Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices

TL;DR

This paper develops a duality framework for optimal investment with random endowments and transaction costs, introducing acceptable portfolios and shadow prices to handle unbounded payoffs and market frictions.

Contribution

It introduces a duality approach using acceptable portfolios and consistent price systems to establish existence and uniqueness of optimal solutions in markets with transaction costs and random endowments.

Findings

Established super-hedging results for markets with transaction costs.

Proved existence and uniqueness of optimal investment strategies.

Provided conditions for the existence of shadow prices in complex market settings.

Abstract

This paper studies the utility maximization on the terminal wealth with random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system (CPS) such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility maximization problem are established using the duality approach. As an important application of the duality theorem, we provide some sufficient conditions for the existence of a shadow price process with random endowments in a generalized form as well as in the usual sense using acceptable portfolios.