We address a portfolio selection problem that combines active (outperformance) and passive (tracking) objectives using techniques from convex analysis. We assume a general semimartingale market model where the assets' growth rate processes are driven by a latent factor. Using techniques from convex analysis we obtain a closed-form solution for the optimal portfolio and provide a theorem establishing its uniqueness. The motivation for incorporating latent factors is to achieve improved growth rate estimation, an otherwise notoriously difficult task. To this end, we focus on a model where growth rates are driven by an unobservable Markov chain. The solution in this case requires a filtering step to obtain posterior probabilities for the state of the Markov chain from asset price information, which are subsequently used to find the optimal allocation. We show the optimal strategy is the posterior average of the optimal strategies the investor would have held in each state assuming the Markov chain remains in that state. Finally, we implement a number of historical backtests to demonstrate the performance of the optimal portfolio.

我们解决了一个组合选择问题，该问题使用凸分析的技术结合了主动（表现优异）和被动（跟踪）目标。我们假设一个一般的半鞅市场模型，其中资产的增长率过程由潜在因素驱动。使用来自凸分析的技术，我们获得了最优投资组合的封闭形式的解决方案，并提供了建立其唯一性的定理。纳入潜在因素的动机是为了实现改进的增长率估计，否则这是一项众所周知的艰巨任务。为此，我们专注于一个模型，其中增长率由不可观察的马尔可夫链驱动。这种情况下的解决方案需要一个过滤步骤，从资产价格信息中获得马尔可夫链状态的后验概率，随后用于寻找最优分配。我们表明最优策略是假设马尔可夫链保持在该状态下投资者在每个状态下持有的最优策略的后验平均值。最后，我们实施了一些历史回测来证明最佳投资组合的表现。