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Non-conservative optimal transport

Gabriela Kováčová, Georg Menz, Niket Patel

TL;DR

Non-conservative OT extends classical optimal transport by introducing a mass-change factor $m(x,y)$ that allows mass to be created or destroyed during transport and by relaxing the exact second marginal constraint to a prescribed shape; existence of optimal transport plans and strong duality are established, and optimal transport maps exist in perturbative ($m\approx1$) and quadratic-mass-loss regimes; a generalized $\ell_p$-cost dynamic formulation is derived, providing a BB-like framework without requiring prior vector-field regularity. The framework connects naturally to unbalanced, entropic, and unnormalized OT and yields direct modeling opportunities for applications such as portfolio rebalancing and spoilage-aware logistics or tariffs.

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.

Non-conservative optimal transport

TL;DR

Non-conservative OT extends classical optimal transport by introducing a mass-change factor that allows mass to be created or destroyed during transport and by relaxing the exact second marginal constraint to a prescribed shape; existence of optimal transport plans and strong duality are established, and optimal transport maps exist in perturbative () and quadratic-mass-loss regimes; a generalized -cost dynamic formulation is derived, providing a BB-like framework without requiring prior vector-field regularity. The framework connects naturally to unbalanced, entropic, and unnormalized OT and yields direct modeling opportunities for applications such as portfolio rebalancing and spoilage-aware logistics or tariffs.

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 costs we derive the analogue of the Benamou-Brenier dynamic formulation.

Paper Structure

This paper contains 18 sections, 19 theorems, 136 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivating example: Optimal rebalancing of portfolios.
  3. Optimal rebalancing problem
  4. Rebalancing and optimal transport
  5. Non-conservative optimal transport
  6. The non-conservative Kantorovich Problem.
  7. The non-conservative Dual Problem.
  8. Strong Duality
  9. Existence of optimal transport map(s)
  10. Dynamic formulation of non-conservative optimal transport
  11. Relation to existing optimal–transport variants
  12. Relation to unbalanced optimal transport
  13. Relation to entropic optimal transport in the sense of liero2018optimal
  14. Relation to unnormalized optimal transport
  15. Proofs of results from Section \ref{['sec:rebalancing_general_graphs']}
  16. ...and 3 more sections

Key Result

Proposition 2.10

Let the financial market $\mathcal{G} = (V, E, P)$ satisfy Assumption A1 (connectedness). A consistent price vector exists if and only if the the market satisfies Assumption A2 of no arbitrage.

Figures (3)

  • Figure 1: Illustration of a general financial market without a single numéraire. The vertices USD, JPY and EUR denote currencies used a numéraires at national markets. Vertices connected to several numéraires denote cross-traded assets, i.e. assets that are traded on several national markets.
  • Figure 2: Illustration of the rebalancing process. The goal is to sell and buy assets such that the wealth of the new portfolio is distributed according to the target proportions (or measure) $\nu$.
  • Figure 3: Uniqueness of the optimal transport map for the quadratic leaky Monge problem (see proof of Proposition \ref{['the:solution_quadratic_leaky_monge']}).

Theorems & Definitions (72)

  • Definition 2.1
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Remark 2.8
  • Example 2.9
  • Proposition 2.10
  • Remark 2.11
  • ...and 62 more