's utility is of CRRA and of CARA type and give an $\alpha$-maximin type expected type utility variation to the robust problem. To the best of our knowledge, this is the first work in deriving explicit solutions in continuous time with time varying uncertainty sets in a nondominated prior setting.
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