On Utility Maximisation Under Model Uncertainty in Discrete-Time Markets

TL;DR

This paper investigates optimal investment strategies under model uncertainty in discrete-time markets, establishing existence results with bounded utility and extending to unbounded utility under integrability conditions, using a novel process-based framework.

Contribution

It introduces a new framework representing all possible stock dynamics as stochastic processes on a single probability space, enabling utility maximisation analysis without multiple probability measures.

Findings

Optimal strategies exist for bounded utility functions.

Boundedness can be relaxed with integrability and no-arbitrage conditions.

Framework simplifies analysis of model uncertainty in discrete markets.

Abstract

We study the problem of maximising terminal utility for an agent facing model uncertainty, in a frictionless discrete-time market with one safe asset and finitely many risky assets. We show that an optimal investment strategy exists if the utility function, defined either over the positive real line or over the whole real line, is bounded from above. We further find that the boundedness assumption can be dropped provided that we impose suitable integrability conditions, related to some strengthened form of no-arbitrage. These results are obtained in an alternative framework for model uncertainty, where all possible dynamics of the stock prices are represented by a collection of stochastic processes on the same filtered probability space, rather than by a family of probability measures.