On the existence of shadow prices for optimal investment with random endowment

TL;DR

This paper proves the existence of shadow prices in a utility maximization framework with transaction costs and random endowment, showing equivalence to a frictionless market and extending to some relaxed constraints.

Contribution

It establishes the existence of shadow prices under specific constraints and demonstrates market equivalence, advancing understanding of optimal investment with transaction costs.

Findings

Existence of primal optimizer under constraints

Market with transaction costs can be replaced by a shadow market

Shadow prices exist even when constraints are relaxed

Abstract

In this paper, we consider a num\'eraire-based utility maximization problem under constant proportional transaction costs and random endowment. Assuming that the agent cannot short sell assets and is endowed with a strictly positive contingent claim, a primal optimizer of this utility maximization problem exists. Moreover, we observe that the original market with transaction costs can be replaced by a frictionless shadow market that yields the same optimality. On the other hand, we present an example to show that in some case when these constraints are relaxed, the existence of shadow prices is still warranted.