Optimal investments for robust utility functionals in complete market models

This paper introduces 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 Q0∈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 Q0. By using the Huber–Strassen theorem from robust statistics, it is shown that Q0 always exists if Q is the s-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.”