Minimal Hellinger Deflators and HARA Forward Utilities with Applications: Hedging with Variable Horizon

Abstract: This thesis develops three major essays on the topic of horizon-dependence for optimal portfolio. The first essay contributes extensively to the newest concept of forward utilities. In this essay, we describe explicitly three classes of forward utilities--that we call HARA forward utilities--as well as their corresponding optimal portfolios. The stochastic tool behind our analysis lies in the concept of Minimal Hellinger Martingale densities (called MHM densities hereafter), introduced and developed recently by Choulli and his collaborators. The obtained results for HARA forward utilities by using MHM densities are derived under assumptions on the market model. The relaxation of some of these assumptions leads to introduce the new concept of Minimal Hellinger Deflator in order to characterize HARA forward utilities. The second essay addresses the problem of finding horizon-unbiased optimal portfolio from the perspective of contract theory. In fact, we consider an agent with classical exponential utility and describe--as explicit as possible--the payoff process for which there exists a horizon-unbiased optimal hedging portfolio. The last essay focuses on the financial problem that we call optimal sale problem. This problem consists of an agent who is investing in stocks and possesses a non-tradable asset that she aims to sell. The goal of this investor is to find the optimal portfolio--from her investment in stock market--and optimal time to liquidate all her assets (tradable or not).