Analytical Approach to Continuous-Time Causal Optimal Transport

TL;DR

This paper develops a novel analytical framework for continuous-time causal optimal transport between Markov processes, utilizing nonlinear PDEs and stochastic control to characterize and approximate the transport value.

Contribution

It introduces a fully nonlinear parabolic master equation and two stochastic control formulations to analyze and numerically approximate causal optimal transport in continuous time.

Findings

Characterization of the transport value via a nonlinear PDE.

Equivalence of the transport problem to two stochastic control problems.

Development of numerical schemes for approximation from above and below.

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.