Specific Wasserstein Divergence Between Continuous Martingales

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.

Funding: Financial support from the Austrian Science Fund [Grant P36835] is gratefully acknowledged. X. Zhang is partially supported by the National Science Foundation [Grant DMS-2508556].