In this thesis we deal with different topics in financial mathematics, that are all related to market imperfections and to the fundamental technique of utility maximization. The work consists of three parts. In the first one, which is based on two papers, we consider the problem of optimal investment on a financial market with proportional transaction costs. We initially study the investment problem in the case where the utility function is multivariate (which is particularly suitable on currency markets) and the agent is endowed with a random claim, which can be interpreted as an option or another derivative contract. After analyzing the properties of the primal and dual problems, we apply those results to investigate, in this context, some aspects of a popular pricing technique in incomplete markets, i.e. utility indifference evaluation. In the second contribution to the transaction costs literature, we investigate the existence problem for a set of prices (called shadow prices) that would provide the same maximal utility to the agent if the market did not have frictions. These results shed some light on the link between the classical theory of frictionless markets and the quickly growing literature on transaction costs. In the second part of this thesis we consider the utility indifference pricing problem in incomplete markets, where the source of incompleteness comes from the fact that some assets in the market cannot be actively traded, which is the case for example in the framework of structural models for electricity prices. We provide a BSDE characterization for the price under mild assumptions, and then focus on the case of European claims by establishing in particular the existence of an optimal hedging strategy even when the claim presents discontinuities and is possibly unbounded. In the last contribution we analyze a simple principal-agent problem in finite time horizon, where the principal is mainly interpreted as a regulator and the agent as a firm producing some kind of polluting emissions. We separately treat both the agent's and the principal's problems and use the BSDE theory for providing necessary and sufficient conditions for optimality. We also perform some sensitivity analyses and give numerical results in order to provide a better understanding of the agents' behavior.
