Convex duality in singular control--Optimal consumption choice with intertemporal substitution and optimal investment in incomplete markets.

摘要

In this thesis we study the problem of optimal consumption choice with investment in incomplete markets. The agent's preferences are modeled using non time-additive utilities of the type proposed by Hindy, Huang and Kreps. For such preferences the period utilities depend on the entire path of consumption up to date.;We show that a dual relationship exists between the utility optimization problem and a carefully chosen dual minimization problem. Time-inhomogeneity of the preferences and the dependence on past consumption leads to utility gradients that in a deterministic setting, have the structure of inhomogeneously convex functions. A stochastic representation theorem is used to extend this concept to apply in the random setting. We find that the appropriate dual variables are not necessarily adapted, but that they do have adapted densities.;We illustrate the techniques by finding explicit solutions in a Wiener driven market with multiple assets. For the explicit solutions we pass to the infinite time-horizon, and show how to use the duality framework as a verification theorem. The optimal solution is to consume whenever the supremum of a certain Brownian motion with drift increases. Thus optimal consumption is singular: there is no period of time in which the agent consumes continuously.