Robust utility maximization of terminal wealth with drift and volatility uncertainty

We give explicit solutions for utility maximization of terminal wealth problem (Formula presented.) in the presence of Knightian uncertainty (Formula presented.) in continuous time (Formula presented.). We assume there is uncertainty on both drift and volatility of the underlying stocks, which induce nonequivalent measures on canonical space of continuous paths Ω. 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. Numerical simulations revealing the effects of uncertainty on the dynamics are also presented.