Non-conservative optimal transport

ArXiv ID: 2510.03332 “View on arXiv”

Authors: Gabriela Kováčová, Georg Menz, Niket Patel

Abstract

Motivated by optimal re-balancing of a portfolio, we formalize an optimal transport problem in which the transported mass is scaled by a mass-change factor depending on the source and destination. This allows direct modeling of the creation or destruction of mass. We discuss applications and position the framework alongside unbalanced, entropic, and unnormalized optimal transport. The existence of optimal transport plans and strong duality are established. The existence of optimal maps are deduced in two central regimes, i.e., perturbative mass-change and quadratic mass-loss. For $\ell_p$ costs we derive the analogue of the Benamou-Brenier dynamic formulation.

Keywords: Optimal Transport, Mass-Change Factor, Strong Duality, Benamou-Brenier Formulation, Portfolio Rebalancing, Multi-Asset

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper introduces a novel mathematical framework for non-conservative optimal transport with advanced proofs and derivations (e.g., strong duality, existence of optimal maps, Benamou-Brenier analogue), but provides only a conceptual financial motivation without any backtesting, datasets, or implementation details.

  flowchart TD
    A["Research Goal<br>Formulate an optimal transport problem<br>modeling mass creation/destruction"] --> B["Methodology<br>Formalize framework with<br>mass-change factor μ(x,y)"]
    B --> C["Inputs<br>Portfolio rebalancing context<br>Unbalanced/Entropic OT benchmarks"]
    C --> D["Computational Process<br>Establish existence & strong duality<br>Derive Benamou-Brenier dynamics"]
    D --> E["Outcomes<br>Existence of optimal maps<br>perturbative & quadratic regimes"]
    E --> F["Applications<br>Multi-asset portfolio optimization<br>General mass-modifying transport"]