Specific Wasserstein Divergence Between Continuous Martingales

• 成果类型: Article; Early Access
• 署名作者: Backhoff-Veraguas, Julio; Zhang, Xin
• 署名单位: University of Vienna; New York University; New York University Tandon School of Engineering
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.2025.0896
• 发表日期: 2026-01-08
• 关键词: entropy; win-martingale; martingale optimal transport; Wasserstein Distance; Schrodinger problem; Optimal Transport; causal transport; DISCRETE-TIME; arbitrage; Duality; entropy

摘要

Defining a divergence between the laws of continuous martingales is a delicate task, owing to the fact that these laws tend to be singular to each other. An important idea, by Gantert, is to instead consider a scaling limit of the relative entropy between such continuous martingales sampled over a finite time grid. This gives rise to the concept of specific relative entropy. In order to develop a general theory of divergences between continuous martingales, it is natural to replace the role of the relative entropy by a different notion of discrepancy between finite-dimensional probability distributions. We take a first step in this direction, taking a power p of the Wasserstein distance. We call the newly obtained scaling limit the specific p-Wasserstein divergence. In this paper, we prove that the specific p-Wasserstein divergence is well-defined, exhibit an explicit expression for it, and compare it with the specific relative entropy and adapted Wasserstein distance on a class of stochastic differential equations. Then, we consider specific p-Wasserstein divergence optimization over the set of win-martingales. Finally, we characterize the solution of such optimization problems for all p > 0, and, surprisingly, we single out the case p = 1/2 as the one with the best probabilistic properties.