Essays on financial markets with frictions

These essays examine optimal consumption and investment problems when an agent is confronted by a financial market consisting of a risk free asset and a risky asset which trades with a bid-ask spread. We allow for quite general specifications of the financial market. The analysis characterizes the optimal investment and consumption strategies using probabilistic and convex analysis. The first essay analyzes a continuous trading market while the second essay analyzes a finite state discrete trading market. First, it is demonstrated that any feasible trading strategy in the transaction cost economy must satisfy a large number of "linear" budget constraints each of which corresponds to a frictionless financial market without transactions costs. We then examine the case where short sales are prohibited. In this case, existence of the optimal trading strategies imply the existence of two supermartingales whose ratio is bounded by the bid and ask prices. If the short sales constraints do not bind, then the supermartingales are local martingales and we can construct a frictionless economy which supports the optimal trading strategies and solution in the transaction cost problem. In the case where short sales are allowed in the discrete time economy but consumption is required to be positive, it is shown that we can always construct a frictionless economy which supports the optimal trading strategies and solution to the transaction cost problem when the terminal consumption is strictly positive. This result is also demonstrated for the continuous trading economy under some rather strigent assumptions on the optimal solution to the transaction cost problem. Many of our results are closely linked to the differentiability properties of the value function and we reach interesting conclusions concerning this issue.