Empirical martingale projections via the adapted Wasserstein distance

Understand

Given a collection of multidimensional pairs $\{(X_i,Y_i):1 \leq i\leq n\}$, we study the problem of projecting the associated suitably smoothed empirical measure onto the space of martingale couplings (i.e.

• distributions satisfying $\mathbb{E}[Y|X]=X$) using the adapted Wasserstein distance.
• We call the resulting distance the smoothed empirical martingale projection distance (SE-MPD), for which we obtain an explicit characterization.
• We also show that the space of martingale couplings remains invariant under the smoothing operation.