Relative arbitrage problem under eigenvalue lower bounds

TL;DR

This paper introduces a new formulation of the relative arbitrage problem in stochastic portfolio theory, linking market volatility bounds to PDEs and mean curvature flow, providing a novel mathematical characterization of arbitrage opportunities.

Contribution

It formulates the relative arbitrage problem using eigenvalue bounds and characterizes the time horizon via a nonlinear PDE, connecting finance with geometric flow concepts.

Findings

Characterizes arbitrage time horizon through viscosity solutions of PDEs.

Links market volatility bounds to mean curvature flow in a special case.

Extends analysis to non-convex domains with bounded covariation matrices.

Abstract

We give a new formulation of the relative arbitrage problem from stochastic portfolio theory that asks for a time horizon beyond which arbitrage relative to the market exists in all ``sufficiently volatile'' markets. In our formulation, ``sufficiently volatile'' is interpreted as a lower bound on an ordered eigenvalue of the instantaneous covariation matrix, a quantity that has been studied extensively in the empirical finance literature. Upon framing the problem in the language of stochastic optimal control, we characterize the time horizon in question through the unique upper semicontinuous viscosity solution of a fully nonlinear elliptic partial differential equation (PDE). In a special case, this PDE amounts to the arrival time formulation of the Ambrosio-Soner co-dimension mean curvature flow. Beyond the setting of stochastic portfolio theory, the stochastic optimal control problem is analyzed for arbitrary compact, possibly non-convex, domains, thanks to a boundedness assumption on the instantaneous covariation matrix.