We introduce a systematic approach to the problem of maximizing the robust utility of the terminal wealth of an admissible strategy in a general complete market model, where the robust utility functional is defined by a set \cQ of probability measures. Our main result shows that this problem can be reduced to determining a "least favorable" measure Q_0\in\cQ, which is universal in the sense that it does not depend on the particular utility function. The robust problem is thus equivalent to a standard utility maximization problem with respect to the "subjective" probability measure Q0. By using the Huber-Strassen theorem from robust statistics, it is shown that Q0 always exists if \cQ is the core of a 2-alternating upper probability. We also discuss the problem of robust utility maximization with uncertain drift in a Black-Scholes market and the case of "weak information" as studied by Baudoin (2002)
