Paper: arXiv 2512.17702

Authors: Jou-Hua Lai, Mykhaylo Shkolnikov, H. Mete Soner

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical formulation of the relative arbitrage problem in stochastic portfolio theory using viscosity solutions of fully nonlinear elliptic PDEs and optimal control. However, it is purely theoretical with no empirical backtesting, datasets, or code implementation provided.

Research Flowchart

  flowchart TD
    A[Research Goal: Find time horizon for relative arbitrage under eigenvalue lower bounds] --> B[Input: Market volatility framed as eigenvalue bound on covariation matrix]
    B --> C[Method: Formulate problem using Stochastic Optimal Control]
    C --> D[Analysis: Generalize to arbitrary compact, non-convex domains]
    D --> E[PDE Formulation: Derive fully nonlinear elliptic PDE for optimal time horizon]
    E --> F[Special Case: Map PDE to Ambrosio-Soner mean curvature flow]
    F --> G[Outcome: Characterize time horizon via unique viscosity solution]