On two utility maximization problems

This dissertation studies two expected utility maximization problems from mathematical finance. The first project (Chapter 2) deals with a single-agent utility maximization under constraints on intertemporal consumption; the second project (Chapter 3) studies Nash equilibria in an N-player game of utility maximization under relative performance criteria. The main and unifying approach to the two problems is convex duality in a semimartingale market model. Utility maximization under ratchet and drawdown constraints on consumption in incomplete semimartingale markets. We consider the value function associated with this concave optimization problem as having two parameters: the initial wealth and the essential lower bound on consumption rate process. In order to rigorously formulate the constraints and define the primal domains for optimization in the general setting of semimartingale markets and optional consumption plans, we introduce a natural extension of the notion of running maximum to arbitrary non-negative optional processes. The dual domains for optimization are characterized in terms of solidity of the set of equivalent martingale deflators with respect to an ordering that is introduced on the set of non-negative optional processes. The abstract duality result obtained in the general incomplete market case is used in order to derive a more detailed characterization of solutions in the complete market case. Utility maximization under relative performance criteria of multiplicative form. In the first part of this project we consider the case of a complete semimartingale market. For a large class of players’ utility functions, a general characterization of Nash equilibria for a given initial wealth vector is provided in terms of invertibility of a map from ℝᴺ to ℝᴺ. As a consequence of the general theorem, we derive existence and uniqueness of Nash equilibria for an arbitrary initial wealth vector, as well as their convergence, in two special cases. The second part studies the case of CRRA utility functions in an incomplete Itô-process market model. We establish a one-to-one correspondence between Nash equilibria satisfying a certain integrability condition and solutions of a BSDE when relative risk aversions of players are small (less than 1) and linear constraints on trading strategies are allowed.Mathematic