In this note we revisit Merton's optimal portfolio selection problem in an exponential Levy market, for an agent acting with a canonical utility function, the logarithm. As we show, the explicit optimal portfolios exhibit features similar to some of the pathological examples given by Kramkov and Schachermayer where certain discounted asset processes fail to be martingales. In our examples, these pathologies are seen to arise from natural shortselling and borrowing constraints imposed by the logarithmic utility.
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