On the dual problem of utility maximization in incomplete markets

TL;DR

This paper investigates the dual problem of utility maximization in incomplete markets with bounded endowment, establishing a representation of the dual optimizer as a supermartingale deflator in the Brownian setting.

Contribution

It extends previous work by showing the dual optimizer's countably additive part can be represented via a supermartingale deflator that is a local martingale.

Findings

Dual optimizer's countably additive part represented by a supermartingale deflator

In the Brownian framework, the deflator is a local martingale

Connects dual problem solutions with supermartingale deflators in incomplete markets

Abstract

In this paper, we study the dual problem of the expected utility maximization in incomplete markets with bounded random endowment. We start with the problem formulated in the paper of Cvitani\'{c}-Schachermayer-Wang (2001) and prove the following statement: in the Brownian framework, the countably additive part $Q^r$ of the dual optimizer $Q\in (L^\infty)^*$ obtained in that paper can be represented by the terminal value of a supermartingale deflator $Y$ defined in the paper of Kramkov-Schachermayer (1999), which is a local martingale.