Robust Utility Maximization with Drift and Volatility Uncertainty

We give explicit solutions for utility maximization of terminal wealth problem $u(X_T)$ in the presence of Knightian uncertainty in continuous time $[0,T]$ in a complete market. We assume there is uncertainty on both drift and volatility of the underlying stocks, which induce nonequivalent measures on canonical space of continuous paths $\text{O}$. We take that the uncertainty set resides in compact sets that are time dependent. In this framework, we solve the robust optimization problem with logarithmic, power and exponential utility functions, explicitly.