Deep Semi-Martingale Optimal Transport

TL;DR

This paper introduces two deep neural network-based methods for solving semi-martingale optimal transport problems, effectively handling high-dimensional cases and applying to portfolio optimization.

Contribution

It presents novel neural network approaches for semi-martingale optimal transport, including a penalized terminal constraint method and a dual saddle point formulation.

Findings

Methods work effectively up to 10 dimensions.

Achieved accurate solutions in test examples.

Applied to portfolio optimization with promising results.

Abstract

We propose two deep neural network-based methods for solving semi-martingale optimal transport problems. The first method is based on a relaxation/penalization of the terminal constraint, and is solved using deep neural networks. The second method is based on the dual formulation of the problem, which we express as a saddle point problem, and is solved using adversarial networks. Both methods are mesh-free and therefore mitigate the curse of dimensionality. We test the performance and accuracy of our methods on several examples up to dimension 10. We also apply the first algorithm to a portfolio optimization problem where the goal is, given an initial wealth distribution, to find an investment strategy leading to a prescribed terminal wealth distribution.