Admissible Strategies in Semimartingale Portfolio Selection

TL;DR

This paper introduces a new, versatile notion of admissible trading strategies in financial markets that simplifies analysis and includes classical and modern preferences, without requiring strict utility or price process conditions.

Contribution

It proposes a novel admissibility concept characterized under the objective measure, encompassing classical and modern preferences with minimal assumptions.

Findings

Admissible strategies can be approximated by simple strategies with finite trading dates.

The wealth process of admissible strategies is a supermartingale under all pricing measures.

The class includes the optimizer under milder conditions than traditional assumptions.

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.