Bicausal optimal transport for SDEs with irregular coefficients

TL;DR

This paper introduces a novel bicausal optimal transport framework for SDEs with irregular coefficients, providing new methods for computing adapted Wasserstein distances and analyzing model uncertainty.

Contribution

It develops a bicausal optimal transport approach for SDEs with irregular coefficients, including a semi-implicit numerical scheme with strong convergence guarantees.

Findings

Synchronous coupling is optimal among bicausal couplings.

Provides a numerical method for computing adapted Wasserstein distances.

Establishes strong convergence for SDEs with irregular, discontinuous drifts.

Abstract

We solve constrained optimal transport problems in which the marginal laws are given by the laws of solutions of stochastic differential equations (SDEs). We consider SDEs with irregular coefficients, making only minimal regularity assumptions. We show that the so-called synchronous coupling is optimal among bicausal couplings, that is couplings that respect the flow of information encoded in the stochastic processes. Our results provide a method to numerically compute the adapted Wasserstein distance between laws of SDEs with irregular coefficients. We show that this can be applied to quantifying model uncertainty in stochastic optimisation problems. Moreover, we introduce a transformation-based semi-implicit numerical scheme and establish the first strong convergence result for SDEs with exponentially growing and discontinuous drift.