Indifference price with general semimartingales

TL;DR

This paper establishes a duality formula for utility maximization in incomplete markets with general semimartingale prices, allowing for non locally bounded processes and broadening the understanding of indifference prices as convex risk measures.

Contribution

It introduces a duality framework for utility maximization with general semimartingales, extending previous results to include non locally bounded processes and weaker claim conditions.

Findings

Indifference price $(B)$ is a convex risk measure.

Duality formula applies to non locally bounded semimartingale prices.

Broader conditions for claims improve practical applicability.

Abstract

For utility functions $u$ finite valued on $\mathbb{R}$, we prove a duality formula for utility maximization with random endowment in general semimartingale incomplete markets. The main novelty of the paper is that possibly non locally bounded semimartingale price processes are allowed. Following Biagini and Frittelli \cite{BiaFri06}, the analysis is based on the duality between the Orlicz spaces $(L^{\widehat{u}}, (L^{\widehat{u}})^*)$ naturally associated to the utility function. This formulation enables several key properties of the indifference price $\pi(B)$ of a claim $B$ satisfying conditions weaker than those assumed in literature. In particular, the indifference price functional $\pi$ turns out to be, apart from a sign, a convex risk measure on the Orlicz space $L^{\widehat{u}}$.