In this paper we investigate a utility maximization problem with drift uncertainty in a continuous-time Black--Scholes type financial market. We impose a constraint on the admissible strategies that prevents a pure bond investment and we include uncertainty by means of ellipsoidal uncertainty sets for the drift. Our main results consist in finding an explicit representation of the optimal strategy and the worst-case parameter and proving a minimax theorem that connects our robust utility maximization problem with the corresponding dual problem. Moreover, we show that, as the degree of model uncertainty increases, the optimal strategy converges to a generalized uniform diversification strategy.
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