Viscosity Characterization of the Arbitrage Function under Model Uncertainty

TL;DR

This paper characterizes the arbitrage function in an equity market with model uncertainty as a viscosity solution of an HJB equation, generalizing previous classical solutions under weaker assumptions.

Contribution

It extends the characterization of the arbitrage function as a viscosity solution under model uncertainty, relaxing previous stronger conditions.

Findings

Arbitrage function is a viscosity solution of the HJB equation.

Generalizes previous classical solution results.

Applicable under weaker boundedness, continuity, and Markovian assumptions.

Abstract

We show that in an equity market model with Knightian uncertainty regarding the relative risk and covariance structure of its assets, the arbitrage function -- defined as the reciprocal of the highest return on investment that can be achieved relative to the market using nonanticipative strategies, and under any admissible market model configuration -- is a viscosity solution of an associated Hamilton-Jacobi-Bellman (HJB) equation under appropriate boundedness, continuity and Markovian assumptions on the uncertainty structure. This result generalizes that of Fernholz and Karatzas (2011), who characterized this arbitrage function as a classical solution of a Cauchy problem for this HJB equation under much stronger conditions than those needed here.