# Computational methods for martingale optimal transport problems

@article{Guo2017ComputationalMF, title={Computational methods for martingale optimal transport problems}, author={Gaoyue Guo and Jan Obł{\'o}j}, journal={arXiv: Probability}, year={2017} }

We establish numerical methods for solving the martingale optimal transport problem (MOT) - a version of the classical optimal transport with an additional martingale constraint on transport's dynamics. We prove that the MOT value can be approximated using linear programming (LP) problems which result from a discretisation of the marginal distributions combined with a suitable relaxation of the martingale constraint. Specialising to dimension one, we provide bounds on the convergence rate of…

## 62 Citations

Computational methods for adapted optimal transport

- Mathematics, Computer Science
- 2022

It is shown that AOT problems are stable with respect to perturbations in the marginals and thus arbitrary AOTblems can be approximated by sequences of linear programs and further study entropic methods to solve Aot problems.

Multidimensional martingale optimal transport.

- Mathematics
- 2018

In this thesis, we study various aspects of martingale optimal transport in dimension greater than one, from duality to local structure, and finally we propose numerical approximation methods.We…

Approximation of optimal transport problems with marginal moments constraints

- Mathematics, Computer ScienceMath. Comput.
- 2021

This work investigates the relaxation of the OT problem when the marginal constraints are replaced by some moment constraints, using Tchakaloff's theorem to show that the Moment Constrained Optimal Transport problem (MCOT) is achieved by a finite discrete measure.

Continuous-time Martingale Optimal Transport and Optimal Skorokhod Embedding

- Mathematics
- 2016

This PhD dissertation presents three research topics, the first two being independent and the last one relating the first two issues in a concrete case.In the first part we focus on the martingale…

Martingale optimal transport in the discrete case via simple linear programming techniques

- MathematicsMath. Methods Oper. Res.
- 2019

This work assumes that the marginal distributions at the two time points the authors consider are discrete probability distributions, and proves the optimality of left-monotone transport plans under this assumption and provides an algorithm for its construction.

Stability of martingale optimal transport and weak optimal transport

- MathematicsThe Annals of Applied Probability
- 2022

Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $\pi_1, \pi_2, \ldots$ converges weakly to a transport plan $\pi$,…

The structure of non-linear martingale optimal transport problems

- Mathematics
- 2019

We explore the structure of solutions to a family of non-linear martingale optimal transport (MOT) problems that involve conditional expectations in the objective functional. En route general results…

Stability of the Weak Martingale Optimal Transport Problem

- Mathematics
- 2021

While many questions in (robust) finance can be posed in the martingale optimal transport (MOT) framework, others require to consider also non-linear cost functionals. Following the terminology of…

Instability of martingale optimal transport in dimension d≥2

- MathematicsElectronic Communications in Probability
- 2022

Stability of the value function and the set of minimizers w.r.t. the given data is a desirable feature of optimal transport problems. For the classical Kantorovich transport problem, stability is…

Approximation of martingale couplings on the line in the weak adapted topology

- Mathematics
- 2021

Our main result is to establish stability of martingale couplings: suppose that π is a martingale coupling with marginals μ, ν. Then, given approximating marginal measures μ̃ ≈ μ, ν̃ ≈ ν in convex…

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