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Optimal Investments for Robust Utility Functionals in Complete Market Models

Institut für Mathematik, MA 7-4 TU Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
schied{at}math.tu-berlin.de, http://www.math.tu-berlin.de/~schied

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 of probability measures. Our main result shows that this problem can often be reduced to determining a "least favorable" measure Q₀ , 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 is the -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."

Key Words: robust utility functional; utility maximization; Knightian uncertainty; robust Savage representation; least favorable measure; uncertain drift; Huber-Strassen theory
History: Received: April 7, 2004; revision received: August 8, 2004;