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

36 Pages Posted: 15 May 2016  

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

Xiang Yu

Hong Kong Polytechnic University - Department of Applied Mathematics

Date Written: April 25, 2016

Abstract

This paper studies the utility maximization problem on the terminal wealth with both 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 the 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 convex duality analysis. As an important application of the duality theory, 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.

Keywords: Proportional Transaction Costs, Unbounded Random Endowments, Acceptable Portfolios, Superhedging Theorem, Utility Maximization, Shadow Prices, Convex Duality

Suggested Citation

Bayraktar, Erhan and Yu, Xiang, Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices (April 25, 2016). Available at SSRN: https://ssrn.com/abstract=2779119 or http://dx.doi.org/10.2139/ssrn.2779119

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Xiang Yu (Contact Author)

Hong Kong Polytechnic University - Department of Applied Mathematics ( email )

Yip Kit Chuen Building
Hung Hom
Kowloon
Hong Kong

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