Optimal investment with random endowments and transaction costs: duality theory and shadow prices
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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 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 similar to Czichowsky and Schachermayer (Ann Appl Probab 26(3):1888–1941, 2016) as well as in the usual sense using acceptable portfolios.
KeywordsProportional transaction costs Unbounded random endowments Acceptable portfolios Utility maximization Convex duality Shadow prices
JEL ClassificationG11 G13
E. Bayraktar is supported in part by the National Science Foundation under Grant DMS-1613170 and the Susan M. Smith Professorship. X. Yu is supported by the Hong Kong Early Career Scheme under Grant 25302116 and the Start-Up Fund of the Hong Kong Polytechnic University under Grant 1-ZE5A.
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