On the Martingale Schrödinger Bridge between Two Distributions

We study a martingale Schrödinger bridge problem: given two probability distributions, find their martingale coupling with minimal relative entropy. Our main result provides Schrödinger potentials for this coupling. Namely, under certain conditions, the log-density of the optimal coupling is given by a triplet of real functions representing the marginal and martingale constraints. The potentials are also described as the solution of a dual problem.