On the Stability of Utility Maximization Problems

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.

Keywords: Utility maximization, incomplete markets, stability, convex analysis for functions from $L^0$ to $L^0$, convex compactness, continuous semimartingales