A super-martingale property of the optimal portfolio process

We show that, for a utility function U : R→ R having reasonable asymptotic elasticity, the optimal investment process Ĥ ·S is a super-martingale under each equivalent martingale measure Q, such that E[V ( dP )] < ∞, 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 (S1 t , S 2 t )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 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 the initial endowment into the bond.