# Robust utility maximization under model uncertainty via a penalization approach

## TL;DR

This paper develops a robust utility maximization framework under model uncertainty using penalization, interpreting it as a stochastic differential game, and demonstrates its effectiveness with real market data.

## Contribution

It introduces a penalization-based robust optimization approach, linking it to a stochastic differential game and providing analytical and numerical solutions.

## Key findings

- Robust portfolios yield higher expected utility.
- Portfolios are more stable during market downturns.
- The approach is validated with real market data.

## Abstract

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.