Optimal arbitrage under model uncertainty

TL;DR

This paper characterizes the maximum achievable outperformance of the market under model uncertainty using nonlinear PDEs, stochastic control, and game theory, providing a framework for optimal arbitrage strategies.

Contribution

It introduces a novel characterization of optimal arbitrage under Knightian uncertainty via PDEs, stochastic control, and game-theoretic approaches.

Findings

The highest relative return can be characterized by a nonlinear PDE.

The value function corresponds to the probability of an auxiliary process remaining positive.

An investment rule can be derived to achieve the optimal outperformance.

Abstract

In an equity market model with "Knightian" uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon, and under any admissible configuration of model parameters that might materialize. One characterization is in terms of the smallest positive supersolution to a fully nonlinear parabolic partial differential equation of the Hamilton--Jacobi--Bellman type. Under appropriate conditions, this smallest supersolution is the value function of an associated stochastic control problem, namely, the maximal probability with which an auxiliary multidimensional diffusion process, controlled in a manner which affects both its drift and covariance structures, stays in the interior of the positive orthant through the end of the time-horizon. This value function is also characterized in terms of a stochastic game, and can be used to generate an investment rule that realizes such best possible outperformance of the market.