Robust utility maximization under model uncertainty via a penalization approach

This paper addresses the problem of utility maximization under uncertain parameters. In contrast with the classical approach, where the parameters of the model evolve freely within a given range, we constrain them via a penalty function. In addition, our paper dedicates in proposing various numerical algorithms to solve for the value function, including finite difference method, Generative Adversarial Networks and Monte Carlo simulation. These methods contribute to the quantitative techniques for solving robust portfolio optimization problems. We show that this robust optimization process can be interpreted as a two-player zero-sum stochastic differential game. We prove that the value function satisfies the Dynamic Programming Principle and that it is the unique viscosity solution of an associated Hamilton-Jacobi-Bellman-Isaacs equation. By testing this robust algorithm on real market data, we show that robust portfolios generally have higher expected utilities and are more stable under strong market downturns.