Utility maximization problem with random endowment and transaction costs: when wealth may become negative

TL;DR

This paper investigates a utility maximization framework with transaction costs and random endowment, allowing for negative wealth, and establishes duality and pricing results using finitely additive measures.

Contribution

It extends utility maximization theory to include negative wealth scenarios with transaction costs, introducing duality results on finitely additive measures and shadow market construction.

Findings

Duality results for utility functions with negative values.

Construction of shadow market via dual optimal process.

Utility-based pricing for random endowment.

Abstract

In this paper we study the problem of maximizing expected utility from the terminal wealth with proportional transaction costs and random endowment. In the context of the existence of consistent price systems, we consider the duality between the primal utility maximization problem and the dual one, which is set up on the domain of finitely additive measures. In particular, we prove duality results for utility functions supporting possibly negative values. Moreover, we construct the shadow market by the dual optimal process and consider the utility based pricing for random endowment.