In this paper we study a robust utility maximization problem in continuous
time under model uncertainty. The model uncertainty is governed by a continuous
semimartingale with uncertain local characteristics. Here, the differential
characteristics are prescribed by a set-valued function that depends on time
and path. We show that the robust utility maximization problem is in duality
with a conjugate problem, and we study the existence of optimal portfolios for
logarithmic, exponential and power utilities.Comment: To appear in "Mathematics and Financial Economics