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Information-Based Martingale Optimal Transport

Georges Kassis, Andrea Macrina

TL;DR

This work introduces the information-based martingale optimal transport (IB-MOT) framework, which selects the worst-case martingale coupling between convexly ordered measures by embedding the coupling into a filtered martingale interpolation (FAM) driven by a randomised arcade process (RAP). The IB-MOT objective quantifies the cumulative, weighted squared error between a final target measure and its FAM, expressed via the innovations process and a conditional martingale representation $M_t = \mathbb{E}_\pi[X_1 \mid X_0, I_t^{(1)}]$, and is shown to admit a finite, unique maximizer; the problem connects to classical OT, Schrödinger bridges, and the martingale Benamou–Brenier framework. The authors develop a gradient-based algorithm for empirical measures, including explicit gradient formulas and a projection step onto the martingale-coupling set, and address the challenge of preserving convex order in sampling through a convex-order convexification strategy. The results establish that, for Gaussian marginals, the Brownian coupling is optimal and that IB-MOT yields a dynamic FAM corresponding to Brownian motion, with potential extensions to non-Gaussian targets via efficient approximation methods. Overall, IB-MOT provides a rigorous, operational approach to introducing noise in MOT and constructing optimal FAMs from given marginals, with clear theoretical guarantees and practical algorithms.

Abstract

Randomised arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between any given ordered set of random variables, at fixed pre-specified times. Utilising these processes as generators of partial information, a class of continuous-time martingale -- the filtered arcade martingales (FAMs) -- are constructed. FAMs interpolate through a sequence of target random variables, which form a discrete-time martingale. The research presented in this paper relaxes the FAM setting to the interpolation between probability measures instead and treats the problem of selecting the worst martingale coupling for given, convexly ordered, probability measures contingent on the paths of FAMs that are constructed using the martingale coupling. This optimisation problem, that we term the information-based martingale optimal transport problem (IB-MOT), can be viewed from different perspectives. It can be understood as a model-free construction of FAMs, in the case where the coupling is not determined a priori. It can also be considered from the vantage point of optimal transport (OT), where the problem is concerned with introducing a noise factor in martingale optimal transport, similarly to how the entropic regularisation of optimal transport introduces noise in OT. The IB-MOT problem is static in its nature, since its aim is to find a coupling. However, a corresponding dynamical solution can be found by considering the FAM constructed with the identified optimal coupling. The existence and uniqueness of its solution are shown and an algorithm for empirical measures is proposed.

Information-Based Martingale Optimal Transport

TL;DR

This work introduces the information-based martingale optimal transport (IB-MOT) framework, which selects the worst-case martingale coupling between convexly ordered measures by embedding the coupling into a filtered martingale interpolation (FAM) driven by a randomised arcade process (RAP). The IB-MOT objective quantifies the cumulative, weighted squared error between a final target measure and its FAM, expressed via the innovations process and a conditional martingale representation , and is shown to admit a finite, unique maximizer; the problem connects to classical OT, Schrödinger bridges, and the martingale Benamou–Brenier framework. The authors develop a gradient-based algorithm for empirical measures, including explicit gradient formulas and a projection step onto the martingale-coupling set, and address the challenge of preserving convex order in sampling through a convex-order convexification strategy. The results establish that, for Gaussian marginals, the Brownian coupling is optimal and that IB-MOT yields a dynamic FAM corresponding to Brownian motion, with potential extensions to non-Gaussian targets via efficient approximation methods. Overall, IB-MOT provides a rigorous, operational approach to introducing noise in MOT and constructing optimal FAMs from given marginals, with clear theoretical guarantees and practical algorithms.

Abstract

Randomised arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between any given ordered set of random variables, at fixed pre-specified times. Utilising these processes as generators of partial information, a class of continuous-time martingale -- the filtered arcade martingales (FAMs) -- are constructed. FAMs interpolate through a sequence of target random variables, which form a discrete-time martingale. The research presented in this paper relaxes the FAM setting to the interpolation between probability measures instead and treats the problem of selecting the worst martingale coupling for given, convexly ordered, probability measures contingent on the paths of FAMs that are constructed using the martingale coupling. This optimisation problem, that we term the information-based martingale optimal transport problem (IB-MOT), can be viewed from different perspectives. It can be understood as a model-free construction of FAMs, in the case where the coupling is not determined a priori. It can also be considered from the vantage point of optimal transport (OT), where the problem is concerned with introducing a noise factor in martingale optimal transport, similarly to how the entropic regularisation of optimal transport introduces noise in OT. The IB-MOT problem is static in its nature, since its aim is to find a coupling. However, a corresponding dynamical solution can be found by considering the FAM constructed with the identified optimal coupling. The existence and uniqueness of its solution are shown and an algorithm for empirical measures is proposed.

Paper Structure

This paper contains 5 sections, 6 theorems, 63 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Simple analogy with several optimal transports
  3. Formulation of the problem and properties
  4. IB-MOT for empirical measures
  5. Conclusion

Key Result

Proposition 1.5

Let $M_t = \mathbb{E} [ X_1 \, | \, X_0, I_t^{(1)} ]$ be a one-arc FAM. If $(I_t^{(1)})$ is a semimartingale with Gauss-Markov driver $(D_t)$, such that $(t,x) \rightarrow V(t,x,X_0)/K(t,x,X_0)$ is $C^2(((T_0,T_1) \setminus N) \times Im(I^{(1)}))$ where $N \subset (T_0,T_1)$ contains finitely many e for $t \in (T_0,T_1)$, where $H_1$ and $H_2$ are differentiable almost everywhere functions given b

Figures (1)

  • Figure 1: Histogram of 10,000 points drawn from $M_{1/2}$ with different densities fitted to these points for comparison. We focus on the interval $[0,3]$ for illustration purposes.

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • ...and 17 more