We study a utility maximization problem in a financial market with a stochastic drift process, combining a worst-case approach with filtering techniques. Drift processes are difficult to estimate from asset prices, and at the same time optimal strategies in portfolio optimization problems depend crucially on the drift. We approach this problem by setting up a worst-case optimization problem with a time-dependent uncertainty set for the drift. Investors assume that the worst possible drift process with values in the uncertainty set will occur. This leads to local optimization problems, and the resulting optimal strategy needs to be updated continuously in time. We prove a minimax theorem for the local optimization problems and derive the optimal strategy. Further, we show how an ellipsoidal uncertainty set can be defined based on filtering techniques and demonstrate that investors need to choose a robust strategy to be able to profit from additional information.

中文翻译：

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具有随机漂移的多变量金融市场中的稳健效用最大化

我们研究了具有随机漂移过程的金融市场中的效用最大化问题，将最坏情况方法与过滤技术相结合。漂移过程很难从资产价格中估计，同时投资组合优化问题中的最优策略关键取决于漂移。我们通过为漂移设置时间相关的不确定性来设置最坏情况优化问题来解决这个问题。投资者假设将发生具有不确定性集合中的值的最坏可能的漂移过程。这会导致局部优化问题，由此产生的最优策略需要及时不断更新。我们证明了局部优化问题的极小极大定理并推导出最优策略。进一步，