  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. 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. We
test this robust algorithm on real market data. The results show that robust
portfolios generally have higher expected utilities and are more stable under
strong market downturns. To solve for the value function, we derive an
analytical solution in the logarithmic utility case and obtain accurate
numerical approximations in the general case by three methods: finite
difference method, Monte Carlo simulation, and Generative Adversarial Networks.
