Robust Utility Maximization in a Multivariate Financial Market with Stochastic Drift

TL;DR

This paper develops a robust utility maximization framework in multivariate financial markets with stochastic drifts, integrating worst-case optimization and filtering to derive strategies resilient to drift estimation uncertainty.

Contribution

It introduces a time-dependent worst-case approach with filtering-based uncertainty sets, providing a novel method to derive optimal, adaptive investment strategies under drift ambiguity.

Findings

Proves a minimax theorem for local optimization problems.

Derives explicit optimal strategies under ellipsoidal uncertainty sets.

Shows robustness of strategies to additional information.

Abstract

We study a utility maximization problem in a financial market with a stochastic drift process, combining a worst-case approach with filtering techniques. Drift processes are difficult to estimate from asset prices, and at the same time optimal strategies in portfolio optimization problems depend crucially on the drift. We approach this problem by setting up a worst-case optimization problem with a time-dependent uncertainty set for the drift. Investors assume that the worst possible drift process with values in the uncertainty set will occur. This leads to local optimization problems, and the resulting optimal strategy needs to be updated continuously in time. We prove a minimax theorem for the local optimization problems and derive the optimal strategy. Further, we show how an ellipsoidal uncertainty set can be defined based on filtering techniques and demonstrate that investors need to choose a robust strategy to be able to profit from additional information.