Optimal consumption from investment and random endowment in incomplete semimartingale markets

Abstract

We consider the problem of maximizing expected utility from consumption in a constrained incomplete semimartingale market with a random endowment process, and establish a general existence and uniqueness result using techniques from convex duality. The notion of "asymptotic elasticity'' of Kramkov and Schachermayer is extended to the time-dependent case. By imposing no smoothness requirements on the utility function in the temporal argument, we can treat both pure consumption and combined consumption--terminal wealth problems in a common framework. To make the duality approach possible, we provide a detailed characterization of the enlarged dual domain which is reminiscent of the enlargement of $\lone$ to its topological bidual $\linfd$, a space of finitely additive measures. As an application, we treat a constrained Itô process market model, as well as a "totally incomplete'' model.