Explicit Description of HARA Forward Utilities and Their Optimal Portfolios

TL;DR

This paper provides a complete explicit characterization of HARA forward utilities and their optimal portfolios in a semimartingale market model, utilizing Hellinger martingale densities and illustrating with discrete-time examples.

Contribution

It introduces a novel explicit parametrization of HARA forward utilities and describes their optimal portfolios using Hellinger martingale densities, advancing the theoretical understanding of forward utility optimization.

Findings

Explicit characterization of HARA forward utilities

Derivation of optimal portfolios using Hellinger martingale densities

Illustrations on discrete-time market models

Abstract

This paper deals with forward performances of HARA type. Precisely, for a market model in which stock price processes are modeled by a locally bounded $d$-dimensional semimartingale, we elaborate a complete and explicit characterization for this type of forward utilities. Furthermore, the optimal portfolios for each of these forward utilities are explicitly described. Our approach is based on the minimal Hellinger martingale densities that are obtained from the important statistical concept of Hellinger process. These martingale densities were introduced recently, and appeared herein tailor-made for these forward utilities. After outlining our parametrization method for the HARA forward, we provide illustrations on discrete-time market models. Finally, we conclude our paper by pointing out a number of related open questions.