How potential investments may change the optimal portfolio for the exponential utility

We show that, for a utility function U: R to R having reasonable asymptotic elasticity, the optimal investment process H. S is a super-martingale under each equivalent martingale measure Q, such that E[V(dQ/dP)] < "unendlich", where V is conjugate to U. Similar results for the special case of the exponential utility were recently obtained by Delbaen, Grandits, RheinlĂ¤nder, 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 the 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 (S_t^1, S_t^2)_0<=t<=T 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 S^1. However, if we pass to the market consisting only of the bond and the first risky asset S^1, and leaving the information structure unchanged, this trading strategy is not optimal any more: in this smaller market it is optimal to invest the initial endowment into the bond. (author's abstract) ; Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

Walter Schachermayer

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