Optimal consumption from investment and random endowment in incomplete semimartingale markets

TL;DR

This paper develops a comprehensive framework for optimal consumption in incomplete semimartingale markets with random endowment, extending duality methods and asymptotic elasticity concepts to time-dependent utility functions.

Contribution

It introduces a unified approach for pure and combined consumption/terminal wealth problems in incomplete markets, extending existing duality and elasticity theories.

Findings

Established existence and uniqueness of optimal consumption strategies.

Extended asymptotic elasticity to time-dependent utility functions.

Characterized the dual domain using finitely-additive measures.

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 $L^1$ to its topological bidual $(L^{\infty})^*$, a space of finitely-additive measures. As an application, we treat the case of a constrained It\^ o-process market-model.