Robust Utility Maximization with Drift and Volatility Uncertainty

TL;DR

This paper provides explicit solutions for utility maximization of terminal wealth under drift and volatility uncertainty in a complete market, considering time-dependent uncertainty sets and multiple utility functions.

Contribution

It introduces a method to explicitly solve the robust utility maximization problem with drift and volatility uncertainty in continuous time.

Findings

Explicit solutions for logarithmic, power, and exponential utilities.

Handles time-dependent uncertainty sets in a complete market.

Addresses uncertainty on both drift and volatility, inducing nonequivalent measures.

Abstract

We give explicit solutions for utility maximization of terminal wealth problem $u(X_T)$ in the presence of Knightian uncertainty in continuous time $[0,T]$ in a complete market. We assume there is uncertainty on both drift and volatility of the underlying stocks, which induce nonequivalent measures on canonical space of continuous paths $\O$. We take that the uncertainty set resides in compact sets that are time dependent. In this framework, we solve the robust optimization problem with logarithmic, power and exponential utility functions, explicitly.