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

Abstract

In this paper, we consider a numéraire-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.

Keywords:

• Utility maximization
• random endowment
• transaction costs
• no-short-selling constraint
• shadow prices

Acknowledgements

This work is partially done during the visit of L. Gu and J. Yang hosted by Prof. N. Touzi at CMAP, École Polytechnique, which is very much appreciated. The authors are grateful to Prof. W. Schachermayer for his suggestions on the Appendix and to the anonymous reviewers for their kind comments.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Austrian Science Fund (FWF) [grant number P25815]; the European Research Council under ERC Advanced [grant number 321111].