Non-concave optimal investment and no-arbitrage: a measure theoretical approach

TL;DR

This paper develops a measure-theoretic framework for optimal investment with non-concave, non-smooth utility functions in incomplete discrete-time markets, establishing no-arbitrage conditions and optimal portfolio existence without assuming a complete probability space.

Contribution

It introduces a novel measure-theoretic approach to characterize no-arbitrage and optimal investment in incomplete markets with non-concave utilities, relaxing the completeness assumption.

Findings

Established no-arbitrage condition characterization.

Proved existence of optimal portfolios under non-concave utilities.

Extended the framework to incomplete probability spaces.

Abstract

We consider non-concave and non-smooth random utility functions with do- main of definition equal to the non-negative half-line. We use a dynamic pro- gramming framework together with measurable selection arguments to establish both the no-arbitrage condition characterization and the existence of an optimal portfolio in a (generically incomplete) discrete-time financial market model with finite time horizon. In contrast to the existing literature, we propose to consider a probability space which is not necessarily complete.