Entropy-Regularized Causal Optimal Transport

Optimal transport (OT) provides a general framework for comparing probability distributions and solving matching problems, but direct discretizations often require large linear programs. Entropy-regularized optimal transport (EOT) replaces this hard problem by a structured convex formulation with a Kullback-Leibler geometry, enabling scalable Bregman-projection updates. In this project, we develop an improved Bregman-projection scheme designed to handle the marginal and non-anticipativity constraints. It decomposes the discretized causal problem into tractable projection steps, replacing a global linear-program solve with scalable iterative updates. A delayed causal construction supplies finite-entropy feasible couplings and, for atomless time marginals and bounded continuous costs, connects the regularized values to causal OT as the entropy parameter vanishes. From a complementary optimal-stopping viewpoint, a time-law Lagrange multiplier yields a finite-horizon dual, whose inner value is characterized by the lower Snell envelope. This dual both clarifies the constraints and provides a complementary computational formulation.