Optimal Investments for Robust Utility Functionals in Complete Market Models

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 Q of probability measures. Our main result shows that this problem can often be reduced to determining a " least favorable " measure Q 0 ∈ Q, 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 Q 0. By using the Huber-Strassen theorem from robust statistics, it is shown that Q 0 always exists if Q is the σ-core of a 2-alternating capacity. Besides other examples, 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).