Problem
Research questions and friction points this paper is trying to address.
martingale Schrödinger bridge
convex order
probability measures
classical Schrödinger bridge
Föllmer martingale
Bridging classical and martingale Schrödinger bridges
🤖 AI Summary
This work investigates the construction of continuous martingales with prescribed marginal distributions in arbitrary dimensions and establishes a profound connection with the classical Schrödinger bridge problem. By introducing a weighted quadratic energy minimization framework, the authors extend the martingale Schrödinger bridge to the multidimensional setting and demonstrate its equivalence to the Föllmer martingale, a variational problem under convex order constraints, and a dual formulation of weak optimal transport. The main contribution lies in the first successful multidimensional generalization of this framework, proving that under irreducibility conditions, the continuous martingale Schrödinger bridge coincides with the Föllmer process. This result provides multiple equivalent characterizations, thereby unifying and significantly deepening the theoretical foundations of both classical and martingale Schrödinger bridges.
📝 Abstract
We investigate the martingale Schrödinger bridge, recently introduced by Nutz and Wiesel as a distinguished martingale transport plan between two probability measures in convex order. We show that this construction extends naturally to arbitrary dimension and admits several equivalent characterizations. In particular, we identify its continuous-time counterpart as the continuous martingale with prescribed marginals that minimizes a weighted quadratic energy measuring the deviation from Brownian motion. In the irreducible case, we prove that this continuous martingale Schrödinger bridge coincides with the Föllmer martingale, that is, with the Doob martingale associated to a suitable Föllmer process. More generally, we relate the martingale Schrödinger bridge to a variational problem over base measures and to the dual formulation of the corresponding weak optimal transport problem, thereby clarifying its connection with the classical Schrödinger bridge.
Innovation
Methods, ideas, or system contributions that make the work stand out.
martingale Schrödinger bridge
Föllmer martingale
convex order
weak optimal transport
quadratic energy minimization
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