Portfolio Optimization under Nonlinear Utility

TL;DR

This paper investigates a utility maximization framework using BSDEs, establishing existence of optimal strategies, and connecting the problem to robust control and duality methods, with implications for financial decision-making.

Contribution

It introduces a novel approach to utility maximization with nonlinear preferences modeled by BSDEs, linking optimal strategies to controlled FBSDEs and saddle points.

Findings

Existence of optimal trading strategies proven.

Utility maximization linked to robust control and saddle points.

Characterization of solutions via BSDE duality and convex analysis.

Abstract

This paper studies the utility maximization problem of an agent with non-trivial endowment, and whose preferences are modeled by the maximal subsolution of a BSDE. We prove existence of an optimal trading strategy and relate our existence result to the existence of a maximal subsolution to a controlled decoupled FBSDE. Using BSDE duality, we show that the utility maximization problem can be seen as a robust control problem admitting a saddle point if the generator of the BSDE additionally satisfies a specific growth condition. We show by convex duality that any saddle point of the robust control problem agrees with a primal and a dual optimizer of the utility maximization problem, and can be characterized in terms of a BSDE solution.