Topics in Financial Mathematics and Stochastic Control

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The project funded by this award consists of a collection of topics in Stochastic Control and Financial Mathematics. The first topic is an optimal investment problem where the investor is paying a proportional share of the profit to the fund manager. The problem involves both modeling and the analysis of the resulting non-standard stochastic control problem, in order to fully characterize the optimal policy. In addition, an asymptotic analysis for small proportional fees will be performed. The second topic represents a characterization of semimartingale models of financial markets where Mutual Fund Theorems hold true in the context of expected utility from consumption. This will provide a tool for the study of mutual fund theorems for markets in equilibrium. The last topic is based on a new definition of admissible strategies for the optimal investment problem for utilities defined on the whole real line and the duality results that derive from it.Incomplete markets are financial models where contingent claims cannot be replicated by trading, so they are not redundant. While, in practice, most models are incomplete, the mathematical analysis of optimal investment and pricing in these markets is usually very difficult. The present project contributes to both modeling of incompleteness arising from different market frictions, as well as to the mathematical analysis of the new stochastic control problems resulting from such models. The second topic of the project is expected to provide a better understanding of incomplete markets in a very general mathematical framework.