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, put forward by N. 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 only natural to replace the role of the relative entropy in this construction by a different notion of discrepancy between finite dimensional probability distributions. In the present work we take a first step in this direction, taking a power $p$ of the Wasserstein distance instead of the relative entropy. We call the newly obtained scaling limit the specific $p$-Wasserstein divergence. In our first main result we prove that the specific $p$-Wasserstein divergence is well-defined, exhibit an explicit expression for it in terms of the quadratic variations of the martingales involved, and compare it with the specific relative entropy and adapted Wasserstein distance on a class of SDEs. Next we consider the specific $p$-Wasserstein divergence optimization over the set of win-martingales. In our second main result we characterize the solution of this optimization problem for all $p>0$ and, somewhat surprisingly, we single out the case $p=1/2$ as the one with the best probabilistic properties. For instance, the optimal martingale in this case is very explicit and can be connected to the solution of a variant of the Schr\"odinger problem.