Abstract.
We show that, for a utility function having reasonable asymptotic elasticity , the optimal investment process is a super-martingale under each equivalent martingale measure , such that , where V is the conjugate function of U. Similar results for the special case of the exponential utility were recently obtained by Delbaen, Grandits, Rheinländer, Samperi, Schweizer, Stricker as well as Kabanov, Stricker.
This result gives rise to a rather delicate analysis of the “good definition” of “allowed” trading strategies H for a financial market S. One offspring of these considerations leads to the subsequent - at first glance paradoxical - example.
There is a financial market consisting of a deterministic bond and two risky financial assets such that, for an agent whose preferences are modeled by expected exponential utility at time T, it is optimal to constantly hold one unit of asset S1. However, if we pass to the market consisting only of the bond and the first risky asset S1, and leaving the information structure unchanged, this trading strategy is not optimal any more: in this smaller market it is optimal to invest t he initial endowment into the bond.
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Manuscript received: January 2001; final version received: September 2002
Support by the Austrian Science Foundation (FWF) under the Wittgenstein-Preis program Z36-MAT and grant SFB#010 and by the Austrian National Bank under grant 'Jubiläumsfondprojekt Number 8699' is gratefully acknowledged. Thanks go to P. Grandits and Th. Rheinländer for discussions on the topic of this paper and specially to Ch. Stricker for pointing out an erroneous argument in a previous version of this paper which circulated since December 2000 under the title “How Potential Investments may Change the Optimal Portfolio for the Exponential Utility”. The present version has greatly benefited from numerous suggestions and comments by Y. Kabanov, Ch. Stricker, M. Schweizer, as well as by two anonymous referees, for which I also thank.
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Schachermayer, W. A super-martingale property of the optimal portfolio process. Finance Stochast 7, 433–456 (2003). https://doi.org/10.1007/s007800200096
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DOI: https://doi.org/10.1007/s007800200096