"This project focuses on a class of portfolio choice problems arising in Mathematical Finance. These problems share a common relevance for financial applications, and lead to novel mathematical questions, mainly in the area of Stochastic Processes. Research is proposed on dynamic portfolio choice with: (i) Random Environments; (ii) Trading Frictions; and (iii) Incentive Fees. Random environments encompass those asset pricing models in which interest rates, risk premia, and covariances may depend on state variables. These model are important because recent empirical investigations show that variables such as the dividend yield, or earnings/price ratios drive expected returns, paving the way to multifactor models. Transaction costs lead to similar technical questions, although for conceptually different reasons. State variables arise in these models implicitly, through the notion of shadow prices. This partial isomorphism between market frictions and random environments suggests that transaction costs have an hidden potential as tools for explaining pricing anomalies. Incentive fees, traditionally neglected as minor imperfections, are now actively studied for their role in shaping the behavior of intermediaries. For example, high-water mark contracts, now prevalent in the hedge-fund industry, have ambiguous effects on fund managers. Understanding these effects is crucial for evaluating the potential consequences of regulation. In summary, the proposed problems share a common relevance, and similar technical features, which require novel mathematical tools. This project aims at developing these tools, and at bringing to life their implications."