In this paper, we present a duality theory for the robust utility maximisation problem in continuous time for utility functions defined on the positive real line. Our results are inspired by – and can be seen as the robust analogues of – the seminal work of Kramkov and Schachermayer (Ann. Appl. Probab. 9:904–950, 1999). Namely, we show that if the set of attainable trading outcomes and the set of pricing measures satisfy a bipolar relation, then the utility maximisation problem is in duality with a conjugate problem. We further discuss the existence of optimal trading strategies. In particular, our general results include the case of logarithmic and power utility, and they apply to drift and volatility uncertainty.

在本文中，我们提出了一个对偶理论，用于在连续时间内对定义在正实线上的效用函数的鲁棒效用最大化问题。我们的结果受到 Kramkov 和 Schachermayer（Ann. Appl. Probab. 9:904–950, 1999）开创性工作的启发，并且可以被视为其强大的类似物。也就是说，我们表明，如果可达到的交易结果集和定价措施集满足双极关系，则效用最大化问题是具有共轭问题的对偶问题。我们进一步讨论了最优交易策略的存在。特别是，我们的一般结果包括对数和幂效用的情况，它们适用于漂移和波动性的不确定性。