Admissible Strategies in Semimartingale Portfolio Selection

Abstract

The choice of admissible trading strategies in mathematical modelling of financial markets is a delicate issue, going back to Harrison and Kreps (1979). In the context of optimal portfolio selection with expected utility preferences this question has been a focus of considerable attention over the last twenty years.

We propose a novel notion of admissibility that has many pleasant features - admissibility is characterized purely under the objective measure; each admissible strategy can be approximated by simple strategies using finite number of trading dates; the wealth of any admissible strategy is a supermartingale under all pricing measures; local boundedness of the price process is not required; neither strict monotonicity, strict concavity nor differentiability of the utility function are necessary; the definition encompasses both the classical mean-variance preferences and the monotone expected utility.

For utility functions finite on the whole real line, our class represents a minimal set containing simple strategies which also contains the optimizer, under conditions that are milder than the celebrated reasonable asymptotic elasticity condition on the utility function.

Keywords: utility maximization, non-locally bounded semimartingale, incomplete market, sigma-localization, sigma-martingale measure, Orlicz space, convex duality

JEL Classification: G11, G12, G13