Analytical Approach to Continuous-Time Causal Optimal Transport
Optimization and Control
2026-05-20 v1 Analysis of PDEs
Probability
Abstract
We study causal optimal transport in continuous time, with Markovian cost, between a finite-state Markov source and a diffusion target. By replacing the source with its conditional law given the observation of the target, we characterize the value of this transport problem through a fully nonlinear parabolic master equation on an enlarged state space. We further show that this value coincides with those of two equivalent stochastic control problems on the simplex: a control of the Kushner--Stratonovich filtering equation with a zero-mean condition, and a state-constrained stochastic optimal control problem. Both formulations give rise to implementable numerical schemes that approximate the value from above and below.
Cite
@article{arxiv.2605.19978,
title = {Analytical Approach to Continuous-Time Causal Optimal Transport},
author = {Julio Backhoff and Erhan Bayraktar and Ibrahim Ekren and Antonios Zitridis},
journal= {arXiv preprint arXiv:2605.19978},
year = {2026}
}