Optimal Execution in a General One-Sided Limit-Order Book and Endogenous Dynamic Completeness of Financial Models

This thesis consists of two parts. The first one is a result obtained under the supervision of Steven Shreve and with the collaboration of Gennady Shaikhet. Our work yielded a detailed description of the optimal strategies for a large investor, when she needed to buy a large amount of shares of a stock over a finite time horizon. The dynamics of the limit order book of the underlying stock is a generalization of known results to continuous time and to arbitrary distributions of the said limit order book. See the introduction section in chapter 2 for a more detailed discussion.;The second part is a result obtained under the supervision of Dmitry Krarnkov. Our work yielded a sufficient condition on the structure of the economic factors, dividends of traded assets and total endowment in a single-agent economy, such that in an Arrow - Debreu - Radner equilibrium the market is complete. The main result is formulated as an integral representation theorem. Our work generalizes and complements fairly recent results in this direction (at the time of this thesis) by requiring less smoothness of the driving diffusion process at the expense of seemingly stronger conditions on the terminal dividends of the assets. See the introduction section in chapter 3 for a more detailed discussion.