On the Stability of Utility Maximization Problems

TL;DR

This paper investigates the stability of utility maximization problems in stochastic settings, extending classical convex analysis to random variable spaces and emphasizing the role of convex compactness.

Contribution

It extends stability results for utility maximization by incorporating convex analysis on $L^0$ spaces and utilizing convex compactness concepts.

Findings

Extended stability results for utility maximization problems.

Developed convex analysis tools for maps from $L^0$ to $L^0$.

Highlighted the importance of convex compactness in stochastic optimization.

Abstract

In this paper we extend the stability results of [4]}. Our utility maximization problem is defined as an essential supremum of conditional expectations of the terminal values of wealth processes, conditioned on the filtration at the stopping time $\tau$. To establish our results, we extend the classical results of convex analysis to maps from $L^0$ to $L^0$. The notion of convex compactness introduced in [7] plays an important role in our analysis.