Causal transport on path space

TL;DR

This paper investigates the structure of causal couplings between stochastic process measures, characterizing bicausal couplings, and showing their relation to stochastic integrals and Monge maps, with implications for optimal transport in stochastic settings.

Contribution

It provides a complete characterization of bicausal couplings for solutions of SDEs and describes their structure via stochastic integrals, advancing the understanding of causal transport.

Findings

Bicausal couplings are characterized for solutions of SDEs.

Bicausal Monge couplings are induced by stochastic integrals of rotation-valued integrands.

Bicausal Monge transports are dense among bicausal couplings for regular SDE coefficients.

Abstract

We study properties of causal couplings for probability measures on the space of continuous functions. We first provide a characterization of bicausal couplings between weak solutions of stochastic differential equations. We then provide a complete description of all such bicausal Monge couplings. In particular, we show that bicausal Monge couplings of $d$-dimensional Wiener measures are induced by stochastic integrals of rotation-valued integrands. As an application, we give necessary and sufficient conditions for bicausal couplings to be induced by Monge maps and show that such bicausal Monge transports are dense in the set of bicausal couplings between laws of SDEs with regular coefficients.